/************************************************************************* Copyright (c) Sergey Bochkanov (ALGLIB project). >>> SOURCE LICENSE >>> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation (www.fsf.org); either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses >>> END OF LICENSE >>> *************************************************************************/ #pragma warning disable 162 #pragma warning disable 219 using System; public partial class alglib { } public partial class alglib { /************************************************************************* This object stores state of the nonlinear CG optimizer. You should use ALGLIB functions to work with this object. *************************************************************************/ public class mincgstate { // // Public declarations // public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } } public bool needfg { get { return _innerobj.needfg; } set { _innerobj.needfg = value; } } public bool xupdated { get { return _innerobj.xupdated; } set { _innerobj.xupdated = value; } } public double f { get { return _innerobj.f; } set { _innerobj.f = value; } } public double[] g { get { return _innerobj.g; } } public double[] x { get { return _innerobj.x; } } public mincgstate() { _innerobj = new mincg.mincgstate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private mincg.mincgstate _innerobj; public mincg.mincgstate innerobj { get { return _innerobj; } } public mincgstate(mincg.mincgstate obj) { _innerobj = obj; } } /************************************************************************* *************************************************************************/ public class mincgreport { // // Public declarations // public int iterationscount { get { return _innerobj.iterationscount; } set { _innerobj.iterationscount = value; } } public int nfev { get { return _innerobj.nfev; } set { _innerobj.nfev = value; } } public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } } public mincgreport() { _innerobj = new mincg.mincgreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private mincg.mincgreport _innerobj; public mincg.mincgreport innerobj { get { return _innerobj; } } public mincgreport(mincg.mincgreport obj) { _innerobj = obj; } } /************************************************************************* NONLINEAR CONJUGATE GRADIENT METHOD DESCRIPTION: The subroutine minimizes function F(x) of N arguments by using one of the nonlinear conjugate gradient methods. These CG methods are globally convergent (even on non-convex functions) as long as grad(f) is Lipschitz continuous in a some neighborhood of the L = { x : f(x)<=f(x0) }. REQUIREMENTS: Algorithm will request following information during its operation: * function value F and its gradient G (simultaneously) at given point X USAGE: 1. User initializes algorithm state with MinCGCreate() call 2. User tunes solver parameters with MinCGSetCond(), MinCGSetStpMax() and other functions 3. User calls MinCGOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 4. User calls MinCGResults() to get solution 5. Optionally, user may call MinCGRestartFrom() to solve another problem with same N but another starting point and/or another function. MinCGRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 25.03.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgcreate(int n, double[] x, out mincgstate state) { state = new mincgstate(); mincg.mincgcreate(n, x, state.innerobj); return; } public static void mincgcreate(double[] x, out mincgstate state) { int n; state = new mincgstate(); n = ap.len(x); mincg.mincgcreate(n, x, state.innerobj); return; } /************************************************************************* The subroutine is finite difference variant of MinCGCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinCGCreate() in order to get more information about creation of CG optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinCGSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. L-BFGS needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ public static void mincgcreatef(int n, double[] x, double diffstep, out mincgstate state) { state = new mincgstate(); mincg.mincgcreatef(n, x, diffstep, state.innerobj); return; } public static void mincgcreatef(double[] x, double diffstep, out mincgstate state) { int n; state = new mincgstate(); n = ap.len(x); mincg.mincgcreatef(n, x, diffstep, state.innerobj); return; } /************************************************************************* This function sets stopping conditions for CG optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinCGSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetcond(mincgstate state, double epsg, double epsf, double epsx, int maxits) { mincg.mincgsetcond(state.innerobj, epsg, epsf, epsx, maxits); return; } /************************************************************************* This function sets scaling coefficients for CG optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of CG optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the CG too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinCGSetPrec...() functions. There is special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void mincgsetscale(mincgstate state, double[] s) { mincg.mincgsetscale(state.innerobj, s); return; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinCGOptimize(). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetxrep(mincgstate state, bool needxrep) { mincg.mincgsetxrep(state.innerobj, needxrep); return; } /************************************************************************* This function sets CG algorithm. INPUT PARAMETERS: State - structure which stores algorithm state CGType - algorithm type: * -1 automatic selection of the best algorithm * 0 DY (Dai and Yuan) algorithm * 1 Hybrid DY-HS algorithm -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetcgtype(mincgstate state, int cgtype) { mincg.mincgsetcgtype(state.innerobj, cgtype); return; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetstpmax(mincgstate state, double stpmax) { mincg.mincgsetstpmax(state.innerobj, stpmax); return; } /************************************************************************* This function allows to suggest initial step length to the CG algorithm. Suggested step length is used as starting point for the line search. It can be useful when you have badly scaled problem, i.e. when ||grad|| (which is used as initial estimate for the first step) is many orders of magnitude different from the desired step. Line search may fail on such problems without good estimate of initial step length. Imagine, for example, problem with ||grad||=10^50 and desired step equal to 0.1 Line search function will use 10^50 as initial step, then it will decrease step length by 2 (up to 20 attempts) and will get 10^44, which is still too large. This function allows us to tell than line search should be started from some moderate step length, like 1.0, so algorithm will be able to detect desired step length in a several searches. Default behavior (when no step is suggested) is to use preconditioner, if it is available, to generate initial estimate of step length. This function influences only first iteration of algorithm. It should be called between MinCGCreate/MinCGRestartFrom() call and MinCGOptimize call. Suggested step is ignored if you have preconditioner. INPUT PARAMETERS: State - structure used to store algorithm state. Stp - initial estimate of the step length. Can be zero (no estimate). -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsuggeststep(mincgstate state, double stp) { mincg.mincgsuggeststep(state.innerobj, stp); return; } /************************************************************************* Modification of the preconditioner: preconditioning is turned off. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetprecdefault(mincgstate state) { mincg.mincgsetprecdefault(state.innerobj); return; } /************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: D[i] should be positive. Exception will be thrown otherwise. NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetprecdiag(mincgstate state, double[] d) { mincg.mincgsetprecdiag(state.innerobj, d); return; } /************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinCGSetScale() call (before or after MinCGSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetprecscale(mincgstate state) { mincg.mincgsetprecscale(state.innerobj); return; } /************************************************************************* This function provides reverse communication interface Reverse communication interface is not documented or recommended to use. See below for functions which provide better documented API *************************************************************************/ public static bool mincgiteration(mincgstate state) { bool result = mincg.mincgiteration(state.innerobj); return result; } /************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be null obj - optional object which is passed to func/grad/hess/jac/rep can be null NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinCGCreate() for analytical gradient or MinCGCreateF() for numerical differentiation) you should choose appropriate variant of MinCGOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinCGOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinCGOptimize() and specific function used to create optimizer. | USER PASSED TO MinCGOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinCGCreateF() | work FAIL MinCGCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinCGOptimize() version. Attemps to use such combination (for example, to create optimizer with MinCGCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/ public static void mincgoptimize(mincgstate state, ndimensional_func func, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'mincgoptimize()' (func is null)"); while( alglib.mincgiteration(state) ) { if( state.needf ) { func(state.x, ref state.innerobj.f, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'mincgoptimize' (some derivatives were not provided?)"); } } public static void mincgoptimize(mincgstate state, ndimensional_grad grad, ndimensional_rep rep, object obj) { if( grad==null ) throw new alglibexception("ALGLIB: error in 'mincgoptimize()' (grad is null)"); while( alglib.mincgiteration(state) ) { if( state.needfg ) { grad(state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'mincgoptimize' (some derivatives were not provided?)"); } } /************************************************************************* Conjugate gradient results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible, we return best X found so far * 8 terminated by user * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/ public static void mincgresults(mincgstate state, out double[] x, out mincgreport rep) { x = new double[0]; rep = new mincgreport(); mincg.mincgresults(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* Conjugate gradient results Buffered implementation of MinCGResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/ public static void mincgresultsbuf(mincgstate state, ref double[] x, mincgreport rep) { mincg.mincgresultsbuf(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* This subroutine restarts CG algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used to store algorithm state. X - new starting point. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgrestartfrom(mincgstate state, double[] x) { mincg.mincgrestartfrom(state.innerobj, x); return; } } public partial class alglib { /************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinBLEIC subpackage to work with this object *************************************************************************/ public class minbleicstate { // // Public declarations // public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } } public bool needfg { get { return _innerobj.needfg; } set { _innerobj.needfg = value; } } public bool xupdated { get { return _innerobj.xupdated; } set { _innerobj.xupdated = value; } } public double f { get { return _innerobj.f; } set { _innerobj.f = value; } } public double[] g { get { return _innerobj.g; } } public double[] x { get { return _innerobj.x; } } public minbleicstate() { _innerobj = new minbleic.minbleicstate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minbleic.minbleicstate _innerobj; public minbleic.minbleicstate innerobj { get { return _innerobj; } } public minbleicstate(minbleic.minbleicstate obj) { _innerobj = obj; } } /************************************************************************* This structure stores optimization report: * InnerIterationsCount number of inner iterations * OuterIterationsCount number of outer iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -10 unsupported combination of algorithm settings: 1) StpMax is set to non-zero value, AND 2) non-default preconditioner is used. You can't use both features at the same moment, so you have to choose one of them (and to turn off another one). -3 inconsistent constraints. Feasible point is either nonexistent or too hard to find. Try to restart optimizer with better initial approximation 4 conditions on constraints are fulfilled with error less than or equal to EpsC 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. ADDITIONAL FIELDS There are additional fields which can be used for debugging: * DebugEqErr error in the equality constraints (2-norm) * DebugFS f, calculated at projection of initial point to the feasible set * DebugFF f, calculated at the final point * DebugDX |X_start-X_final| *************************************************************************/ public class minbleicreport { // // Public declarations // public int inneriterationscount { get { return _innerobj.inneriterationscount; } set { _innerobj.inneriterationscount = value; } } public int outeriterationscount { get { return _innerobj.outeriterationscount; } set { _innerobj.outeriterationscount = value; } } public int nfev { get { return _innerobj.nfev; } set { _innerobj.nfev = value; } } public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } } public double debugeqerr { get { return _innerobj.debugeqerr; } set { _innerobj.debugeqerr = value; } } public double debugfs { get { return _innerobj.debugfs; } set { _innerobj.debugfs = value; } } public double debugff { get { return _innerobj.debugff; } set { _innerobj.debugff = value; } } public double debugdx { get { return _innerobj.debugdx; } set { _innerobj.debugdx = value; } } public int debugfeasqpits { get { return _innerobj.debugfeasqpits; } set { _innerobj.debugfeasqpits = value; } } public int debugfeasgpaits { get { return _innerobj.debugfeasgpaits; } set { _innerobj.debugfeasgpaits = value; } } public minbleicreport() { _innerobj = new minbleic.minbleicreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minbleic.minbleicreport _innerobj; public minbleic.minbleicreport innerobj { get { return _innerobj; } } public minbleicreport(minbleic.minbleicreport obj) { _innerobj = obj; } } /************************************************************************* BOUND CONSTRAINED OPTIMIZATION WITH ADDITIONAL LINEAR EQUALITY AND INEQUALITY CONSTRAINTS DESCRIPTION: The subroutine minimizes function F(x) of N arguments subject to any combination of: * bound constraints * linear inequality constraints * linear equality constraints REQUIREMENTS: * user must provide function value and gradient * starting point X0 must be feasible or not too far away from the feasible set * grad(f) must be Lipschitz continuous on a level set: L = { x : f(x)<=f(x0) } * function must be defined everywhere on the feasible set F USAGE: Constrained optimization if far more complex than the unconstrained one. Here we give very brief outline of the BLEIC optimizer. We strongly recommend you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide on optimization, which is available at http://www.alglib.net/optimization/ 1. User initializes algorithm state with MinBLEICCreate() call 2. USer adds boundary and/or linear constraints by calling MinBLEICSetBC() and MinBLEICSetLC() functions. 3. User sets stopping conditions for underlying unconstrained solver with MinBLEICSetInnerCond() call. This function controls accuracy of underlying optimization algorithm. 4. User sets stopping conditions for outer iteration by calling MinBLEICSetOuterCond() function. This function controls handling of boundary and inequality constraints. 5. Additionally, user may set limit on number of internal iterations by MinBLEICSetMaxIts() call. This function allows to prevent algorithm from looping forever. 6. User calls MinBLEICOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 7. User calls MinBLEICResults() to get solution 8. Optionally user may call MinBLEICRestartFrom() to solve another problem with same N but another starting point. MinBLEICRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size ofX X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleiccreate(int n, double[] x, out minbleicstate state) { state = new minbleicstate(); minbleic.minbleiccreate(n, x, state.innerobj); return; } public static void minbleiccreate(double[] x, out minbleicstate state) { int n; state = new minbleicstate(); n = ap.len(x); minbleic.minbleiccreate(n, x, state.innerobj); return; } /************************************************************************* The subroutine is finite difference variant of MinBLEICCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinBLEICCreate() in order to get more information about creation of BLEIC optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinBLEICSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. CG needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ public static void minbleiccreatef(int n, double[] x, double diffstep, out minbleicstate state) { state = new minbleicstate(); minbleic.minbleiccreatef(n, x, diffstep, state.innerobj); return; } public static void minbleiccreatef(double[] x, double diffstep, out minbleicstate state) { int n; state = new minbleicstate(); n = ap.len(x); minbleic.minbleiccreatef(n, x, diffstep, state.innerobj); return; } /************************************************************************* This function sets boundary constraints for BLEIC optimizer. Boundary constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinBLEICRestartFrom(). INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF. BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF. NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints, even when numerical differentiation is used (algorithm adjusts nodes according to boundary constraints) -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetbc(minbleicstate state, double[] bndl, double[] bndu) { minbleic.minbleicsetbc(state.innerobj, bndl, bndu); return; } /************************************************************************* This function sets linear constraints for BLEIC optimizer. Linear constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinBLEICRestartFrom(). INPUT PARAMETERS: State - structure previously allocated with MinBLEICCreate call. C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT NOTE 1: linear (non-bound) constraints are satisfied only approximately: * there always exists some minor violation (about Epsilon in magnitude) due to rounding errors * numerical differentiation, if used, may lead to function evaluations outside of the feasible area, because algorithm does NOT change numerical differentiation formula according to linear constraints. If you want constraints to be satisfied exactly, try to reformulate your problem in such manner that all constraints will become boundary ones (this kind of constraints is always satisfied exactly, both in the final solution and in all intermediate points). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetlc(minbleicstate state, double[,] c, int[] ct, int k) { minbleic.minbleicsetlc(state.innerobj, c, ct, k); return; } public static void minbleicsetlc(minbleicstate state, double[,] c, int[] ct) { int k; if( (ap.rows(c)!=ap.len(ct))) throw new alglibexception("Error while calling 'minbleicsetlc': looks like one of arguments has wrong size"); k = ap.rows(c); minbleic.minbleicsetlc(state.innerobj, c, ct, k); return; } /************************************************************************* This function sets stopping conditions for the underlying nonlinear CG optimizer. It controls overall accuracy of solution. These conditions should be strict enough in order for algorithm to converge. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinBLEICSetScale() Passing EpsG=0, EpsF=0 and EpsX=0 (simultaneously) will lead to automatic stopping criterion selection. These conditions are used to terminate inner iterations. However, you need to tune termination conditions for outer iterations too. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetinnercond(minbleicstate state, double epsg, double epsf, double epsx) { minbleic.minbleicsetinnercond(state.innerobj, epsg, epsf, epsx); return; } /************************************************************************* This function sets stopping conditions for outer iteration of BLEIC algo. These conditions control accuracy of constraint handling and amount of infeasibility allowed in the solution. INPUT PARAMETERS: State - structure which stores algorithm state EpsX - >0, stopping condition on outer iteration step length EpsI - >0, stopping condition on infeasibility Both EpsX and EpsI must be non-zero. MEANING OF EpsX EpsX is a stopping condition for outer iterations. Algorithm will stop when solution of the current modified subproblem will be within EpsX (using 2-norm) of the previous solution. MEANING OF EpsI EpsI controls feasibility properties - algorithm won't stop until all inequality constraints will be satisfied with error (distance from current point to the feasible area) at most EpsI. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetoutercond(minbleicstate state, double epsx, double epsi) { minbleic.minbleicsetoutercond(state.innerobj, epsx, epsi); return; } /************************************************************************* This function sets scaling coefficients for BLEIC optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the BLEIC too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinBLEICSetPrec...() functions. There is a special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetscale(minbleicstate state, double[] s) { minbleic.minbleicsetscale(state.innerobj, s); return; } /************************************************************************* Modification of the preconditioner: preconditioning is turned off. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetprecdefault(minbleicstate state) { minbleic.minbleicsetprecdefault(state.innerobj); return; } /************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE 1: D[i] should be positive. Exception will be thrown otherwise. NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetprecdiag(minbleicstate state, double[] d) { minbleic.minbleicsetprecdiag(state.innerobj, d); return; } /************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinBLEICSetScale() call (before or after MinBLEICSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetprecscale(minbleicstate state) { minbleic.minbleicsetprecscale(state.innerobj); return; } /************************************************************************* This function allows to stop algorithm after specified number of inner iterations. INPUT PARAMETERS: State - structure which stores algorithm state MaxIts - maximum number of inner iterations. If MaxIts=0, the number of iterations is unlimited. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetmaxits(minbleicstate state, int maxits) { minbleic.minbleicsetmaxits(state.innerobj, maxits); return; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinBLEICOptimize(). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetxrep(minbleicstate state, bool needxrep) { minbleic.minbleicsetxrep(state.innerobj, needxrep); return; } /************************************************************************* This function sets maximum step length IMPORTANT: this feature is hard to combine with preconditioning. You can't set upper limit on step length, when you solve optimization problem with linear (non-boundary) constraints AND preconditioner turned on. When non-boundary constraints are present, you have to either a) use preconditioner, or b) use upper limit on step length. YOU CAN'T USE BOTH! In this case algorithm will terminate with appropriate error code. INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which lead to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetstpmax(minbleicstate state, double stpmax) { minbleic.minbleicsetstpmax(state.innerobj, stpmax); return; } /************************************************************************* This function provides reverse communication interface Reverse communication interface is not documented or recommended to use. See below for functions which provide better documented API *************************************************************************/ public static bool minbleiciteration(minbleicstate state) { bool result = minbleic.minbleiciteration(state.innerobj); return result; } /************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be null obj - optional object which is passed to func/grad/hess/jac/rep can be null NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinBLEICCreate() for analytical gradient or MinBLEICCreateF() for numerical differentiation) you should choose appropriate variant of MinBLEICOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinBLEICOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinBLEICOptimize() and specific function used to create optimizer. | USER PASSED TO MinBLEICOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinBLEICCreateF() | work FAIL MinBLEICCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinBLEICOptimize() version. Attemps to use such combination (for example, to create optimizer with MinBLEICCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicoptimize(minbleicstate state, ndimensional_func func, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'minbleicoptimize()' (func is null)"); while( alglib.minbleiciteration(state) ) { if( state.needf ) { func(state.x, ref state.innerobj.f, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minbleicoptimize' (some derivatives were not provided?)"); } } public static void minbleicoptimize(minbleicstate state, ndimensional_grad grad, ndimensional_rep rep, object obj) { if( grad==null ) throw new alglibexception("ALGLIB: error in 'minbleicoptimize()' (grad is null)"); while( alglib.minbleiciteration(state) ) { if( state.needfg ) { grad(state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minbleicoptimize' (some derivatives were not provided?)"); } } /************************************************************************* BLEIC results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one. More information about fields of this structure can be found in the comments on MinBLEICReport datatype. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicresults(minbleicstate state, out double[] x, out minbleicreport rep) { x = new double[0]; rep = new minbleicreport(); minbleic.minbleicresults(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* BLEIC results Buffered implementation of MinBLEICResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicresultsbuf(minbleicstate state, ref double[] x, minbleicreport rep) { minbleic.minbleicresultsbuf(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* This subroutine restarts algorithm from new point. All optimization parameters (including constraints) are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure previously allocated with MinBLEICCreate call. X - new starting point. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicrestartfrom(minbleicstate state, double[] x) { minbleic.minbleicrestartfrom(state.innerobj, x); return; } } public partial class alglib { /************************************************************************* *************************************************************************/ public class minlbfgsstate { // // Public declarations // public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } } public bool needfg { get { return _innerobj.needfg; } set { _innerobj.needfg = value; } } public bool xupdated { get { return _innerobj.xupdated; } set { _innerobj.xupdated = value; } } public double f { get { return _innerobj.f; } set { _innerobj.f = value; } } public double[] g { get { return _innerobj.g; } } public double[] x { get { return _innerobj.x; } } public minlbfgsstate() { _innerobj = new minlbfgs.minlbfgsstate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minlbfgs.minlbfgsstate _innerobj; public minlbfgs.minlbfgsstate innerobj { get { return _innerobj; } } public minlbfgsstate(minlbfgs.minlbfgsstate obj) { _innerobj = obj; } } /************************************************************************* *************************************************************************/ public class minlbfgsreport { // // Public declarations // public int iterationscount { get { return _innerobj.iterationscount; } set { _innerobj.iterationscount = value; } } public int nfev { get { return _innerobj.nfev; } set { _innerobj.nfev = value; } } public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } } public minlbfgsreport() { _innerobj = new minlbfgs.minlbfgsreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minlbfgs.minlbfgsreport _innerobj; public minlbfgs.minlbfgsreport innerobj { get { return _innerobj; } } public minlbfgsreport(minlbfgs.minlbfgsreport obj) { _innerobj = obj; } } /************************************************************************* LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION DESCRIPTION: The subroutine minimizes function F(x) of N arguments by using a quasi- Newton method (LBFGS scheme) which is optimized to use a minimum amount of memory. The subroutine generates the approximation of an inverse Hessian matrix by using information about the last M steps of the algorithm (instead of N). It lessens a required amount of memory from a value of order N^2 to a value of order 2*N*M. REQUIREMENTS: Algorithm will request following information during its operation: * function value F and its gradient G (simultaneously) at given point X USAGE: 1. User initializes algorithm state with MinLBFGSCreate() call 2. User tunes solver parameters with MinLBFGSSetCond() MinLBFGSSetStpMax() and other functions 3. User calls MinLBFGSOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 4. User calls MinLBFGSResults() to get solution 5. Optionally user may call MinLBFGSRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLBFGSRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension. N>0 M - number of corrections in the BFGS scheme of Hessian approximation update. Recommended value: 3<=M<=7. The smaller value causes worse convergence, the bigger will not cause a considerably better convergence, but will cause a fall in the performance. M<=N. X - initial solution approximation, array[0..N-1]. OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLBFGSSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLBFGSSetStpMax() function to bound algorithm's steps. However, L-BFGS rarely needs such a tuning. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgscreate(int n, int m, double[] x, out minlbfgsstate state) { state = new minlbfgsstate(); minlbfgs.minlbfgscreate(n, m, x, state.innerobj); return; } public static void minlbfgscreate(int m, double[] x, out minlbfgsstate state) { int n; state = new minlbfgsstate(); n = ap.len(x); minlbfgs.minlbfgscreate(n, m, x, state.innerobj); return; } /************************************************************************* The subroutine is finite difference variant of MinLBFGSCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinLBFGSCreate() in order to get more information about creation of LBFGS optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of corrections in the BFGS scheme of Hessian approximation update. Recommended value: 3<=M<=7. The smaller value causes worse convergence, the bigger will not cause a considerably better convergence, but will cause a fall in the performance. M<=N. X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinLBFGSSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. LBFGS needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ public static void minlbfgscreatef(int n, int m, double[] x, double diffstep, out minlbfgsstate state) { state = new minlbfgsstate(); minlbfgs.minlbfgscreatef(n, m, x, diffstep, state.innerobj); return; } public static void minlbfgscreatef(int m, double[] x, double diffstep, out minlbfgsstate state) { int n; state = new minlbfgsstate(); n = ap.len(x); minlbfgs.minlbfgscreatef(n, m, x, diffstep, state.innerobj); return; } /************************************************************************* This function sets stopping conditions for L-BFGS optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinLBFGSSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetcond(minlbfgsstate state, double epsg, double epsf, double epsx, int maxits) { minlbfgs.minlbfgssetcond(state.innerobj, epsg, epsf, epsx, maxits); return; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinLBFGSOptimize(). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetxrep(minlbfgsstate state, bool needxrep) { minlbfgs.minlbfgssetxrep(state.innerobj, needxrep); return; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0 (default), if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetstpmax(minlbfgsstate state, double stpmax) { minlbfgs.minlbfgssetstpmax(state.innerobj, stpmax); return; } /************************************************************************* This function sets scaling coefficients for LBFGS optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the LBFGS too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinLBFGSSetPrec...() functions. There is special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetscale(minlbfgsstate state, double[] s) { minlbfgs.minlbfgssetscale(state.innerobj, s); return; } /************************************************************************* Modification of the preconditioner: default preconditioner (simple scaling, same for all elements of X) is used. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetprecdefault(minlbfgsstate state) { minlbfgs.minlbfgssetprecdefault(state.innerobj); return; } /************************************************************************* Modification of the preconditioner: Cholesky factorization of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state P - triangular preconditioner, Cholesky factorization of the approximate Hessian. array[0..N-1,0..N-1], (if larger, only leading N elements are used). IsUpper - whether upper or lower triangle of P is given (other triangle is not referenced) After call to this function preconditioner is changed to P (P is copied into the internal buffer). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: P should be nonsingular. Exception will be thrown otherwise. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetpreccholesky(minlbfgsstate state, double[,] p, bool isupper) { minlbfgs.minlbfgssetpreccholesky(state.innerobj, p, isupper); return; } /************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: D[i] should be positive. Exception will be thrown otherwise. NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetprecdiag(minlbfgsstate state, double[] d) { minlbfgs.minlbfgssetprecdiag(state.innerobj, d); return; } /************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinLBFGSSetScale() call (before or after MinLBFGSSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetprecscale(minlbfgsstate state) { minlbfgs.minlbfgssetprecscale(state.innerobj); return; } /************************************************************************* This function provides reverse communication interface Reverse communication interface is not documented or recommended to use. See below for functions which provide better documented API *************************************************************************/ public static bool minlbfgsiteration(minlbfgsstate state) { bool result = minlbfgs.minlbfgsiteration(state.innerobj); return result; } /************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be null obj - optional object which is passed to func/grad/hess/jac/rep can be null NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinLBFGSCreate() for analytical gradient or MinLBFGSCreateF() for numerical differentiation) you should choose appropriate variant of MinLBFGSOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinLBFGSOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinLBFGSOptimize() and specific function used to create optimizer. | USER PASSED TO MinLBFGSOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinLBFGSCreateF() | work FAIL MinLBFGSCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinLBFGSOptimize() version. Attemps to use such combination (for example, to create optimizer with MinLBFGSCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlbfgsoptimize(minlbfgsstate state, ndimensional_func func, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'minlbfgsoptimize()' (func is null)"); while( alglib.minlbfgsiteration(state) ) { if( state.needf ) { func(state.x, ref state.innerobj.f, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlbfgsoptimize' (some derivatives were not provided?)"); } } public static void minlbfgsoptimize(minlbfgsstate state, ndimensional_grad grad, ndimensional_rep rep, object obj) { if( grad==null ) throw new alglibexception("ALGLIB: error in 'minlbfgsoptimize()' (grad is null)"); while( alglib.minlbfgsiteration(state) ) { if( state.needfg ) { grad(state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlbfgsoptimize' (some derivatives were not provided?)"); } } /************************************************************************* L-BFGS algorithm results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * -2 rounding errors prevent further improvement. X contains best point found. * -1 incorrect parameters were specified * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgsresults(minlbfgsstate state, out double[] x, out minlbfgsreport rep) { x = new double[0]; rep = new minlbfgsreport(); minlbfgs.minlbfgsresults(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* L-BFGS algorithm results Buffered implementation of MinLBFGSResults which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.08.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgsresultsbuf(minlbfgsstate state, ref double[] x, minlbfgsreport rep) { minlbfgs.minlbfgsresultsbuf(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* This subroutine restarts LBFGS algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used to store algorithm state X - new starting point. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgsrestartfrom(minlbfgsstate state, double[] x) { minlbfgs.minlbfgsrestartfrom(state.innerobj, x); return; } } public partial class alglib { /************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinQP subpackage to work with this object *************************************************************************/ public class minqpstate { // // Public declarations // public minqpstate() { _innerobj = new minqp.minqpstate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minqp.minqpstate _innerobj; public minqp.minqpstate innerobj { get { return _innerobj; } } public minqpstate(minqp.minqpstate obj) { _innerobj = obj; } } /************************************************************************* This structure stores optimization report: * InnerIterationsCount number of inner iterations * OuterIterationsCount number of outer iterations * NCholesky number of Cholesky decomposition * NMV number of matrix-vector products (only products calculated as part of iterative process are counted) * TerminationType completion code (see below) Completion codes: * -5 inappropriate solver was used: * Cholesky solver for semidefinite or indefinite problems * Cholesky solver for problems with non-boundary constraints * -3 inconsistent constraints (or, maybe, feasible point is too hard to find). If you are sure that constraints are feasible, try to restart optimizer with better initial approximation. * -1 solver error * 4 successful completion * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. *************************************************************************/ public class minqpreport { // // Public declarations // public int inneriterationscount { get { return _innerobj.inneriterationscount; } set { _innerobj.inneriterationscount = value; } } public int outeriterationscount { get { return _innerobj.outeriterationscount; } set { _innerobj.outeriterationscount = value; } } public int nmv { get { return _innerobj.nmv; } set { _innerobj.nmv = value; } } public int ncholesky { get { return _innerobj.ncholesky; } set { _innerobj.ncholesky = value; } } public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } } public minqpreport() { _innerobj = new minqp.minqpreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minqp.minqpreport _innerobj; public minqp.minqpreport innerobj { get { return _innerobj; } } public minqpreport(minqp.minqpreport obj) { _innerobj = obj; } } /************************************************************************* CONSTRAINED QUADRATIC PROGRAMMING The subroutine creates QP optimizer. After initial creation, it contains default optimization problem with zero quadratic and linear terms and no constraints. You should set quadratic/linear terms with calls to functions provided by MinQP subpackage. INPUT PARAMETERS: N - problem size OUTPUT PARAMETERS: State - optimizer with zero quadratic/linear terms and no constraints -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpcreate(int n, out minqpstate state) { state = new minqpstate(); minqp.minqpcreate(n, state.innerobj); return; } /************************************************************************* This function sets linear term for QP solver. By default, linear term is zero. INPUT PARAMETERS: State - structure which stores algorithm state B - linear term, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetlinearterm(minqpstate state, double[] b) { minqp.minqpsetlinearterm(state.innerobj, b); return; } /************************************************************************* This function sets quadratic term for QP solver. By default quadratic term is zero. IMPORTANT: this solver minimizes following function: f(x) = 0.5*x'*A*x + b'*x. Note that quadratic term has 0.5 before it. So if you want to minimize f(x) = x^2 + x you should rewrite your problem as follows: f(x) = 0.5*(2*x^2) + x and your matrix A will be equal to [[2.0]], not to [[1.0]] INPUT PARAMETERS: State - structure which stores algorithm state A - matrix, array[N,N] IsUpper - (optional) storage type: * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn’t used * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn’t used * if not given, both lower and upper triangles must be filled. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetquadraticterm(minqpstate state, double[,] a, bool isupper) { minqp.minqpsetquadraticterm(state.innerobj, a, isupper); return; } public static void minqpsetquadraticterm(minqpstate state, double[,] a) { bool isupper; if( !alglib.ap.issymmetric(a) ) throw new alglibexception("'a' parameter is not symmetric matrix"); isupper = false; minqp.minqpsetquadraticterm(state.innerobj, a, isupper); return; } /************************************************************************* This function sets starting point for QP solver. It is useful to have good initial approximation to the solution, because it will increase speed of convergence and identification of active constraints. INPUT PARAMETERS: State - structure which stores algorithm state X - starting point, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetstartingpoint(minqpstate state, double[] x) { minqp.minqpsetstartingpoint(state.innerobj, x); return; } /************************************************************************* This function sets origin for QP solver. By default, following QP program is solved: min(0.5*x'*A*x+b'*x) This function allows to solve different problem: min(0.5*(x-x_origin)'*A*(x-x_origin)+b'*(x-x_origin)) INPUT PARAMETERS: State - structure which stores algorithm state XOrigin - origin, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetorigin(minqpstate state, double[] xorigin) { minqp.minqpsetorigin(state.innerobj, xorigin); return; } /************************************************************************* This function tells solver to use Cholesky-based algorithm. Cholesky-based algorithm can be used when: * problem is convex * there is no constraints or only boundary constraints are present This algorithm has O(N^3) complexity for unconstrained problem and is up to several times slower on bound constrained problems (these additional iterations are needed to identify active constraints). INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetalgocholesky(minqpstate state) { minqp.minqpsetalgocholesky(state.innerobj); return; } /************************************************************************* This function sets boundary constraints for QP solver Boundary constraints are inactive by default (after initial creation). After being set, they are preserved until explicitly turned off with another SetBC() call. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF (latter is recommended because it will allow solver to use better algorithm). BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF (latter is recommended because it will allow solver to use better algorithm). NOTE: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetbc(minqpstate state, double[] bndl, double[] bndu) { minqp.minqpsetbc(state.innerobj, bndl, bndu); return; } /************************************************************************* This function solves quadratic programming problem. You should call it after setting solver options with MinQPSet...() calls. INPUT PARAMETERS: State - algorithm state You should use MinQPResults() function to access results after calls to this function. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpoptimize(minqpstate state) { minqp.minqpoptimize(state.innerobj); return; } /************************************************************************* QP solver results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType, which contains completion code, and you may check another fields which contain another information about algorithm functioning. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpresults(minqpstate state, out double[] x, out minqpreport rep) { x = new double[0]; rep = new minqpreport(); minqp.minqpresults(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* QP results Buffered implementation of MinQPResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpresultsbuf(minqpstate state, ref double[] x, minqpreport rep) { minqp.minqpresultsbuf(state.innerobj, ref x, rep.innerobj); return; } } public partial class alglib { /************************************************************************* Levenberg-Marquardt optimizer. This structure should be created using one of the MinLMCreate???() functions. You should not access its fields directly; use ALGLIB functions to work with it. *************************************************************************/ public class minlmstate { // // Public declarations // public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } } public bool needfg { get { return _innerobj.needfg; } set { _innerobj.needfg = value; } } public bool needfgh { get { return _innerobj.needfgh; } set { _innerobj.needfgh = value; } } public bool needfi { get { return _innerobj.needfi; } set { _innerobj.needfi = value; } } public bool needfij { get { return _innerobj.needfij; } set { _innerobj.needfij = value; } } public bool xupdated { get { return _innerobj.xupdated; } set { _innerobj.xupdated = value; } } public double f { get { return _innerobj.f; } set { _innerobj.f = value; } } public double[] fi { get { return _innerobj.fi; } } public double[] g { get { return _innerobj.g; } } public double[,] h { get { return _innerobj.h; } } public double[,] j { get { return _innerobj.j; } } public double[] x { get { return _innerobj.x; } } public minlmstate() { _innerobj = new minlm.minlmstate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minlm.minlmstate _innerobj; public minlm.minlmstate innerobj { get { return _innerobj; } } public minlmstate(minlm.minlmstate obj) { _innerobj = obj; } } /************************************************************************* Optimization report, filled by MinLMResults() function FIELDS: * TerminationType, completetion code: * -9 derivative correctness check failed; see Rep.WrongNum, Rep.WrongI, Rep.WrongJ for more information. * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient is no more than EpsG. * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * IterationsCount, contains iterations count * NFunc, number of function calculations * NJac, number of Jacobi matrix calculations * NGrad, number of gradient calculations * NHess, number of Hessian calculations * NCholesky, number of Cholesky decomposition calculations *************************************************************************/ public class minlmreport { // // Public declarations // public int iterationscount { get { return _innerobj.iterationscount; } set { _innerobj.iterationscount = value; } } public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } } public int nfunc { get { return _innerobj.nfunc; } set { _innerobj.nfunc = value; } } public int njac { get { return _innerobj.njac; } set { _innerobj.njac = value; } } public int ngrad { get { return _innerobj.ngrad; } set { _innerobj.ngrad = value; } } public int nhess { get { return _innerobj.nhess; } set { _innerobj.nhess = value; } } public int ncholesky { get { return _innerobj.ncholesky; } set { _innerobj.ncholesky = value; } } public minlmreport() { _innerobj = new minlm.minlmreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private minlm.minlmreport _innerobj; public minlm.minlmreport innerobj { get { return _innerobj; } } public minlmreport(minlm.minlmreport obj) { _innerobj = obj; } } /************************************************************************* IMPROVED LEVENBERG-MARQUARDT METHOD FOR NON-LINEAR LEAST SQUARES OPTIMIZATION DESCRIPTION: This function is used to find minimum of function which is represented as sum of squares: F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1]) using value of function vector f[] and Jacobian of f[]. REQUIREMENTS: This algorithm will request following information during its operation: * function vector f[] at given point X * function vector f[] and Jacobian of f[] (simultaneously) at given point There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts fvec() and jac() callbacks. First one is used to calculate f[] at given point, second one calculates f[] and Jacobian df[i]/dx[j]. You can try to initialize MinLMState structure with VJ function and then use incorrect version of MinLMOptimize() (for example, version which works with general form function and does not provide Jacobian), but it will lead to exception being thrown after first attempt to calculate Jacobian. USAGE: 1. User initializes algorithm state with MinLMCreateVJ() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of functions f[i] X - initial solution, array[0..N-1] OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatevj(int n, int m, double[] x, out minlmstate state) { state = new minlmstate(); minlm.minlmcreatevj(n, m, x, state.innerobj); return; } public static void minlmcreatevj(int m, double[] x, out minlmstate state) { int n; state = new minlmstate(); n = ap.len(x); minlm.minlmcreatevj(n, m, x, state.innerobj); return; } /************************************************************************* IMPROVED LEVENBERG-MARQUARDT METHOD FOR NON-LINEAR LEAST SQUARES OPTIMIZATION DESCRIPTION: This function is used to find minimum of function which is represented as sum of squares: F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1]) using value of function vector f[] only. Finite differences are used to calculate Jacobian. REQUIREMENTS: This algorithm will request following information during its operation: * function vector f[] at given point X There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts fvec() callback. You can try to initialize MinLMState structure with VJ function and then use incorrect version of MinLMOptimize() (for example, version which works with general form function and does not accept function vector), but it will lead to exception being thrown after first attempt to calculate Jacobian. USAGE: 1. User initializes algorithm state with MinLMCreateV() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of functions f[i] X - initial solution, array[0..N-1] DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state See also MinLMIteration, MinLMResults. NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatev(int n, int m, double[] x, double diffstep, out minlmstate state) { state = new minlmstate(); minlm.minlmcreatev(n, m, x, diffstep, state.innerobj); return; } public static void minlmcreatev(int m, double[] x, double diffstep, out minlmstate state) { int n; state = new minlmstate(); n = ap.len(x); minlm.minlmcreatev(n, m, x, diffstep, state.innerobj); return; } /************************************************************************* LEVENBERG-MARQUARDT-LIKE METHOD FOR NON-LINEAR OPTIMIZATION DESCRIPTION: This function is used to find minimum of general form (not "sum-of- -squares") function F = F(x[0], ..., x[n-1]) using its gradient and Hessian. Levenberg-Marquardt modification with L-BFGS pre-optimization and internal pre-conditioned L-BFGS optimization after each Levenberg-Marquardt step is used. REQUIREMENTS: This algorithm will request following information during its operation: * function value F at given point X * F and gradient G (simultaneously) at given point X * F, G and Hessian H (simultaneously) at given point X There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts func(), grad() and hess() function pointers. First pointer is used to calculate F at given point, second one calculates F(x) and grad F(x), third one calculates F(x), grad F(x), hess F(x). You can try to initialize MinLMState structure with FGH-function and then use incorrect version of MinLMOptimize() (for example, version which does not provide Hessian matrix), but it will lead to exception being thrown after first attempt to calculate Hessian. USAGE: 1. User initializes algorithm state with MinLMCreateFGH() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and pointers (delegates, etc.) to callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - initial solution, array[0..N-1] OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatefgh(int n, double[] x, out minlmstate state) { state = new minlmstate(); minlm.minlmcreatefgh(n, x, state.innerobj); return; } public static void minlmcreatefgh(double[] x, out minlmstate state) { int n; state = new minlmstate(); n = ap.len(x); minlm.minlmcreatefgh(n, x, state.innerobj); return; } /************************************************************************* This function sets stopping conditions for Levenberg-Marquardt optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinLMSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Only Levenberg-Marquardt iterations are counted (L-BFGS/CG iterations are NOT counted because their cost is very low compared to that of LM). Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetcond(minlmstate state, double epsg, double epsf, double epsx, int maxits) { minlm.minlmsetcond(state.innerobj, epsg, epsf, epsx, maxits); return; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinLMOptimize(). Both Levenberg-Marquardt and internal L-BFGS iterations are reported. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetxrep(minlmstate state, bool needxrep) { minlm.minlmsetxrep(state.innerobj, needxrep); return; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. NOTE: non-zero StpMax leads to moderate performance degradation because intermediate step of preconditioned L-BFGS optimization is incompatible with limits on step size. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetstpmax(minlmstate state, double stpmax) { minlm.minlmsetstpmax(state.innerobj, stpmax); return; } /************************************************************************* This function sets scaling coefficients for LM optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Generally, scale is NOT considered to be a form of preconditioner. But LM optimizer is unique in that it uses scaling matrix both in the stopping condition tests and as Marquardt damping factor. Proper scaling is very important for the algorithm performance. It is less important for the quality of results, but still has some influence (it is easier to converge when variables are properly scaled, so premature stopping is possible when very badly scalled variables are combined with relaxed stopping conditions). INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minlmsetscale(minlmstate state, double[] s) { minlm.minlmsetscale(state.innerobj, s); return; } /************************************************************************* This function sets boundary constraints for LM optimizer Boundary constraints are inactive by default (after initial creation). They are preserved until explicitly turned off with another SetBC() call. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF (latter is recommended because it will allow solver to use better algorithm). BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF (latter is recommended because it will allow solver to use better algorithm). NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints or at its boundary -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minlmsetbc(minlmstate state, double[] bndl, double[] bndu) { minlm.minlmsetbc(state.innerobj, bndl, bndu); return; } /************************************************************************* This function is used to change acceleration settings You can choose between three acceleration strategies: * AccType=0, no acceleration. * AccType=1, secant updates are used to update quadratic model after each iteration. After fixed number of iterations (or after model breakdown) we recalculate quadratic model using analytic Jacobian or finite differences. Number of secant-based iterations depends on optimization settings: about 3 iterations - when we have analytic Jacobian, up to 2*N iterations - when we use finite differences to calculate Jacobian. AccType=1 is recommended when Jacobian calculation cost is prohibitive high (several Mx1 function vector calculations followed by several NxN Cholesky factorizations are faster than calculation of one M*N Jacobian). It should also be used when we have no Jacobian, because finite difference approximation takes too much time to compute. Table below list optimization protocols (XYZ protocol corresponds to MinLMCreateXYZ) and acceleration types they support (and use by default). ACCELERATION TYPES SUPPORTED BY OPTIMIZATION PROTOCOLS: protocol 0 1 comment V + + VJ + + FGH + DAFAULT VALUES: protocol 0 1 comment V x without acceleration it is so slooooooooow VJ x FGH x NOTE: this function should be called before optimization. Attempt to call it during algorithm iterations may result in unexpected behavior. NOTE: attempt to call this function with unsupported protocol/acceleration combination will result in exception being thrown. -- ALGLIB -- Copyright 14.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetacctype(minlmstate state, int acctype) { minlm.minlmsetacctype(state.innerobj, acctype); return; } /************************************************************************* This function provides reverse communication interface Reverse communication interface is not documented or recommended to use. See below for functions which provide better documented API *************************************************************************/ public static bool minlmiteration(minlmstate state) { bool result = minlm.minlmiteration(state.innerobj); return result; } /************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x hess - callback which calculates function (or merit function) value func, gradient grad and Hessian hess at given point x fvec - callback which calculates function vector fi[] at given point x jac - callback which calculates function vector fi[] and Jacobian jac at given point x rep - optional callback which is called after each iteration can be null obj - optional object which is passed to func/grad/hess/jac/rep can be null NOTES: 1. Depending on function used to create state structure, this algorithm may accept Jacobian and/or Hessian and/or gradient. According to the said above, there ase several versions of this function, which accept different sets of callbacks. This flexibility opens way to subtle errors - you may create state with MinLMCreateFGH() (optimization using Hessian), but call function which does not accept Hessian. So when algorithm will request Hessian, there will be no callback to call. In this case exception will be thrown. Be careful to avoid such errors because there is no way to find them at compile time - you can see them at runtime only. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmoptimize(minlmstate state, ndimensional_fvec fvec, ndimensional_rep rep, object obj) { if( fvec==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (fvec is null)"); while( alglib.minlmiteration(state) ) { if( state.needfi ) { fvec(state.x, state.innerobj.fi, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlmoptimize' (some derivatives were not provided?)"); } } public static void minlmoptimize(minlmstate state, ndimensional_fvec fvec, ndimensional_jac jac, ndimensional_rep rep, object obj) { if( fvec==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (fvec is null)"); if( jac==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (jac is null)"); while( alglib.minlmiteration(state) ) { if( state.needfi ) { fvec(state.x, state.innerobj.fi, obj); continue; } if( state.needfij ) { jac(state.x, state.innerobj.fi, state.innerobj.j, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlmoptimize' (some derivatives were not provided?)"); } } public static void minlmoptimize(minlmstate state, ndimensional_func func, ndimensional_grad grad, ndimensional_hess hess, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (func is null)"); if( grad==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (grad is null)"); if( hess==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (hess is null)"); while( alglib.minlmiteration(state) ) { if( state.needf ) { func(state.x, ref state.innerobj.f, obj); continue; } if( state.needfg ) { grad(state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.needfgh ) { hess(state.x, ref state.innerobj.f, state.innerobj.g, state.innerobj.h, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlmoptimize' (some derivatives were not provided?)"); } } public static void minlmoptimize(minlmstate state, ndimensional_func func, ndimensional_jac jac, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (func is null)"); if( jac==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (jac is null)"); while( alglib.minlmiteration(state) ) { if( state.needf ) { func(state.x, ref state.innerobj.f, obj); continue; } if( state.needfij ) { jac(state.x, state.innerobj.fi, state.innerobj.j, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlmoptimize' (some derivatives were not provided?)"); } } public static void minlmoptimize(minlmstate state, ndimensional_func func, ndimensional_grad grad, ndimensional_jac jac, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (func is null)"); if( grad==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (grad is null)"); if( jac==null ) throw new alglibexception("ALGLIB: error in 'minlmoptimize()' (jac is null)"); while( alglib.minlmiteration(state) ) { if( state.needf ) { func(state.x, ref state.innerobj.f, obj); continue; } if( state.needfg ) { grad(state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.needfij ) { jac(state.x, state.innerobj.fi, state.innerobj.j, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minlmoptimize' (some derivatives were not provided?)"); } } /************************************************************************* Levenberg-Marquardt algorithm results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report; see comments for this structure for more info. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmresults(minlmstate state, out double[] x, out minlmreport rep) { x = new double[0]; rep = new minlmreport(); minlm.minlmresults(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* Levenberg-Marquardt algorithm results Buffered implementation of MinLMResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmresultsbuf(minlmstate state, ref double[] x, minlmreport rep) { minlm.minlmresultsbuf(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* This subroutine restarts LM algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used for reverse communication previously allocated with MinLMCreateXXX call. X - new starting point. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmrestartfrom(minlmstate state, double[] x) { minlm.minlmrestartfrom(state.innerobj, x); return; } /************************************************************************* This is obsolete function. Since ALGLIB 3.3 it is equivalent to MinLMCreateVJ(). -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatevgj(int n, int m, double[] x, out minlmstate state) { state = new minlmstate(); minlm.minlmcreatevgj(n, m, x, state.innerobj); return; } public static void minlmcreatevgj(int m, double[] x, out minlmstate state) { int n; state = new minlmstate(); n = ap.len(x); minlm.minlmcreatevgj(n, m, x, state.innerobj); return; } /************************************************************************* This is obsolete function. Since ALGLIB 3.3 it is equivalent to MinLMCreateFJ(). -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatefgj(int n, int m, double[] x, out minlmstate state) { state = new minlmstate(); minlm.minlmcreatefgj(n, m, x, state.innerobj); return; } public static void minlmcreatefgj(int m, double[] x, out minlmstate state) { int n; state = new minlmstate(); n = ap.len(x); minlm.minlmcreatefgj(n, m, x, state.innerobj); return; } /************************************************************************* This function is considered obsolete since ALGLIB 3.1.0 and is present for backward compatibility only. We recommend to use MinLMCreateVJ, which provides similar, but more consistent and feature-rich interface. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatefj(int n, int m, double[] x, out minlmstate state) { state = new minlmstate(); minlm.minlmcreatefj(n, m, x, state.innerobj); return; } public static void minlmcreatefj(int m, double[] x, out minlmstate state) { int n; state = new minlmstate(); n = ap.len(x); minlm.minlmcreatefj(n, m, x, state.innerobj); return; } } public partial class alglib { /************************************************************************* *************************************************************************/ public class minasastate { // // Public declarations // public bool needfg { get { return _innerobj.needfg; } set { _innerobj.needfg = value; } } public bool xupdated { get { return _innerobj.xupdated; } set { _innerobj.xupdated = value; } } public double f { get { return _innerobj.f; } set { _innerobj.f = value; } } public double[] g { get { return _innerobj.g; } } public double[] x { get { return _innerobj.x; } } public minasastate() { _innerobj = new mincomp.minasastate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private mincomp.minasastate _innerobj; public mincomp.minasastate innerobj { get { return _innerobj; } } public minasastate(mincomp.minasastate obj) { _innerobj = obj; } } /************************************************************************* *************************************************************************/ public class minasareport { // // Public declarations // public int iterationscount { get { return _innerobj.iterationscount; } set { _innerobj.iterationscount = value; } } public int nfev { get { return _innerobj.nfev; } set { _innerobj.nfev = value; } } public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } } public int activeconstraints { get { return _innerobj.activeconstraints; } set { _innerobj.activeconstraints = value; } } public minasareport() { _innerobj = new mincomp.minasareport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private mincomp.minasareport _innerobj; public mincomp.minasareport innerobj { get { return _innerobj; } } public minasareport(mincomp.minasareport obj) { _innerobj = obj; } } /************************************************************************* Obsolete function, use MinLBFGSSetPrecDefault() instead. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetdefaultpreconditioner(minlbfgsstate state) { mincomp.minlbfgssetdefaultpreconditioner(state.innerobj); return; } /************************************************************************* Obsolete function, use MinLBFGSSetCholeskyPreconditioner() instead. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetcholeskypreconditioner(minlbfgsstate state, double[,] p, bool isupper) { mincomp.minlbfgssetcholeskypreconditioner(state.innerobj, p, isupper); return; } /************************************************************************* This is obsolete function which was used by previous version of the BLEIC optimizer. It does nothing in the current version of BLEIC. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetbarrierwidth(minbleicstate state, double mu) { mincomp.minbleicsetbarrierwidth(state.innerobj, mu); return; } /************************************************************************* This is obsolete function which was used by previous version of the BLEIC optimizer. It does nothing in the current version of BLEIC. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetbarrierdecay(minbleicstate state, double mudecay) { mincomp.minbleicsetbarrierdecay(state.innerobj, mudecay); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 25.03.2010 by Bochkanov Sergey *************************************************************************/ public static void minasacreate(int n, double[] x, double[] bndl, double[] bndu, out minasastate state) { state = new minasastate(); mincomp.minasacreate(n, x, bndl, bndu, state.innerobj); return; } public static void minasacreate(double[] x, double[] bndl, double[] bndu, out minasastate state) { int n; if( (ap.len(x)!=ap.len(bndl)) || (ap.len(x)!=ap.len(bndu))) throw new alglibexception("Error while calling 'minasacreate': looks like one of arguments has wrong size"); state = new minasastate(); n = ap.len(x); mincomp.minasacreate(n, x, bndl, bndu, state.innerobj); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetcond(minasastate state, double epsg, double epsf, double epsx, int maxits) { mincomp.minasasetcond(state.innerobj, epsg, epsf, epsx, maxits); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetxrep(minasastate state, bool needxrep) { mincomp.minasasetxrep(state.innerobj, needxrep); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetalgorithm(minasastate state, int algotype) { mincomp.minasasetalgorithm(state.innerobj, algotype); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetstpmax(minasastate state, double stpmax) { mincomp.minasasetstpmax(state.innerobj, stpmax); return; } /************************************************************************* This function provides reverse communication interface Reverse communication interface is not documented or recommended to use. See below for functions which provide better documented API *************************************************************************/ public static bool minasaiteration(minasastate state) { bool result = mincomp.minasaiteration(state.innerobj); return result; } /************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be null obj - optional object which is passed to func/grad/hess/jac/rep can be null -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minasaoptimize(minasastate state, ndimensional_grad grad, ndimensional_rep rep, object obj) { if( grad==null ) throw new alglibexception("ALGLIB: error in 'minasaoptimize()' (grad is null)"); while( alglib.minasaiteration(state) ) { if( state.needfg ) { grad(state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.x, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'minasaoptimize' (some derivatives were not provided?)"); } } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minasaresults(minasastate state, out double[] x, out minasareport rep) { x = new double[0]; rep = new minasareport(); mincomp.minasaresults(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minasaresultsbuf(minasastate state, ref double[] x, minasareport rep) { mincomp.minasaresultsbuf(state.innerobj, ref x, rep.innerobj); return; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/ public static void minasarestartfrom(minasastate state, double[] x, double[] bndl, double[] bndu) { mincomp.minasarestartfrom(state.innerobj, x, bndl, bndu); return; } } public partial class alglib { public class optserv { /************************************************************************* This subroutine is used to prepare threshold value which will be used for trimming of the target function (see comments on TrimFunction() for more information). This function accepts only one parameter: function value at the starting point. It returns threshold which will be used for trimming. -- ALGLIB -- Copyright 10.05.2011 by Bochkanov Sergey *************************************************************************/ public static void trimprepare(double f, ref double threshold) { threshold = 0; threshold = 10*(Math.Abs(f)+1); } /************************************************************************* This subroutine is used to "trim" target function, i.e. to do following transformation: { {F,G} if F=Threshold Such transformation allows us to solve problems with singularities by redefining function in such way that it becomes bounded from above. -- ALGLIB -- Copyright 10.05.2011 by Bochkanov Sergey *************************************************************************/ public static void trimfunction(ref double f, ref double[] g, int n, double threshold) { int i = 0; if( (double)(f)>=(double)(threshold) ) { f = threshold; for(i=0; i<=n-1; i++) { g[i] = 0.0; } } } /************************************************************************* This function enforces boundary constraints in the X. This function correctly (although a bit inefficient) handles BL[i] which are -INF and BU[i] which are +INF. We have NMain+NSlack dimensional X, with first NMain components bounded by BL/BU, and next NSlack ones bounded by non-negativity constraints. INPUT PARAMETERS X - array[NMain+NSlack], point BL - array[NMain], lower bounds (may contain -INF, when bound is not present) HaveBL - array[NMain], if HaveBL[i] is False, then i-th bound is not present BU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBU - array[NMain], if HaveBU[i] is False, then i-th bound is not present OUTPUT PARAMETERS X - X with all constraints being enforced It returns True when constraints are consistent, False - when constraints are inconsistent. -- ALGLIB -- Copyright 10.01.2012 by Bochkanov Sergey *************************************************************************/ public static bool enforceboundaryconstraints(ref double[] x, double[] bl, bool[] havebl, double[] bu, bool[] havebu, int nmain, int nslack) { bool result = new bool(); int i = 0; result = false; for(i=0; i<=nmain-1; i++) { if( (havebl[i] && havebu[i]) && (double)(bl[i])>(double)(bu[i]) ) { return result; } if( havebl[i] && (double)(x[i])<(double)(bl[i]) ) { x[i] = bl[i]; } if( havebu[i] && (double)(x[i])>(double)(bu[i]) ) { x[i] = bu[i]; } } for(i=0; i<=nslack-1; i++) { if( (double)(x[nmain+i])<(double)(0) ) { x[nmain+i] = 0; } } result = true; return result; } /************************************************************************* This function projects gradient into feasible area of boundary constrained optimization problem. X can be infeasible with respect to boundary constraints. We have NMain+NSlack dimensional X, with first NMain components bounded by BL/BU, and next NSlack ones bounded by non-negativity constraints. INPUT PARAMETERS X - array[NMain+NSlack], point G - array[NMain+NSlack], gradient BL - lower bounds (may contain -INF, when bound is not present) HaveBL - if HaveBL[i] is False, then i-th bound is not present BU - upper bounds (may contain +INF, when bound is not present) HaveBU - if HaveBU[i] is False, then i-th bound is not present OUTPUT PARAMETERS G - projection of G. Components of G which satisfy one of the following (1) (X[I]<=BndL[I]) and (G[I]>0), OR (2) (X[I]>=BndU[I]) and (G[I]<0) are replaced by zeros. NOTE 1: this function assumes that constraints are feasible. It throws exception otherwise. NOTE 2: in fact, projection of ANTI-gradient is calculated, because this function trims components of -G which points outside of the feasible area. However, working with -G is considered confusing, because all optimization source work with G. -- ALGLIB -- Copyright 10.01.2012 by Bochkanov Sergey *************************************************************************/ public static void projectgradientintobc(double[] x, ref double[] g, double[] bl, bool[] havebl, double[] bu, bool[] havebu, int nmain, int nslack) { int i = 0; for(i=0; i<=nmain-1; i++) { alglib.ap.assert((!havebl[i] || !havebu[i]) || (double)(bl[i])<=(double)(bu[i]), "ProjectGradientIntoBC: internal error (infeasible constraints)"); if( (havebl[i] && (double)(x[i])<=(double)(bl[i])) && (double)(g[i])>(double)(0) ) { g[i] = 0; } if( (havebu[i] && (double)(x[i])>=(double)(bu[i])) && (double)(g[i])<(double)(0) ) { g[i] = 0; } } for(i=0; i<=nslack-1; i++) { if( (double)(x[nmain+i])<=(double)(0) && (double)(g[nmain+i])>(double)(0) ) { g[nmain+i] = 0; } } } /************************************************************************* Given a) initial point X0[NMain+NSlack] (feasible with respect to bound constraints) b) step vector alpha*D[NMain+NSlack] c) boundary constraints BndL[NMain], BndU[NMain] d) implicit non-negativity constraints for slack variables this function calculates bound on the step length subject to boundary constraints. It returns: * MaxStepLen - such step length that X0+MaxStepLen*alpha*D is exactly at the boundary given by constraints * VariableToFreeze - index of the constraint to be activated, 0 <= VariableToFreeze < NMain+NSlack * ValueToFreeze - value of the corresponding constraint. Notes: * it is possible that several constraints can be activated by the step at once. In such cases only one constraint is returned. It is caller responsibility to check other constraints. This function makes sure that we activate at least one constraint, and everything else is the responsibility of the caller. * steps smaller than MaxStepLen still can activate constraints due to numerical errors. Thus purpose of this function is not to guard against accidental activation of the constraints - quite the reverse, its purpose is to activate at least constraint upon performing step which is too long. * in case there is no constraints to activate, we return negative VariableToFreeze and zero MaxStepLen and ValueToFreeze. * this function assumes that constraints are consistent; it throws exception otherwise. INPUT PARAMETERS X - array[NMain+NSlack], point. Must be feasible with respect to bound constraints (exception will be thrown otherwise) D - array[NMain+NSlack], step direction alpha - scalar multiplier before D, alpha<>0 BndL - lower bounds, array[NMain] (may contain -INF, when bound is not present) HaveBndL - array[NMain], if HaveBndL[i] is False, then i-th bound is not present BndU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBndU - array[NMain], if HaveBndU[i] is False, then i-th bound is not present NMain - number of main variables NSlack - number of slack variables OUTPUT PARAMETERS VariableToFreeze: * negative value = step is unbounded, ValueToFreeze=0, MaxStepLen=0. * non-negative value = at least one constraint, given by this parameter, will be activated upon performing maximum step. ValueToFreeze- value of the variable which will be constrained MaxStepLen - maximum length of the step. Can be zero when step vector looks outside of the feasible area. -- ALGLIB -- Copyright 10.01.2012 by Bochkanov Sergey *************************************************************************/ public static void calculatestepbound(double[] x, double[] d, double alpha, double[] bndl, bool[] havebndl, double[] bndu, bool[] havebndu, int nmain, int nslack, ref int variabletofreeze, ref double valuetofreeze, ref double maxsteplen) { int i = 0; double prevmax = 0; double initval = 0; variabletofreeze = 0; valuetofreeze = 0; maxsteplen = 0; alglib.ap.assert((double)(alpha)!=(double)(0), "CalculateStepBound: zero alpha"); variabletofreeze = -1; initval = math.maxrealnumber; maxsteplen = initval; for(i=0; i<=nmain-1; i++) { if( havebndl[i] && (double)(alpha*d[i])<(double)(0) ) { alglib.ap.assert((double)(x[i])>=(double)(bndl[i]), "CalculateStepBound: infeasible X"); prevmax = maxsteplen; maxsteplen = apserv.safeminposrv(x[i]-bndl[i], -(alpha*d[i]), maxsteplen); if( (double)(maxsteplen)<(double)(prevmax) ) { variabletofreeze = i; valuetofreeze = bndl[i]; } } if( havebndu[i] && (double)(alpha*d[i])>(double)(0) ) { alglib.ap.assert((double)(x[i])<=(double)(bndu[i]), "CalculateStepBound: infeasible X"); prevmax = maxsteplen; maxsteplen = apserv.safeminposrv(bndu[i]-x[i], alpha*d[i], maxsteplen); if( (double)(maxsteplen)<(double)(prevmax) ) { variabletofreeze = i; valuetofreeze = bndu[i]; } } } for(i=0; i<=nslack-1; i++) { if( (double)(alpha*d[nmain+i])<(double)(0) ) { alglib.ap.assert((double)(x[nmain+i])>=(double)(0), "CalculateStepBound: infeasible X"); prevmax = maxsteplen; maxsteplen = apserv.safeminposrv(x[nmain+i], -(alpha*d[nmain+i]), maxsteplen); if( (double)(maxsteplen)<(double)(prevmax) ) { variabletofreeze = nmain+i; valuetofreeze = 0; } } } if( (double)(maxsteplen)==(double)(initval) ) { valuetofreeze = 0; maxsteplen = 0; } } /************************************************************************* This function postprocesses bounded step by: * analysing step length (whether it is equal to MaxStepLen) and activating constraint given by VariableToFreeze if needed * checking for additional bound constraints to activate This function uses final point of the step, quantities calculated by the CalculateStepBound() function. As result, it returns point which is exactly feasible with respect to boundary constraints. NOTE 1: this function does NOT handle and check linear equality constraints NOTE 2: when StepTaken=MaxStepLen we always activate at least one constraint INPUT PARAMETERS X - array[NMain+NSlack], final point to postprocess XPrev - array[NMain+NSlack], initial point BndL - lower bounds, array[NMain] (may contain -INF, when bound is not present) HaveBndL - array[NMain], if HaveBndL[i] is False, then i-th bound is not present BndU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBndU - array[NMain], if HaveBndU[i] is False, then i-th bound is not present NMain - number of main variables NSlack - number of slack variables VariableToFreeze-result of CalculateStepBound() ValueToFreeze- result of CalculateStepBound() StepTaken - actual step length (actual step is equal to the possibly non-unit step direction vector times this parameter). StepTaken<=MaxStepLen. MaxStepLen - result of CalculateStepBound() OUTPUT PARAMETERS X - point bounded with respect to constraints. components corresponding to active constraints are exactly equal to the boundary values. RESULT: number of constraints activated in addition to previously active ones. Constraints which were DEACTIVATED are ignored (do not influence function value). -- ALGLIB -- Copyright 10.01.2012 by Bochkanov Sergey *************************************************************************/ public static int postprocessboundedstep(ref double[] x, double[] xprev, double[] bndl, bool[] havebndl, double[] bndu, bool[] havebndu, int nmain, int nslack, int variabletofreeze, double valuetofreeze, double steptaken, double maxsteplen) { int result = 0; int i = 0; alglib.ap.assert(variabletofreeze<0 || (double)(steptaken)<=(double)(maxsteplen)); // // Activate constraints // if( variabletofreeze>=0 && (double)(steptaken)==(double)(maxsteplen) ) { x[variabletofreeze] = valuetofreeze; } for(i=0; i<=nmain-1; i++) { if( havebndl[i] && (double)(x[i])<(double)(bndl[i]) ) { x[i] = bndl[i]; } if( havebndu[i] && (double)(x[i])>(double)(bndu[i]) ) { x[i] = bndu[i]; } } for(i=0; i<=nslack-1; i++) { if( (double)(x[nmain+i])<=(double)(0) ) { x[nmain+i] = 0; } } // // Calculate number of constraints being activated // result = 0; for(i=0; i<=nmain-1; i++) { if( (double)(x[i])!=(double)(xprev[i]) && ((havebndl[i] && (double)(x[i])==(double)(bndl[i])) || (havebndu[i] && (double)(x[i])==(double)(bndu[i]))) ) { result = result+1; } } for(i=0; i<=nslack-1; i++) { if( (double)(x[nmain+i])!=(double)(xprev[nmain+i]) && (double)(x[nmain+i])==(double)(0.0) ) { result = result+1; } } return result; } /************************************************************************* The purpose of this function is to prevent algorithm from "unsticking" from the active bound constraints because of numerical noise in the gradient or Hessian. It is done by zeroing some components of the search direction D. D[i] is zeroed when both (a) and (b) are true: a) corresponding X[i] is exactly at the boundary b) |D[i]*S[i]| <= DropTol*Sqrt(SUM(D[i]^2*S[I]^2)) D can be step direction , antigradient, gradient, or anything similar. Sign of D does not matter, nor matters step length. NOTE 1: boundary constraints are expected to be consistent, as well as X is expected to be feasible. Exception will be thrown otherwise. INPUT PARAMETERS D - array[NMain+NSlack], direction X - array[NMain+NSlack], current point BndL - lower bounds, array[NMain] (may contain -INF, when bound is not present) HaveBndL - array[NMain], if HaveBndL[i] is False, then i-th bound is not present BndU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBndU - array[NMain], if HaveBndU[i] is False, then i-th bound is not present S - array[NMain+NSlack], scaling of the variables NMain - number of main variables NSlack - number of slack variables DropTol - drop tolerance, >=0 OUTPUT PARAMETERS X - point bounded with respect to constraints. components corresponding to active constraints are exactly equal to the boundary values. -- ALGLIB -- Copyright 10.01.2012 by Bochkanov Sergey *************************************************************************/ public static void filterdirection(ref double[] d, double[] x, double[] bndl, bool[] havebndl, double[] bndu, bool[] havebndu, double[] s, int nmain, int nslack, double droptol) { int i = 0; double scalednorm = 0; bool isactive = new bool(); scalednorm = 0.0; for(i=0; i<=nmain+nslack-1; i++) { scalednorm = scalednorm+math.sqr(d[i]*s[i]); } scalednorm = Math.Sqrt(scalednorm); for(i=0; i<=nmain-1; i++) { alglib.ap.assert(!havebndl[i] || (double)(x[i])>=(double)(bndl[i]), "FilterDirection: infeasible point"); alglib.ap.assert(!havebndu[i] || (double)(x[i])<=(double)(bndu[i]), "FilterDirection: infeasible point"); isactive = (havebndl[i] && (double)(x[i])==(double)(bndl[i])) || (havebndu[i] && (double)(x[i])==(double)(bndu[i])); if( isactive && (double)(Math.Abs(d[i]*s[i]))<=(double)(droptol*scalednorm) ) { d[i] = 0.0; } } for(i=0; i<=nslack-1; i++) { alglib.ap.assert((double)(x[nmain+i])>=(double)(0), "FilterDirection: infeasible point"); if( (double)(x[nmain+i])==(double)(0) && (double)(Math.Abs(d[nmain+i]*s[nmain+i]))<=(double)(droptol*scalednorm) ) { d[nmain+i] = 0.0; } } } /************************************************************************* This function returns number of bound constraints whose state was changed (either activated or deactivated) when making step from XPrev to X. Constraints are considered: * active - when we are exactly at the boundary * inactive - when we are not at the boundary You should note that antigradient direction is NOT taken into account when we make decions on the constraint status. INPUT PARAMETERS X - array[NMain+NSlack], final point. Must be feasible with respect to bound constraints. XPrev - array[NMain+NSlack], initial point. Must be feasible with respect to bound constraints. BndL - lower bounds, array[NMain] (may contain -INF, when bound is not present) HaveBndL - array[NMain], if HaveBndL[i] is False, then i-th bound is not present BndU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBndU - array[NMain], if HaveBndU[i] is False, then i-th bound is not present NMain - number of main variables NSlack - number of slack variables RESULT: number of constraints whose state was changed. -- ALGLIB -- Copyright 10.01.2012 by Bochkanov Sergey *************************************************************************/ public static int numberofchangedconstraints(double[] x, double[] xprev, double[] bndl, bool[] havebndl, double[] bndu, bool[] havebndu, int nmain, int nslack) { int result = 0; int i = 0; bool statuschanged = new bool(); result = 0; for(i=0; i<=nmain-1; i++) { if( (double)(x[i])!=(double)(xprev[i]) ) { statuschanged = false; if( havebndl[i] && ((double)(x[i])==(double)(bndl[i]) || (double)(xprev[i])==(double)(bndl[i])) ) { statuschanged = true; } if( havebndu[i] && ((double)(x[i])==(double)(bndu[i]) || (double)(xprev[i])==(double)(bndu[i])) ) { statuschanged = true; } if( statuschanged ) { result = result+1; } } } for(i=0; i<=nslack-1; i++) { if( (double)(x[nmain+i])!=(double)(xprev[nmain+i]) && ((double)(x[nmain+i])==(double)(0) || (double)(xprev[nmain+i])==(double)(0)) ) { result = result+1; } } return result; } /************************************************************************* This function finds feasible point of (NMain+NSlack)-dimensional problem subject to NMain explicit boundary constraints (some constraints can be omitted), NSlack implicit non-negativity constraints, K linear equality constraints. INPUT PARAMETERS X - array[NMain+NSlack], initial point. BndL - lower bounds, array[NMain] (may contain -INF, when bound is not present) HaveBndL - array[NMain], if HaveBndL[i] is False, then i-th bound is not present BndU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBndU - array[NMain], if HaveBndU[i] is False, then i-th bound is not present NMain - number of main variables NSlack - number of slack variables CE - array[K,NMain+NSlack+1], equality constraints CE*x=b. Rows contain constraints, first NMain+NSlack columns contain coefficients before X[], last column contain right part. K - number of linear constraints EpsI - infeasibility (error in the right part) allowed in the solution OUTPUT PARAMETERS: X - feasible point or best infeasible point found before algorithm termination QPIts - number of QP iterations (for debug purposes) GPAIts - number of GPA iterations (for debug purposes) RESULT: True in case X is feasible, False - if it is infeasible. -- ALGLIB -- Copyright 20.01.2012 by Bochkanov Sergey *************************************************************************/ public static bool findfeasiblepoint(ref double[] x, double[] bndl, bool[] havebndl, double[] bndu, bool[] havebndu, int nmain, int nslack, double[,] ce, int k, double epsi, ref int qpits, ref int gpaits) { bool result = new bool(); int i = 0; int j = 0; int idx0 = 0; int idx1 = 0; double[] permx = new double[0]; double[] xn = new double[0]; double[] xa = new double[0]; double[] newtonstep = new double[0]; double[] g = new double[0]; double[] pg = new double[0]; double[,] a = new double[0,0]; double armijostep = 0; double armijobeststep = 0; double armijobestfeas = 0; double v = 0; double mx = 0; double feaserr = 0; double feasold = 0; double feasnew = 0; double pgnorm = 0; double vn = 0; double vd = 0; double stp = 0; int vartofreeze = 0; double valtofreeze = 0; double maxsteplen = 0; bool werechangesinconstraints = new bool(); bool stage1isover = new bool(); bool converged = new bool(); double[] activeconstraints = new double[0]; double[] tmpk = new double[0]; double[] colnorms = new double[0]; int nactive = 0; int nfree = 0; int nsvd = 0; int[] p1 = new int[0]; int[] p2 = new int[0]; apserv.apbuffers buf = new apserv.apbuffers(); double[] w = new double[0]; double[] s = new double[0]; double[,] u = new double[0,0]; double[,] vt = new double[0,0]; int itscount = 0; int itswithintolerance = 0; int maxitswithintolerance = 0; int gparuns = 0; int maxgparuns = 0; int maxarmijoruns = 0; int i_ = 0; ce = (double[,])ce.Clone(); qpits = 0; gpaits = 0; maxitswithintolerance = 3; maxgparuns = 3; maxarmijoruns = 5; qpits = 0; gpaits = 0; // // Initial enforcement of the feasibility with respect to boundary constraints // NOTE: after this block we assume that boundary constraints are consistent. // if( !enforceboundaryconstraints(ref x, bndl, havebndl, bndu, havebndu, nmain, nslack) ) { result = false; return result; } if( k==0 ) { // // No linear constraints, we can exit right now // result = true; return result; } // // Scale rows of CE in such way that max(CE[i,0..nmain+nslack-1])=1 for any i=0..k-1 // for(i=0; i<=k-1; i++) { v = 0.0; for(j=0; j<=nmain+nslack-1; j++) { v = Math.Max(v, Math.Abs(ce[i,j])); } if( (double)(v)!=(double)(0) ) { v = 1/v; for(i_=0; i_<=nmain+nslack;i_++) { ce[i,i_] = v*ce[i,i_]; } } } // // Allocate temporaries // xn = new double[nmain+nslack]; xa = new double[nmain+nslack]; permx = new double[nmain+nslack]; g = new double[nmain+nslack]; pg = new double[nmain+nslack]; tmpk = new double[k]; a = new double[k, nmain+nslack]; activeconstraints = new double[nmain+nslack]; newtonstep = new double[nmain+nslack]; s = new double[nmain+nslack]; colnorms = new double[nmain+nslack]; for(i=0; i<=nmain+nslack-1; i++) { s[i] = 1.0; colnorms[i] = 0.0; for(j=0; j<=k-1; j++) { colnorms[i] = colnorms[i]+math.sqr(ce[j,i]); } } // // K>0, we have linear equality constraints combined with bound constraints. // // Try to find feasible point as minimizer of the quadratic function // F(x) = 0.5*||CE*x-b||^2 = 0.5*x'*(CE'*CE)*x - (b'*CE)*x + 0.5*b'*b // subject to boundary constraints given by BL, BU and non-negativity of // the slack variables. BTW, we drop constant term because it does not // actually influences on the solution. // // Below we will assume that K>0. // itswithintolerance = 0; itscount = 0; while( true ) { // // Stage 0: check for exact convergence // converged = true; feaserr = 0; for(i=0; i<=k-1; i++) { // // Calculate: // * V - error in the right part // * MX - maximum term in the left part // // Terminate if error in the right part is not greater than 100*Eps*MX. // // IMPORTANT: we must perform check for non-strict inequality, i.e. to use <= instead of <. // it will allow us to easily handle situations with zero rows of CE. // mx = 0; v = -ce[i,nmain+nslack]; for(j=0; j<=nmain+nslack-1; j++) { mx = Math.Max(mx, Math.Abs(ce[i,j]*x[j])); v = v+ce[i,j]*x[j]; } feaserr = feaserr+math.sqr(v); converged = converged && (double)(Math.Abs(v))<=(double)(100*math.machineepsilon*mx); } feaserr = Math.Sqrt(feaserr); if( converged ) { result = (double)(feaserr)<=(double)(epsi); return result; } // // Stage 1: equality constrained quadratic programming // // * treat active bound constraints as equality ones (constraint is considered // active when we are at the boundary, independently of the antigradient direction) // * calculate unrestricted Newton step to point XM (which may be infeasible) // calculate MaxStepLen = largest step in direction of XM which retains feasibility. // * perform bounded step from X to XN: // a) XN=XM (if XM is feasible) // b) XN=X-MaxStepLen*(XM-X) (otherwise) // * X := XN // * if XM (Newton step subject to currently active constraints) was feasible, goto Stage 2 // * repeat Stage 1 // // NOTE 1: in order to solve constrained qudratic subproblem we will have to reorder // variables in such way that ones corresponding to inactive constraints will // be first, and active ones will be last in the list. CE and X are now // [ xi ] // separated into two parts: CE = [CEi CEa], x = [ ], where CEi/Xi correspond // [ xa ] // to INACTIVE constraints, and CEa/Xa correspond to the ACTIVE ones. // // Now, instead of F=0.5*x'*(CE'*CE)*x - (b'*CE)*x + 0.5*b'*b, we have // F(xi) = 0.5*(CEi*xi,CEi*xi) + (CEa*xa-b,CEi*xi) + (0.5*CEa*xa-b,CEa*xa). // Here xa is considered constant, i.e. we optimize with respect to xi, leaving xa fixed. // // We can solve it by performing SVD of CEi and calculating pseudoinverse of the // Hessian matrix. Of course, we do NOT calculate pseudoinverse explicitly - we // just use singular vectors to perform implicit multiplication by it. // // while( true ) { // // Calculate G - gradient subject to equality constraints, // multiply it by inverse of the Hessian diagonal to obtain initial // step vector. // // Bound step subject to constraints which can be activated, // run Armijo search with increasing step size. // Search is terminated when feasibility error stops to decrease. // // NOTE: it is important to test for "stops to decrease" instead // of "starts to increase" in order to correctly handle cases with // zero CE. // armijobeststep = 0.0; armijobestfeas = 0.0; for(i=0; i<=nmain+nslack-1; i++) { g[i] = 0; } for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*x[i_]; } v = v-ce[i,nmain+nslack]; armijobestfeas = armijobestfeas+math.sqr(v); for(i_=0; i_<=nmain+nslack-1;i_++) { g[i_] = g[i_] + v*ce[i,i_]; } } armijobestfeas = Math.Sqrt(armijobestfeas); for(i=0; i<=nmain-1; i++) { if( havebndl[i] && (double)(x[i])==(double)(bndl[i]) ) { g[i] = 0.0; } if( havebndu[i] && (double)(x[i])==(double)(bndu[i]) ) { g[i] = 0.0; } } for(i=0; i<=nslack-1; i++) { if( (double)(x[nmain+i])==(double)(0.0) ) { g[nmain+i] = 0.0; } } v = 0.0; for(i=0; i<=nmain+nslack-1; i++) { if( (double)(math.sqr(colnorms[i]))!=(double)(0) ) { newtonstep[i] = -(g[i]/math.sqr(colnorms[i])); } else { newtonstep[i] = 0.0; } v = v+math.sqr(newtonstep[i]); } if( (double)(v)==(double)(0) ) { // // Constrained gradient is zero, QP iterations are over // break; } calculatestepbound(x, newtonstep, 1.0, bndl, havebndl, bndu, havebndu, nmain, nslack, ref vartofreeze, ref valtofreeze, ref maxsteplen); if( vartofreeze>=0 && (double)(maxsteplen)==(double)(0) ) { // // Can not perform step, QP iterations are over // break; } if( vartofreeze>=0 ) { armijostep = Math.Min(1.0, maxsteplen); } else { armijostep = 1; } while( true ) { for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = x[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = xa[i_] + armijostep*newtonstep[i_]; } enforceboundaryconstraints(ref xa, bndl, havebndl, bndu, havebndu, nmain, nslack); feaserr = 0.0; for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*xa[i_]; } v = v-ce[i,nmain+nslack]; feaserr = feaserr+math.sqr(v); } feaserr = Math.Sqrt(feaserr); if( (double)(feaserr)>=(double)(armijobestfeas) ) { break; } armijobestfeas = feaserr; armijobeststep = armijostep; armijostep = 2.0*armijostep; } for(i_=0; i_<=nmain+nslack-1;i_++) { x[i_] = x[i_] + armijobeststep*newtonstep[i_]; } enforceboundaryconstraints(ref x, bndl, havebndl, bndu, havebndu, nmain, nslack); // // Determine number of active and free constraints // nactive = 0; for(i=0; i<=nmain-1; i++) { activeconstraints[i] = 0; if( havebndl[i] && (double)(x[i])==(double)(bndl[i]) ) { activeconstraints[i] = 1; } if( havebndu[i] && (double)(x[i])==(double)(bndu[i]) ) { activeconstraints[i] = 1; } if( (double)(activeconstraints[i])>(double)(0) ) { nactive = nactive+1; } } for(i=0; i<=nslack-1; i++) { activeconstraints[nmain+i] = 0; if( (double)(x[nmain+i])==(double)(0.0) ) { activeconstraints[nmain+i] = 1; } if( (double)(activeconstraints[nmain+i])>(double)(0) ) { nactive = nactive+1; } } nfree = nmain+nslack-nactive; if( nfree==0 ) { break; } qpits = qpits+1; // // Reorder variables // tsort.tagsortbuf(ref activeconstraints, nmain+nslack, ref p1, ref p2, buf); for(i=0; i<=k-1; i++) { for(j=0; j<=nmain+nslack-1; j++) { a[i,j] = ce[i,j]; } } for(j=0; j<=nmain+nslack-1; j++) { permx[j] = x[j]; } for(j=0; j<=nmain+nslack-1; j++) { if( p2[j]!=j ) { idx0 = p2[j]; idx1 = j; for(i=0; i<=k-1; i++) { v = a[i,idx0]; a[i,idx0] = a[i,idx1]; a[i,idx1] = v; } v = permx[idx0]; permx[idx0] = permx[idx1]; permx[idx1] = v; } } // // Calculate (unprojected) gradient: // G(xi) = CEi'*(CEi*xi + CEa*xa - b) // for(i=0; i<=nfree-1; i++) { g[i] = 0; } for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += a[i,i_]*permx[i_]; } tmpk[i] = v-ce[i,nmain+nslack]; } for(i=0; i<=k-1; i++) { v = tmpk[i]; for(i_=0; i_<=nfree-1;i_++) { g[i_] = g[i_] + v*a[i,i_]; } } // // Calculate Newton step using SVD of CEi: // F(xi) = 0.5*xi'*H*xi + g'*xi (Taylor decomposition) // XN = -H^(-1)*g (new point, solution of the QP subproblem) // H = CEi'*CEi // CEi = U*W*V' (SVD of CEi) // H = V*W^2*V' // H^(-1) = V*W^(-2)*V' // step = -V*W^(-2)*V'*g (it is better to perform multiplication from right to left) // // NOTE 1: we do NOT need left singular vectors to perform Newton step. // nsvd = Math.Min(k, nfree); if( !svd.rmatrixsvd(a, k, nfree, 0, 1, 2, ref w, ref u, ref vt) ) { result = false; return result; } for(i=0; i<=nsvd-1; i++) { v = 0.0; for(i_=0; i_<=nfree-1;i_++) { v += vt[i,i_]*g[i_]; } tmpk[i] = v; } for(i=0; i<=nsvd-1; i++) { // // It is important to have strict ">" in order to correctly // handle zero singular values. // if( (double)(math.sqr(w[i]))>(double)(math.sqr(w[0])*(nmain+nslack)*math.machineepsilon) ) { tmpk[i] = tmpk[i]/math.sqr(w[i]); } else { tmpk[i] = 0; } } for(i=0; i<=nmain+nslack-1; i++) { newtonstep[i] = 0; } for(i=0; i<=nsvd-1; i++) { v = tmpk[i]; for(i_=0; i_<=nfree-1;i_++) { newtonstep[i_] = newtonstep[i_] - v*vt[i,i_]; } } for(j=nmain+nslack-1; j>=0; j--) { if( p2[j]!=j ) { idx0 = p2[j]; idx1 = j; v = newtonstep[idx0]; newtonstep[idx0] = newtonstep[idx1]; newtonstep[idx1] = v; } } // // NewtonStep contains Newton step subject to active bound constraints. // // Such step leads us to the minimizer of the equality constrained F, // but such minimizer may be infeasible because some constraints which // are inactive at the initial point can be violated at the solution. // // Thus, we perform optimization in two stages: // a) perform bounded Newton step, i.e. step in the Newton direction // until activation of the first constraint // b) in case (MaxStepLen>0)and(MaxStepLen<1), perform additional iteration // of the Armijo line search in the rest of the Newton direction. // calculatestepbound(x, newtonstep, 1.0, bndl, havebndl, bndu, havebndu, nmain, nslack, ref vartofreeze, ref valtofreeze, ref maxsteplen); if( vartofreeze>=0 && (double)(maxsteplen)==(double)(0) ) { // // Activation of the constraints prevent us from performing step, // QP iterations are over // break; } if( vartofreeze>=0 ) { v = Math.Min(1.0, maxsteplen); } else { v = 1.0; } for(i_=0; i_<=nmain+nslack-1;i_++) { xn[i_] = v*newtonstep[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { xn[i_] = xn[i_] + x[i_]; } postprocessboundedstep(ref xn, x, bndl, havebndl, bndu, havebndu, nmain, nslack, vartofreeze, valtofreeze, v, maxsteplen); if( (double)(maxsteplen)>(double)(0) && (double)(maxsteplen)<(double)(1) ) { // // Newton step was restricted by activation of the constraints, // perform Armijo iteration. // // Initial estimate for best step is zero step. We try different // step sizes, from the 1-MaxStepLen (residual of the full Newton // step) to progressively smaller and smaller steps. // armijobeststep = 0.0; armijobestfeas = 0.0; for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*xn[i_]; } v = v-ce[i,nmain+nslack]; armijobestfeas = armijobestfeas+math.sqr(v); } armijobestfeas = Math.Sqrt(armijobestfeas); armijostep = 1-maxsteplen; for(j=0; j<=maxarmijoruns-1; j++) { for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = xn[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = xa[i_] + armijostep*newtonstep[i_]; } enforceboundaryconstraints(ref xa, bndl, havebndl, bndu, havebndu, nmain, nslack); feaserr = 0.0; for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*xa[i_]; } v = v-ce[i,nmain+nslack]; feaserr = feaserr+math.sqr(v); } feaserr = Math.Sqrt(feaserr); if( (double)(feaserr)<(double)(armijobestfeas) ) { armijobestfeas = feaserr; armijobeststep = armijostep; } armijostep = 0.5*armijostep; } for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = xn[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = xa[i_] + armijobeststep*newtonstep[i_]; } enforceboundaryconstraints(ref xa, bndl, havebndl, bndu, havebndu, nmain, nslack); } else { // // Armijo iteration is not performed // for(i_=0; i_<=nmain+nslack-1;i_++) { xa[i_] = xn[i_]; } } stage1isover = (double)(maxsteplen)>=(double)(1) || (double)(maxsteplen)==(double)(0); // // Calculate feasibility errors for old and new X. // These quantinies are used for debugging purposes only. // However, we can leave them in release code because performance impact is insignificant. // // Update X. Exit if needed. // feasold = 0; feasnew = 0; for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*x[i_]; } feasold = feasold+math.sqr(v-ce[i,nmain+nslack]); v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*xa[i_]; } feasnew = feasnew+math.sqr(v-ce[i,nmain+nslack]); } feasold = Math.Sqrt(feasold); feasnew = Math.Sqrt(feasnew); if( (double)(feasnew)>=(double)(feasold) ) { break; } for(i_=0; i_<=nmain+nslack-1;i_++) { x[i_] = xa[i_]; } if( stage1isover ) { break; } } // // Stage 2: gradient projection algorithm (GPA) // // * calculate feasibility error (with respect to linear equality constraints) // * calculate gradient G of F, project it into feasible area (G => PG) // * exit if norm(PG) is exactly zero or feasibility error is smaller than EpsC // * let XM be exact minimum of F along -PG (XM may be infeasible). // calculate MaxStepLen = largest step in direction of -PG which retains feasibility. // * perform bounded step from X to XN: // a) XN=XM (if XM is feasible) // b) XN=X-MaxStepLen*PG (otherwise) // * X := XN // * stop after specified number of iterations or when no new constraints was activated // // NOTES: // * grad(F) = (CE'*CE)*x - (b'*CE)^T // * CE[i] denotes I-th row of CE // * XM = X+stp*(-PG) where stp=(grad(F(X)),PG)/(CE*PG,CE*PG). // Here PG is a projected gradient, but in fact it can be arbitrary non-zero // direction vector - formula for minimum of F along PG still will be correct. // werechangesinconstraints = false; for(gparuns=1; gparuns<=k; gparuns++) { // // calculate feasibility error and G // feaserr = 0; for(i=0; i<=nmain+nslack-1; i++) { g[i] = 0; } for(i=0; i<=k-1; i++) { // // G += CE[i]^T * (CE[i]*x-b[i]) // v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*x[i_]; } v = v-ce[i,nmain+nslack]; feaserr = feaserr+math.sqr(v); for(i_=0; i_<=nmain+nslack-1;i_++) { g[i_] = g[i_] + v*ce[i,i_]; } } // // project G, filter it (strip numerical noise) // for(i_=0; i_<=nmain+nslack-1;i_++) { pg[i_] = g[i_]; } projectgradientintobc(x, ref pg, bndl, havebndl, bndu, havebndu, nmain, nslack); filterdirection(ref pg, x, bndl, havebndl, bndu, havebndu, s, nmain, nslack, 1.0E-9); for(i=0; i<=nmain+nslack-1; i++) { if( (double)(math.sqr(colnorms[i]))!=(double)(0) ) { pg[i] = pg[i]/math.sqr(colnorms[i]); } else { pg[i] = 0.0; } } // // Check GNorm and feasibility. // Exit when GNorm is exactly zero. // pgnorm = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { pgnorm += pg[i_]*pg[i_]; } feaserr = Math.Sqrt(feaserr); pgnorm = Math.Sqrt(pgnorm); if( (double)(pgnorm)==(double)(0) ) { result = (double)(feaserr)<=(double)(epsi); return result; } // // calculate planned step length // vn = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { vn += g[i_]*pg[i_]; } vd = 0; for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*pg[i_]; } vd = vd+math.sqr(v); } stp = vn/vd; // // Calculate step bound. // Perform bounded step and post-process it // calculatestepbound(x, pg, -1.0, bndl, havebndl, bndu, havebndu, nmain, nslack, ref vartofreeze, ref valtofreeze, ref maxsteplen); if( vartofreeze>=0 && (double)(maxsteplen)==(double)(0) ) { result = false; return result; } if( vartofreeze>=0 ) { v = Math.Min(stp, maxsteplen); } else { v = stp; } for(i_=0; i_<=nmain+nslack-1;i_++) { xn[i_] = x[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { xn[i_] = xn[i_] - v*pg[i_]; } postprocessboundedstep(ref xn, x, bndl, havebndl, bndu, havebndu, nmain, nslack, vartofreeze, valtofreeze, v, maxsteplen); // // update X // check stopping criteria // werechangesinconstraints = werechangesinconstraints || numberofchangedconstraints(xn, x, bndl, havebndl, bndu, havebndu, nmain, nslack)>0; for(i_=0; i_<=nmain+nslack-1;i_++) { x[i_] = xn[i_]; } gpaits = gpaits+1; if( !werechangesinconstraints ) { break; } } // // Stage 3: decide to stop algorithm or not to stop // // 1. we can stop when last GPA run did NOT changed constraints status. // It means that we've found final set of the active constraints even // before GPA made its run. And it means that Newton step moved us to // the minimum subject to the present constraints. // Depending on feasibility error, True or False is returned. // feaserr = 0; for(i=0; i<=k-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += ce[i,i_]*x[i_]; } v = v-ce[i,nmain+nslack]; feaserr = feaserr+math.sqr(v); } feaserr = Math.Sqrt(feaserr); if( (double)(feaserr)<=(double)(epsi) ) { itswithintolerance = itswithintolerance+1; } else { itswithintolerance = 0; } if( !werechangesinconstraints || itswithintolerance>=maxitswithintolerance ) { result = (double)(feaserr)<=(double)(epsi); return result; } itscount = itscount+1; } return result; } } public class mincg { /************************************************************************* This object stores state of the nonlinear CG optimizer. You should use ALGLIB functions to work with this object. *************************************************************************/ public class mincgstate { public int n; public double epsg; public double epsf; public double epsx; public int maxits; public double stpmax; public double suggestedstep; public bool xrep; public bool drep; public int cgtype; public int prectype; public double[] diagh; public double[] diaghl2; public double[,] vcorr; public int vcnt; public double[] s; public double diffstep; public int nfev; public int mcstage; public int k; public double[] xk; public double[] dk; public double[] xn; public double[] dn; public double[] d; public double fold; public double stp; public double curstpmax; public double[] yk; public double lastgoodstep; public double lastscaledstep; public int mcinfo; public bool innerresetneeded; public bool terminationneeded; public double trimthreshold; public int rstimer; public double[] x; public double f; public double[] g; public bool needf; public bool needfg; public bool xupdated; public bool algpowerup; public bool lsstart; public bool lsend; public rcommstate rstate; public int repiterationscount; public int repnfev; public int repterminationtype; public int debugrestartscount; public linmin.linminstate lstate; public double fbase; public double fm2; public double fm1; public double fp1; public double fp2; public double betahs; public double betady; public double[] work0; public double[] work1; public mincgstate() { diagh = new double[0]; diaghl2 = new double[0]; vcorr = new double[0,0]; s = new double[0]; xk = new double[0]; dk = new double[0]; xn = new double[0]; dn = new double[0]; d = new double[0]; yk = new double[0]; x = new double[0]; g = new double[0]; rstate = new rcommstate(); lstate = new linmin.linminstate(); work0 = new double[0]; work1 = new double[0]; } }; public class mincgreport { public int iterationscount; public int nfev; public int terminationtype; }; public const int rscountdownlen = 10; public const double gtol = 0.3; /************************************************************************* NONLINEAR CONJUGATE GRADIENT METHOD DESCRIPTION: The subroutine minimizes function F(x) of N arguments by using one of the nonlinear conjugate gradient methods. These CG methods are globally convergent (even on non-convex functions) as long as grad(f) is Lipschitz continuous in a some neighborhood of the L = { x : f(x)<=f(x0) }. REQUIREMENTS: Algorithm will request following information during its operation: * function value F and its gradient G (simultaneously) at given point X USAGE: 1. User initializes algorithm state with MinCGCreate() call 2. User tunes solver parameters with MinCGSetCond(), MinCGSetStpMax() and other functions 3. User calls MinCGOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 4. User calls MinCGResults() to get solution 5. Optionally, user may call MinCGRestartFrom() to solve another problem with same N but another starting point and/or another function. MinCGRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 25.03.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgcreate(int n, double[] x, mincgstate state) { alglib.ap.assert(n>=1, "MinCGCreate: N too small!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinCGCreate: Length(X)0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinCGSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. L-BFGS needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ public static void mincgcreatef(int n, double[] x, double diffstep, mincgstate state) { alglib.ap.assert(n>=1, "MinCGCreateF: N too small!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinCGCreateF: Length(X)(double)(0), "MinCGCreateF: DiffStep is non-positive!"); mincginitinternal(n, diffstep, state); mincgrestartfrom(state, x); } /************************************************************************* This function sets stopping conditions for CG optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinCGSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetcond(mincgstate state, double epsg, double epsf, double epsx, int maxits) { alglib.ap.assert(math.isfinite(epsg), "MinCGSetCond: EpsG is not finite number!"); alglib.ap.assert((double)(epsg)>=(double)(0), "MinCGSetCond: negative EpsG!"); alglib.ap.assert(math.isfinite(epsf), "MinCGSetCond: EpsF is not finite number!"); alglib.ap.assert((double)(epsf)>=(double)(0), "MinCGSetCond: negative EpsF!"); alglib.ap.assert(math.isfinite(epsx), "MinCGSetCond: EpsX is not finite number!"); alglib.ap.assert((double)(epsx)>=(double)(0), "MinCGSetCond: negative EpsX!"); alglib.ap.assert(maxits>=0, "MinCGSetCond: negative MaxIts!"); if( (((double)(epsg)==(double)(0) && (double)(epsf)==(double)(0)) && (double)(epsx)==(double)(0)) && maxits==0 ) { epsx = 1.0E-6; } state.epsg = epsg; state.epsf = epsf; state.epsx = epsx; state.maxits = maxits; } /************************************************************************* This function sets scaling coefficients for CG optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of CG optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the CG too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinCGSetPrec...() functions. There is special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void mincgsetscale(mincgstate state, double[] s) { int i = 0; alglib.ap.assert(alglib.ap.len(s)>=state.n, "MinCGSetScale: Length(S)=-1 && cgtype<=1, "MinCGSetCGType: incorrect CGType!"); if( cgtype==-1 ) { cgtype = 1; } state.cgtype = cgtype; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetstpmax(mincgstate state, double stpmax) { alglib.ap.assert(math.isfinite(stpmax), "MinCGSetStpMax: StpMax is not finite!"); alglib.ap.assert((double)(stpmax)>=(double)(0), "MinCGSetStpMax: StpMax<0!"); state.stpmax = stpmax; } /************************************************************************* This function allows to suggest initial step length to the CG algorithm. Suggested step length is used as starting point for the line search. It can be useful when you have badly scaled problem, i.e. when ||grad|| (which is used as initial estimate for the first step) is many orders of magnitude different from the desired step. Line search may fail on such problems without good estimate of initial step length. Imagine, for example, problem with ||grad||=10^50 and desired step equal to 0.1 Line search function will use 10^50 as initial step, then it will decrease step length by 2 (up to 20 attempts) and will get 10^44, which is still too large. This function allows us to tell than line search should be started from some moderate step length, like 1.0, so algorithm will be able to detect desired step length in a several searches. Default behavior (when no step is suggested) is to use preconditioner, if it is available, to generate initial estimate of step length. This function influences only first iteration of algorithm. It should be called between MinCGCreate/MinCGRestartFrom() call and MinCGOptimize call. Suggested step is ignored if you have preconditioner. INPUT PARAMETERS: State - structure used to store algorithm state. Stp - initial estimate of the step length. Can be zero (no estimate). -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsuggeststep(mincgstate state, double stp) { alglib.ap.assert(math.isfinite(stp), "MinCGSuggestStep: Stp is infinite or NAN"); alglib.ap.assert((double)(stp)>=(double)(0), "MinCGSuggestStep: Stp<0"); state.suggestedstep = stp; } /************************************************************************* Modification of the preconditioner: preconditioning is turned off. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetprecdefault(mincgstate state) { state.prectype = 0; state.innerresetneeded = true; } /************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: D[i] should be positive. Exception will be thrown otherwise. NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetprecdiag(mincgstate state, double[] d) { int i = 0; alglib.ap.assert(alglib.ap.len(d)>=state.n, "MinCGSetPrecDiag: D is too short"); for(i=0; i<=state.n-1; i++) { alglib.ap.assert(math.isfinite(d[i]), "MinCGSetPrecDiag: D contains infinite or NAN elements"); alglib.ap.assert((double)(d[i])>(double)(0), "MinCGSetPrecDiag: D contains non-positive elements"); } mincgsetprecdiagfast(state, d); } /************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinCGSetScale() call (before or after MinCGSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void mincgsetprecscale(mincgstate state) { state.prectype = 3; state.innerresetneeded = true; } /************************************************************************* NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinCGCreate() for analytical gradient or MinCGCreateF() for numerical differentiation) you should choose appropriate variant of MinCGOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinCGOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinCGOptimize() and specific function used to create optimizer. | USER PASSED TO MinCGOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinCGCreateF() | work FAIL MinCGCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinCGOptimize() version. Attemps to use such combination (for example, to create optimizer with MinCGCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/ public static bool mincgiteration(mincgstate state) { bool result = new bool(); int n = 0; int i = 0; double betak = 0; double v = 0; double vv = 0; int i_ = 0; // // Reverse communication preparations // I know it looks ugly, but it works the same way // anywhere from C++ to Python. // // This code initializes locals by: // * random values determined during code // generation - on first subroutine call // * values from previous call - on subsequent calls // if( state.rstate.stage>=0 ) { n = state.rstate.ia[0]; i = state.rstate.ia[1]; betak = state.rstate.ra[0]; v = state.rstate.ra[1]; vv = state.rstate.ra[2]; } else { n = -983; i = -989; betak = -834; v = 900; vv = -287; } if( state.rstate.stage==0 ) { goto lbl_0; } if( state.rstate.stage==1 ) { goto lbl_1; } if( state.rstate.stage==2 ) { goto lbl_2; } if( state.rstate.stage==3 ) { goto lbl_3; } if( state.rstate.stage==4 ) { goto lbl_4; } if( state.rstate.stage==5 ) { goto lbl_5; } if( state.rstate.stage==6 ) { goto lbl_6; } if( state.rstate.stage==7 ) { goto lbl_7; } if( state.rstate.stage==8 ) { goto lbl_8; } if( state.rstate.stage==9 ) { goto lbl_9; } if( state.rstate.stage==10 ) { goto lbl_10; } if( state.rstate.stage==11 ) { goto lbl_11; } if( state.rstate.stage==12 ) { goto lbl_12; } if( state.rstate.stage==13 ) { goto lbl_13; } if( state.rstate.stage==14 ) { goto lbl_14; } if( state.rstate.stage==15 ) { goto lbl_15; } if( state.rstate.stage==16 ) { goto lbl_16; } // // Routine body // // // Prepare // n = state.n; state.repterminationtype = 0; state.repiterationscount = 0; state.repnfev = 0; state.debugrestartscount = 0; // // Preparations continue: // * set XK // * calculate F/G // * set DK to -G // * powerup algo (it may change preconditioner) // * apply preconditioner to DK // * report update of X // * check stopping conditions for G // for(i_=0; i_<=n-1;i_++) { state.xk[i_] = state.x[i_]; } state.terminationneeded = false; clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_17; } state.needfg = true; state.rstate.stage = 0; goto lbl_rcomm; lbl_0: state.needfg = false; goto lbl_18; lbl_17: state.needf = true; state.rstate.stage = 1; goto lbl_rcomm; lbl_1: state.fbase = state.f; i = 0; lbl_19: if( i>n-1 ) { goto lbl_21; } v = state.x[i]; state.x[i] = v-state.diffstep*state.s[i]; state.rstate.stage = 2; goto lbl_rcomm; lbl_2: state.fm2 = state.f; state.x[i] = v-0.5*state.diffstep*state.s[i]; state.rstate.stage = 3; goto lbl_rcomm; lbl_3: state.fm1 = state.f; state.x[i] = v+0.5*state.diffstep*state.s[i]; state.rstate.stage = 4; goto lbl_rcomm; lbl_4: state.fp1 = state.f; state.x[i] = v+state.diffstep*state.s[i]; state.rstate.stage = 5; goto lbl_rcomm; lbl_5: state.fp2 = state.f; state.x[i] = v; state.g[i] = (8*(state.fp1-state.fm1)-(state.fp2-state.fm2))/(6*state.diffstep*state.s[i]); i = i+1; goto lbl_19; lbl_21: state.f = state.fbase; state.needf = false; lbl_18: if( !state.drep ) { goto lbl_22; } // // Report algorithm powerup (if needed) // clearrequestfields(state); state.algpowerup = true; state.rstate.stage = 6; goto lbl_rcomm; lbl_6: state.algpowerup = false; lbl_22: optserv.trimprepare(state.f, ref state.trimthreshold); for(i_=0; i_<=n-1;i_++) { state.dk[i_] = -state.g[i_]; } preconditionedmultiply(state, ref state.dk, ref state.work0, ref state.work1); if( !state.xrep ) { goto lbl_24; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 7; goto lbl_rcomm; lbl_7: state.xupdated = false; lbl_24: if( state.terminationneeded ) { for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.xk[i_]; } state.repterminationtype = 8; result = false; return result; } v = 0; for(i=0; i<=n-1; i++) { v = v+math.sqr(state.g[i]*state.s[i]); } if( (double)(Math.Sqrt(v))<=(double)(state.epsg) ) { for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.xk[i_]; } state.repterminationtype = 4; result = false; return result; } state.repnfev = 1; state.k = 0; state.fold = state.f; // // Choose initial step. // Apply preconditioner, if we have something other than default. // if( state.prectype==2 || state.prectype==3 ) { // // because we use preconditioner, step length must be equal // to the norm of DK // v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.dk[i_]*state.dk[i_]; } state.lastgoodstep = Math.Sqrt(v); } else { // // No preconditioner is used, we try to use suggested step // if( (double)(state.suggestedstep)>(double)(0) ) { state.lastgoodstep = state.suggestedstep; } else { state.lastgoodstep = 1.0; } } // // Main cycle // state.rstimer = rscountdownlen; lbl_26: if( false ) { goto lbl_27; } // // * clear reset flag // * clear termination flag // * store G[k] for later calculation of Y[k] // * prepare starting point and direction and step length for line search // state.innerresetneeded = false; state.terminationneeded = false; for(i_=0; i_<=n-1;i_++) { state.yk[i_] = -state.g[i_]; } for(i_=0; i_<=n-1;i_++) { state.d[i_] = state.dk[i_]; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xk[i_]; } state.mcstage = 0; state.stp = 1.0; linmin.linminnormalized(ref state.d, ref state.stp, n); if( (double)(state.lastgoodstep)!=(double)(0) ) { state.stp = state.lastgoodstep; } state.curstpmax = state.stpmax; // // Report beginning of line search (if needed) // Terminate algorithm, if user request was detected // if( !state.drep ) { goto lbl_28; } clearrequestfields(state); state.lsstart = true; state.rstate.stage = 8; goto lbl_rcomm; lbl_8: state.lsstart = false; lbl_28: if( state.terminationneeded ) { for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.x[i_]; } state.repterminationtype = 8; result = false; return result; } // // Minimization along D // linmin.mcsrch(n, ref state.x, ref state.f, ref state.g, state.d, ref state.stp, state.curstpmax, gtol, ref state.mcinfo, ref state.nfev, ref state.work0, state.lstate, ref state.mcstage); lbl_30: if( state.mcstage==0 ) { goto lbl_31; } // // Calculate function/gradient using either // analytical gradient supplied by user // or finite difference approximation. // // "Trim" function in order to handle near-singularity points. // clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_32; } state.needfg = true; state.rstate.stage = 9; goto lbl_rcomm; lbl_9: state.needfg = false; goto lbl_33; lbl_32: state.needf = true; state.rstate.stage = 10; goto lbl_rcomm; lbl_10: state.fbase = state.f; i = 0; lbl_34: if( i>n-1 ) { goto lbl_36; } v = state.x[i]; state.x[i] = v-state.diffstep*state.s[i]; state.rstate.stage = 11; goto lbl_rcomm; lbl_11: state.fm2 = state.f; state.x[i] = v-0.5*state.diffstep*state.s[i]; state.rstate.stage = 12; goto lbl_rcomm; lbl_12: state.fm1 = state.f; state.x[i] = v+0.5*state.diffstep*state.s[i]; state.rstate.stage = 13; goto lbl_rcomm; lbl_13: state.fp1 = state.f; state.x[i] = v+state.diffstep*state.s[i]; state.rstate.stage = 14; goto lbl_rcomm; lbl_14: state.fp2 = state.f; state.x[i] = v; state.g[i] = (8*(state.fp1-state.fm1)-(state.fp2-state.fm2))/(6*state.diffstep*state.s[i]); i = i+1; goto lbl_34; lbl_36: state.f = state.fbase; state.needf = false; lbl_33: optserv.trimfunction(ref state.f, ref state.g, n, state.trimthreshold); // // Call MCSRCH again // linmin.mcsrch(n, ref state.x, ref state.f, ref state.g, state.d, ref state.stp, state.curstpmax, gtol, ref state.mcinfo, ref state.nfev, ref state.work0, state.lstate, ref state.mcstage); goto lbl_30; lbl_31: // // * report end of line search // * store current point to XN // * report iteration // * terminate algorithm if user request was detected // if( !state.drep ) { goto lbl_37; } // // Report end of line search (if needed) // clearrequestfields(state); state.lsend = true; state.rstate.stage = 15; goto lbl_rcomm; lbl_15: state.lsend = false; lbl_37: for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.x[i_]; } if( !state.xrep ) { goto lbl_39; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 16; goto lbl_rcomm; lbl_16: state.xupdated = false; lbl_39: if( state.terminationneeded ) { for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.x[i_]; } state.repterminationtype = 8; result = false; return result; } // // Line search is finished. // * calculate BetaK // * calculate DN // * update timers // * calculate step length: // * LastScaledStep is ALWAYS calculated because it is used in the stopping criteria // * LastGoodStep is updated only when MCINFO is equal to 1 (Wolfe conditions hold). // See below for more explanation. // if( state.mcinfo==1 && !state.innerresetneeded ) { // // Standard Wolfe conditions hold // Calculate Y[K] and D[K]'*Y[K] // for(i_=0; i_<=n-1;i_++) { state.yk[i_] = state.yk[i_] + state.g[i_]; } vv = 0.0; for(i_=0; i_<=n-1;i_++) { vv += state.yk[i_]*state.dk[i_]; } // // Calculate BetaK according to DY formula // v = preconditionedmultiply2(state, ref state.g, ref state.g, ref state.work0, ref state.work1); state.betady = v/vv; // // Calculate BetaK according to HS formula // v = preconditionedmultiply2(state, ref state.g, ref state.yk, ref state.work0, ref state.work1); state.betahs = v/vv; // // Choose BetaK // if( state.cgtype==0 ) { betak = state.betady; } if( state.cgtype==1 ) { betak = Math.Max(0, Math.Min(state.betady, state.betahs)); } } else { // // Something is wrong (may be function is too wild or too flat) // or we just have to restart algo. // // We'll set BetaK=0, which will restart CG algorithm. // We can stop later (during normal checks) if stopping conditions are met. // betak = 0; state.debugrestartscount = state.debugrestartscount+1; } if( state.repiterationscount>0 && state.repiterationscount%(3+n)==0 ) { // // clear Beta every N iterations // betak = 0; } if( state.mcinfo==1 || state.mcinfo==5 ) { state.rstimer = rscountdownlen; } else { state.rstimer = state.rstimer-1; } for(i_=0; i_<=n-1;i_++) { state.dn[i_] = -state.g[i_]; } preconditionedmultiply(state, ref state.dn, ref state.work0, ref state.work1); for(i_=0; i_<=n-1;i_++) { state.dn[i_] = state.dn[i_] + betak*state.dk[i_]; } state.lastscaledstep = 0.0; for(i=0; i<=n-1; i++) { state.lastscaledstep = state.lastscaledstep+math.sqr(state.d[i]/state.s[i]); } state.lastscaledstep = state.stp*Math.Sqrt(state.lastscaledstep); if( state.mcinfo==1 ) { // // Step is good (Wolfe conditions hold), update LastGoodStep. // // This check for MCINFO=1 is essential because sometimes in the // constrained optimization setting we may take very short steps // (like 1E-15) because we were very close to boundary of the // feasible area. Such short step does not mean that we've converged // to the solution - it was so short because we were close to the // boundary and there was a limit on step length. // // So having such short step is quite normal situation. However, we // should NOT start next iteration from step whose initial length is // estimated as 1E-15 because it may lead to the failure of the // linear minimizer (step is too short, function does not changes, // line search stagnates). // state.lastgoodstep = 0; for(i=0; i<=n-1; i++) { state.lastgoodstep = state.lastgoodstep+math.sqr(state.d[i]); } state.lastgoodstep = state.stp*Math.Sqrt(state.lastgoodstep); } // // Update information. // Check stopping conditions. // state.repnfev = state.repnfev+state.nfev; state.repiterationscount = state.repiterationscount+1; if( state.repiterationscount>=state.maxits && state.maxits>0 ) { // // Too many iterations // state.repterminationtype = 5; result = false; return result; } v = 0; for(i=0; i<=n-1; i++) { v = v+math.sqr(state.g[i]*state.s[i]); } if( (double)(Math.Sqrt(v))<=(double)(state.epsg) ) { // // Gradient is small enough // state.repterminationtype = 4; result = false; return result; } if( !state.innerresetneeded ) { // // These conditions are checked only when no inner reset was requested by user // if( (double)(state.fold-state.f)<=(double)(state.epsf*Math.Max(Math.Abs(state.fold), Math.Max(Math.Abs(state.f), 1.0))) ) { // // F(k+1)-F(k) is small enough // state.repterminationtype = 1; result = false; return result; } if( (double)(state.lastscaledstep)<=(double)(state.epsx) ) { // // X(k+1)-X(k) is small enough // state.repterminationtype = 2; result = false; return result; } } if( state.rstimer<=0 ) { // // Too many subsequent restarts // state.repterminationtype = 7; result = false; return result; } // // Shift Xk/Dk, update other information // for(i_=0; i_<=n-1;i_++) { state.xk[i_] = state.xn[i_]; } for(i_=0; i_<=n-1;i_++) { state.dk[i_] = state.dn[i_]; } state.fold = state.f; state.k = state.k+1; goto lbl_26; lbl_27: result = false; return result; // // Saving state // lbl_rcomm: result = true; state.rstate.ia[0] = n; state.rstate.ia[1] = i; state.rstate.ra[0] = betak; state.rstate.ra[1] = v; state.rstate.ra[2] = vv; return result; } /************************************************************************* Conjugate gradient results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible, we return best X found so far * 8 terminated by user * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/ public static void mincgresults(mincgstate state, ref double[] x, mincgreport rep) { x = new double[0]; mincgresultsbuf(state, ref x, rep); } /************************************************************************* Conjugate gradient results Buffered implementation of MinCGResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/ public static void mincgresultsbuf(mincgstate state, ref double[] x, mincgreport rep) { int i_ = 0; if( alglib.ap.len(x)=state.n, "MinCGRestartFrom: Length(X)0 // for(i=0; i<=n-1; i++) { x[i] = x[i]/(state.diagh[i]+state.diaghl2[i]); } // // if VCnt>0 // if( vcnt>0 ) { for(i=0; i<=vcnt-1; i++) { v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.vcorr[i,i_]*x[i_]; } work0[i] = v; } for(i=0; i<=n-1; i++) { work1[i] = 0; } for(i=0; i<=vcnt-1; i++) { v = work0[i]; for(i_=0; i_<=n-1;i_++) { state.work1[i_] = state.work1[i_] + v*state.vcorr[i,i_]; } } for(i=0; i<=n-1; i++) { x[i] = x[i]-state.work1[i]/(state.diagh[i]+state.diaghl2[i]); } } } /************************************************************************* This function calculates preconditioned product x'*H^(-1)*y. Work0[] and Work1[] are used as temporaries (size must be at least N; this function doesn't allocate arrays). -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ private static double preconditionedmultiply2(mincgstate state, ref double[] x, ref double[] y, ref double[] work0, ref double[] work1) { double result = 0; int i = 0; int n = 0; int vcnt = 0; double v0 = 0; double v1 = 0; int i_ = 0; n = state.n; vcnt = state.vcnt; // // no preconditioning // if( state.prectype==0 ) { v0 = 0.0; for(i_=0; i_<=n-1;i_++) { v0 += x[i_]*y[i_]; } result = v0; return result; } if( state.prectype==3 ) { result = 0; for(i=0; i<=n-1; i++) { result = result+x[i]*state.s[i]*state.s[i]*y[i]; } return result; } alglib.ap.assert(state.prectype==2, "MinCG: internal error (unexpected PrecType)"); // // low rank preconditioning // result = 0.0; for(i=0; i<=n-1; i++) { result = result+x[i]*y[i]/(state.diagh[i]+state.diaghl2[i]); } if( vcnt>0 ) { for(i=0; i<=n-1; i++) { work0[i] = x[i]/(state.diagh[i]+state.diaghl2[i]); work1[i] = y[i]/(state.diagh[i]+state.diaghl2[i]); } for(i=0; i<=vcnt-1; i++) { v0 = 0.0; for(i_=0; i_<=n-1;i_++) { v0 += work0[i_]*state.vcorr[i,i_]; } v1 = 0.0; for(i_=0; i_<=n-1;i_++) { v1 += work1[i_]*state.vcorr[i,i_]; } result = result-v0*v1; } } return result; } /************************************************************************* Internal initialization subroutine -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ private static void mincginitinternal(int n, double diffstep, mincgstate state) { int i = 0; state.n = n; state.diffstep = diffstep; mincgsetcond(state, 0, 0, 0, 0); mincgsetxrep(state, false); mincgsetdrep(state, false); mincgsetstpmax(state, 0); mincgsetcgtype(state, -1); mincgsetprecdefault(state); state.xk = new double[n]; state.dk = new double[n]; state.xn = new double[n]; state.dn = new double[n]; state.x = new double[n]; state.d = new double[n]; state.g = new double[n]; state.work0 = new double[n]; state.work1 = new double[n]; state.yk = new double[n]; state.s = new double[n]; for(i=0; i<=n-1; i++) { state.s[i] = 1.0; } } } public class minbleic { /************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinBLEIC subpackage to work with this object *************************************************************************/ public class minbleicstate { public int nmain; public int nslack; public double innerepsg; public double innerepsf; public double innerepsx; public double outerepsx; public double outerepsi; public int maxits; public bool xrep; public double stpmax; public double diffstep; public int prectype; public double[] diaghoriginal; public double[] diagh; public double[] x; public double f; public double[] g; public bool needf; public bool needfg; public bool xupdated; public rcommstate rstate; public int repinneriterationscount; public int repouteriterationscount; public int repnfev; public int repterminationtype; public double repdebugeqerr; public double repdebugfs; public double repdebugff; public double repdebugdx; public int repdebugfeasqpits; public int repdebugfeasgpaits; public double[] xcur; public double[] xprev; public double[] xstart; public int itsleft; public double[] xend; public double[] lastg; public double trimthreshold; public double[,] ceoriginal; public double[,] ceeffective; public double[,] cecurrent; public int[] ct; public int cecnt; public int cedim; public double[] xe; public bool[] hasbndl; public bool[] hasbndu; public double[] bndloriginal; public double[] bnduoriginal; public double[] bndleffective; public double[] bndueffective; public bool[] activeconstraints; public double[] constrainedvalues; public double[] transforms; public double[] seffective; public double[] soriginal; public double[] w; public double[] tmp0; public double[] tmp1; public double[] tmp2; public double[] r; public double[,] lmmatrix; public double v0; public double v1; public double v2; public double t; public double errfeas; public double gnorm; public double mpgnorm; public double mba; public int variabletofreeze; public double valuetofreeze; public double fbase; public double fm2; public double fm1; public double fp1; public double fp2; public double xm1; public double xp1; public mincg.mincgstate cgstate; public mincg.mincgreport cgrep; public int optdim; public minbleicstate() { diaghoriginal = new double[0]; diagh = new double[0]; x = new double[0]; g = new double[0]; rstate = new rcommstate(); xcur = new double[0]; xprev = new double[0]; xstart = new double[0]; xend = new double[0]; lastg = new double[0]; ceoriginal = new double[0,0]; ceeffective = new double[0,0]; cecurrent = new double[0,0]; ct = new int[0]; xe = new double[0]; hasbndl = new bool[0]; hasbndu = new bool[0]; bndloriginal = new double[0]; bnduoriginal = new double[0]; bndleffective = new double[0]; bndueffective = new double[0]; activeconstraints = new bool[0]; constrainedvalues = new double[0]; transforms = new double[0]; seffective = new double[0]; soriginal = new double[0]; w = new double[0]; tmp0 = new double[0]; tmp1 = new double[0]; tmp2 = new double[0]; r = new double[0]; lmmatrix = new double[0,0]; cgstate = new mincg.mincgstate(); cgrep = new mincg.mincgreport(); } }; /************************************************************************* This structure stores optimization report: * InnerIterationsCount number of inner iterations * OuterIterationsCount number of outer iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -10 unsupported combination of algorithm settings: 1) StpMax is set to non-zero value, AND 2) non-default preconditioner is used. You can't use both features at the same moment, so you have to choose one of them (and to turn off another one). -3 inconsistent constraints. Feasible point is either nonexistent or too hard to find. Try to restart optimizer with better initial approximation 4 conditions on constraints are fulfilled with error less than or equal to EpsC 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. ADDITIONAL FIELDS There are additional fields which can be used for debugging: * DebugEqErr error in the equality constraints (2-norm) * DebugFS f, calculated at projection of initial point to the feasible set * DebugFF f, calculated at the final point * DebugDX |X_start-X_final| *************************************************************************/ public class minbleicreport { public int inneriterationscount; public int outeriterationscount; public int nfev; public int terminationtype; public double debugeqerr; public double debugfs; public double debugff; public double debugdx; public int debugfeasqpits; public int debugfeasgpaits; }; public const double svdtol = 100; public const double maxouterits = 20; public const double maxnonmonotoniclen = 1.0E5; /************************************************************************* BOUND CONSTRAINED OPTIMIZATION WITH ADDITIONAL LINEAR EQUALITY AND INEQUALITY CONSTRAINTS DESCRIPTION: The subroutine minimizes function F(x) of N arguments subject to any combination of: * bound constraints * linear inequality constraints * linear equality constraints REQUIREMENTS: * user must provide function value and gradient * starting point X0 must be feasible or not too far away from the feasible set * grad(f) must be Lipschitz continuous on a level set: L = { x : f(x)<=f(x0) } * function must be defined everywhere on the feasible set F USAGE: Constrained optimization if far more complex than the unconstrained one. Here we give very brief outline of the BLEIC optimizer. We strongly recommend you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide on optimization, which is available at http://www.alglib.net/optimization/ 1. User initializes algorithm state with MinBLEICCreate() call 2. USer adds boundary and/or linear constraints by calling MinBLEICSetBC() and MinBLEICSetLC() functions. 3. User sets stopping conditions for underlying unconstrained solver with MinBLEICSetInnerCond() call. This function controls accuracy of underlying optimization algorithm. 4. User sets stopping conditions for outer iteration by calling MinBLEICSetOuterCond() function. This function controls handling of boundary and inequality constraints. 5. Additionally, user may set limit on number of internal iterations by MinBLEICSetMaxIts() call. This function allows to prevent algorithm from looping forever. 6. User calls MinBLEICOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 7. User calls MinBLEICResults() to get solution 8. Optionally user may call MinBLEICRestartFrom() to solve another problem with same N but another starting point. MinBLEICRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size ofX X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleiccreate(int n, double[] x, minbleicstate state) { double[,] c = new double[0,0]; int[] ct = new int[0]; alglib.ap.assert(n>=1, "MinBLEICCreate: N<1"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinBLEICCreate: Length(X)0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinBLEICSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. CG needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ public static void minbleiccreatef(int n, double[] x, double diffstep, minbleicstate state) { double[,] c = new double[0,0]; int[] ct = new int[0]; alglib.ap.assert(n>=1, "MinBLEICCreateF: N<1"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinBLEICCreateF: Length(X)(double)(0), "MinBLEICCreateF: DiffStep is non-positive!"); minbleicinitinternal(n, x, diffstep, state); } /************************************************************************* This function sets boundary constraints for BLEIC optimizer. Boundary constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinBLEICRestartFrom(). INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF. BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF. NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints, even when numerical differentiation is used (algorithm adjusts nodes according to boundary constraints) -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetbc(minbleicstate state, double[] bndl, double[] bndu) { int i = 0; int n = 0; n = state.nmain; alglib.ap.assert(alglib.ap.len(bndl)>=n, "MinBLEICSetBC: Length(BndL)=n, "MinBLEICSetBC: Length(BndU)0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT NOTE 1: linear (non-bound) constraints are satisfied only approximately: * there always exists some minor violation (about Epsilon in magnitude) due to rounding errors * numerical differentiation, if used, may lead to function evaluations outside of the feasible area, because algorithm does NOT change numerical differentiation formula according to linear constraints. If you want constraints to be satisfied exactly, try to reformulate your problem in such manner that all constraints will become boundary ones (this kind of constraints is always satisfied exactly, both in the final solution and in all intermediate points). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetlc(minbleicstate state, double[,] c, int[] ct, int k) { int nmain = 0; int i = 0; int i_ = 0; nmain = state.nmain; // // First, check for errors in the inputs // alglib.ap.assert(k>=0, "MinBLEICSetLC: K<0"); alglib.ap.assert(alglib.ap.cols(c)>=nmain+1 || k==0, "MinBLEICSetLC: Cols(C)=k, "MinBLEICSetLC: Rows(C)=k, "MinBLEICSetLC: Length(CT)=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinBLEICSetScale() Passing EpsG=0, EpsF=0 and EpsX=0 (simultaneously) will lead to automatic stopping criterion selection. These conditions are used to terminate inner iterations. However, you need to tune termination conditions for outer iterations too. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetinnercond(minbleicstate state, double epsg, double epsf, double epsx) { alglib.ap.assert(math.isfinite(epsg), "MinBLEICSetInnerCond: EpsG is not finite number"); alglib.ap.assert((double)(epsg)>=(double)(0), "MinBLEICSetInnerCond: negative EpsG"); alglib.ap.assert(math.isfinite(epsf), "MinBLEICSetInnerCond: EpsF is not finite number"); alglib.ap.assert((double)(epsf)>=(double)(0), "MinBLEICSetInnerCond: negative EpsF"); alglib.ap.assert(math.isfinite(epsx), "MinBLEICSetInnerCond: EpsX is not finite number"); alglib.ap.assert((double)(epsx)>=(double)(0), "MinBLEICSetInnerCond: negative EpsX"); state.innerepsg = epsg; state.innerepsf = epsf; state.innerepsx = epsx; } /************************************************************************* This function sets stopping conditions for outer iteration of BLEIC algo. These conditions control accuracy of constraint handling and amount of infeasibility allowed in the solution. INPUT PARAMETERS: State - structure which stores algorithm state EpsX - >0, stopping condition on outer iteration step length EpsI - >0, stopping condition on infeasibility Both EpsX and EpsI must be non-zero. MEANING OF EpsX EpsX is a stopping condition for outer iterations. Algorithm will stop when solution of the current modified subproblem will be within EpsX (using 2-norm) of the previous solution. MEANING OF EpsI EpsI controls feasibility properties - algorithm won't stop until all inequality constraints will be satisfied with error (distance from current point to the feasible area) at most EpsI. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetoutercond(minbleicstate state, double epsx, double epsi) { alglib.ap.assert(math.isfinite(epsx), "MinBLEICSetOuterCond: EpsX is not finite number"); alglib.ap.assert((double)(epsx)>(double)(0), "MinBLEICSetOuterCond: non-positive EpsX"); alglib.ap.assert(math.isfinite(epsi), "MinBLEICSetOuterCond: EpsI is not finite number"); alglib.ap.assert((double)(epsi)>(double)(0), "MinBLEICSetOuterCond: non-positive EpsI"); state.outerepsx = epsx; state.outerepsi = epsi; } /************************************************************************* This function sets scaling coefficients for BLEIC optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the BLEIC too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinBLEICSetPrec...() functions. There is a special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetscale(minbleicstate state, double[] s) { int i = 0; alglib.ap.assert(alglib.ap.len(s)>=state.nmain, "MinBLEICSetScale: Length(S)=state.nmain, "MinBLEICSetPrecDiag: D is too short"); for(i=0; i<=state.nmain-1; i++) { alglib.ap.assert(math.isfinite(d[i]), "MinBLEICSetPrecDiag: D contains infinite or NAN elements"); alglib.ap.assert((double)(d[i])>(double)(0), "MinBLEICSetPrecDiag: D contains non-positive elements"); } apserv.rvectorsetlengthatleast(ref state.diaghoriginal, state.nmain); state.prectype = 2; for(i=0; i<=state.nmain-1; i++) { state.diaghoriginal[i] = d[i]; } } /************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinBLEICSetScale() call (before or after MinBLEICSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetprecscale(minbleicstate state) { state.prectype = 3; } /************************************************************************* This function allows to stop algorithm after specified number of inner iterations. INPUT PARAMETERS: State - structure which stores algorithm state MaxIts - maximum number of inner iterations. If MaxIts=0, the number of iterations is unlimited. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetmaxits(minbleicstate state, int maxits) { alglib.ap.assert(maxits>=0, "MinBLEICSetMaxIts: negative MaxIts!"); state.maxits = maxits; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinBLEICOptimize(). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetxrep(minbleicstate state, bool needxrep) { state.xrep = needxrep; } /************************************************************************* This function sets maximum step length IMPORTANT: this feature is hard to combine with preconditioning. You can't set upper limit on step length, when you solve optimization problem with linear (non-boundary) constraints AND preconditioner turned on. When non-boundary constraints are present, you have to either a) use preconditioner, or b) use upper limit on step length. YOU CAN'T USE BOTH! In this case algorithm will terminate with appropriate error code. INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which lead to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetstpmax(minbleicstate state, double stpmax) { alglib.ap.assert(math.isfinite(stpmax), "MinBLEICSetStpMax: StpMax is not finite!"); alglib.ap.assert((double)(stpmax)>=(double)(0), "MinBLEICSetStpMax: StpMax<0!"); state.stpmax = stpmax; } /************************************************************************* NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinBLEICCreate() for analytical gradient or MinBLEICCreateF() for numerical differentiation) you should choose appropriate variant of MinBLEICOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinBLEICOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinBLEICOptimize() and specific function used to create optimizer. | USER PASSED TO MinBLEICOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinBLEICCreateF() | work FAIL MinBLEICCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinBLEICOptimize() version. Attemps to use such combination (for example, to create optimizer with MinBLEICCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static bool minbleiciteration(minbleicstate state) { bool result = new bool(); int nmain = 0; int nslack = 0; int m = 0; int i = 0; int j = 0; double v = 0; double vv = 0; bool b = new bool(); int i_ = 0; // // Reverse communication preparations // I know it looks ugly, but it works the same way // anywhere from C++ to Python. // // This code initializes locals by: // * random values determined during code // generation - on first subroutine call // * values from previous call - on subsequent calls // if( state.rstate.stage>=0 ) { nmain = state.rstate.ia[0]; nslack = state.rstate.ia[1]; m = state.rstate.ia[2]; i = state.rstate.ia[3]; j = state.rstate.ia[4]; b = state.rstate.ba[0]; v = state.rstate.ra[0]; vv = state.rstate.ra[1]; } else { nmain = -983; nslack = -989; m = -834; i = 900; j = -287; b = false; v = 214; vv = -338; } if( state.rstate.stage==0 ) { goto lbl_0; } if( state.rstate.stage==1 ) { goto lbl_1; } if( state.rstate.stage==2 ) { goto lbl_2; } if( state.rstate.stage==3 ) { goto lbl_3; } if( state.rstate.stage==4 ) { goto lbl_4; } if( state.rstate.stage==5 ) { goto lbl_5; } if( state.rstate.stage==6 ) { goto lbl_6; } if( state.rstate.stage==7 ) { goto lbl_7; } if( state.rstate.stage==8 ) { goto lbl_8; } if( state.rstate.stage==9 ) { goto lbl_9; } if( state.rstate.stage==10 ) { goto lbl_10; } if( state.rstate.stage==11 ) { goto lbl_11; } if( state.rstate.stage==12 ) { goto lbl_12; } // // Routine body // // // Prepare: // * calculate number of slack variables // * initialize locals // * initialize debug fields // * make quick check // nmain = state.nmain; nslack = 0; for(i=0; i<=state.cecnt-1; i++) { if( state.ct[i]!=0 ) { nslack = nslack+1; } } state.nslack = nslack; state.repterminationtype = 0; state.repinneriterationscount = 0; state.repouteriterationscount = 0; state.repnfev = 0; state.repdebugeqerr = 0.0; state.repdebugfs = Double.NaN; state.repdebugff = Double.NaN; state.repdebugdx = Double.NaN; if( (double)(state.stpmax)!=(double)(0) && state.prectype!=0 ) { state.repterminationtype = -10; result = false; return result; } // // allocate // apserv.rvectorsetlengthatleast(ref state.r, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.diagh, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.tmp0, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.tmp1, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.tmp2, nmain+nslack); apserv.rmatrixsetlengthatleast(ref state.cecurrent, state.cecnt, nmain+nslack+1); apserv.bvectorsetlengthatleast(ref state.activeconstraints, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.constrainedvalues, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.lastg, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.xe, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.xcur, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.xprev, nmain+nslack); apserv.rvectorsetlengthatleast(ref state.xend, nmain); // // Create/restart optimizer. // // State.OptDim is used to determine current state of optimizer. // if( state.optdim!=nmain+nslack ) { for(i=0; i<=nmain+nslack-1; i++) { state.tmp1[i] = 0.0; } mincg.mincgcreate(nmain+nslack, state.tmp1, state.cgstate); state.optdim = nmain+nslack; } // // Prepare transformation. // // MinBLEIC's handling of preconditioner matrix is somewhat unusual - // instead of incorporating it into algorithm and making implicit // scaling (as most optimizers do) BLEIC optimizer uses explicit // scaling - it solves problem in the scaled parameters space S, // making transition between scaled (S) and unscaled (X) variables // every time we ask for function value. // // Following fields are calculated here: // * TransformS X[i] = TransformS[i]*S[i], array[NMain] // * SEffective "effective" scale of the variables after // transformation, array[NMain+NSlack] // apserv.rvectorsetlengthatleast(ref state.transforms, nmain); for(i=0; i<=nmain-1; i++) { if( state.prectype==2 ) { state.transforms[i] = 1/Math.Sqrt(state.diaghoriginal[i]); continue; } if( state.prectype==3 ) { state.transforms[i] = state.soriginal[i]; continue; } state.transforms[i] = 1; } apserv.rvectorsetlengthatleast(ref state.seffective, nmain+nslack); for(i=0; i<=nmain-1; i++) { state.seffective[i] = state.soriginal[i]/state.transforms[i]; } for(i=0; i<=nslack-1; i++) { state.seffective[nmain+i] = 1; } mincg.mincgsetscale(state.cgstate, state.seffective); // // Pre-process constraints // * check consistency of bound constraints // * add slack vars, convert problem to the bound/equality // constrained one // // We calculate here: // * BndLEffective - lower bounds after transformation of variables (see above) // * BndUEffective - upper bounds after transformation of variables (see above) // * CEEffective - matrix of equality constraints for transformed variables // for(i=0; i<=nmain-1; i++) { if( state.hasbndl[i] ) { state.bndleffective[i] = state.bndloriginal[i]/state.transforms[i]; } if( state.hasbndu[i] ) { state.bndueffective[i] = state.bnduoriginal[i]/state.transforms[i]; } } for(i=0; i<=nmain-1; i++) { if( state.hasbndl[i] && state.hasbndu[i] ) { if( (double)(state.bndleffective[i])>(double)(state.bndueffective[i]) ) { state.repterminationtype = -3; result = false; return result; } } } apserv.rmatrixsetlengthatleast(ref state.ceeffective, state.cecnt, nmain+nslack+1); m = 0; for(i=0; i<=state.cecnt-1; i++) { // // NOTE: when we add slack variable, we use V = max(abs(CE[i,...])) as // coefficient before it in order to make linear equations better // conditioned. // v = 0; for(j=0; j<=nmain-1; j++) { state.ceeffective[i,j] = state.ceoriginal[i,j]*state.transforms[j]; v = Math.Max(v, Math.Abs(state.ceeffective[i,j])); } if( (double)(v)==(double)(0) ) { v = 1; } for(j=0; j<=nslack-1; j++) { state.ceeffective[i,nmain+j] = 0.0; } state.ceeffective[i,nmain+nslack] = state.ceoriginal[i,nmain]; if( state.ct[i]<0 ) { state.ceeffective[i,nmain+m] = v; m = m+1; } if( state.ct[i]>0 ) { state.ceeffective[i,nmain+m] = -v; m = m+1; } } // // Find feasible point. // // 0. Convert from unscaled values (as stored in XStart) to scaled // ones // 1. calculate values of slack variables such that starting // point satisfies inequality constraints (after conversion to // equality ones) as much as possible. // 2. use PrepareConstraintMatrix() function, which forces X // to be strictly feasible. // for(i=0; i<=nmain-1; i++) { state.tmp0[i] = state.xstart[i]/state.transforms[i]; } m = 0; for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain-1;i_++) { v += state.ceeffective[i,i_]*state.tmp0[i_]; } if( state.ct[i]<0 ) { state.tmp0[nmain+m] = state.ceeffective[i,nmain+nslack]-v; m = m+1; } if( state.ct[i]>0 ) { state.tmp0[nmain+m] = v-state.ceeffective[i,nmain+nslack]; m = m+1; } } for(i=0; i<=nmain+nslack-1; i++) { state.tmp1[i] = 0; } for(i=0; i<=nmain+nslack-1; i++) { state.activeconstraints[i] = false; } b = optserv.findfeasiblepoint(ref state.tmp0, state.bndleffective, state.hasbndl, state.bndueffective, state.hasbndu, nmain, nslack, state.ceeffective, state.cecnt, state.outerepsi, ref state.repdebugfeasqpits, ref state.repdebugfeasgpaits); for(i_=0; i_<=nmain+nslack-1;i_++) { state.xcur[i_] = state.tmp0[i_]; } state.repdebugeqerr = 0.0; for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += state.ceeffective[i,i_]*state.xcur[i_]; } state.repdebugeqerr = state.repdebugeqerr+math.sqr(v-state.ceeffective[i,nmain+nslack]); } state.repdebugeqerr = Math.Sqrt(state.repdebugeqerr); if( !b ) { state.repterminationtype = -3; result = false; return result; } // // Initialize RepDebugFS with function value at initial point // unscalepoint(state, state.xcur, ref state.x); clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_13; } state.needfg = true; state.rstate.stage = 0; goto lbl_rcomm; lbl_0: state.needfg = false; goto lbl_14; lbl_13: state.needf = true; state.rstate.stage = 1; goto lbl_rcomm; lbl_1: state.needf = false; lbl_14: optserv.trimprepare(state.f, ref state.trimthreshold); state.repnfev = state.repnfev+1; state.repdebugfs = state.f; // // Outer cycle // state.itsleft = state.maxits; for(i_=0; i_<=nmain+nslack-1;i_++) { state.xprev[i_] = state.xcur[i_]; } lbl_15: if( false ) { goto lbl_16; } alglib.ap.assert(state.prectype==0 || (double)(state.stpmax)==(double)(0), "MinBLEIC: internal error (-10)"); // // Inner cycle: CG with projections and penalty functions // for(i_=0; i_<=nmain+nslack-1;i_++) { state.tmp0[i_] = state.xcur[i_]; } for(i=0; i<=nmain+nslack-1; i++) { state.tmp1[i] = 0; state.activeconstraints[i] = false; } if( !prepareconstraintmatrix(state, state.tmp0, state.tmp1, ref state.xcur, ref state.tmp2) ) { state.repterminationtype = -3; result = false; return result; } for(i=0; i<=nmain+nslack-1; i++) { state.activeconstraints[i] = false; } rebuildcexe(state); mincg.mincgrestartfrom(state.cgstate, state.xcur); mincg.mincgsetcond(state.cgstate, state.innerepsg, state.innerepsf, state.innerepsx, state.itsleft); mincg.mincgsetxrep(state.cgstate, state.xrep); mincg.mincgsetdrep(state.cgstate, true); mincg.mincgsetstpmax(state.cgstate, state.stpmax); lbl_17: if( !mincg.mincgiteration(state.cgstate) ) { goto lbl_18; } // // process different requests/reports of inner optimizer // if( state.cgstate.algpowerup ) { for(i=0; i<=nmain+nslack-1; i++) { state.activeconstraints[i] = false; } do { rebuildcexe(state); for(i_=0; i_<=nmain+nslack-1;i_++) { state.tmp1[i_] = state.cgstate.g[i_]; } makegradientprojection(state, ref state.tmp1); b = false; for(i=0; i<=nmain-1; i++) { if( !state.activeconstraints[i] ) { if( state.hasbndl[i] ) { if( (double)(state.cgstate.x[i])==(double)(state.bndleffective[i]) && (double)(state.tmp1[i])>=(double)(0) ) { state.activeconstraints[i] = true; state.constrainedvalues[i] = state.bndleffective[i]; b = true; } } if( state.hasbndu[i] ) { if( (double)(state.cgstate.x[i])==(double)(state.bndueffective[i]) && (double)(state.tmp1[i])<=(double)(0) ) { state.activeconstraints[i] = true; state.constrainedvalues[i] = state.bndueffective[i]; b = true; } } } } for(i=0; i<=nslack-1; i++) { if( !state.activeconstraints[nmain+i] ) { if( (double)(state.cgstate.x[nmain+i])==(double)(0) && (double)(state.tmp1[nmain+i])>=(double)(0) ) { state.activeconstraints[nmain+i] = true; state.constrainedvalues[nmain+i] = 0; b = true; } } } } while( b ); for(i_=0; i_<=nmain+nslack-1;i_++) { state.cgstate.g[i_] = state.tmp1[i_]; } goto lbl_17; } if( state.cgstate.lsstart ) { // // Beginning of the line search: set upper limit on step size // to prevent algo from leaving feasible area. // if( (double)(state.cgstate.curstpmax)==(double)(0) ) { state.cgstate.curstpmax = 1.0E50; } state.variabletofreeze = -1; for(i=0; i<=nmain-1; i++) { if( state.hasbndl[i] && (double)(state.cgstate.d[i])<(double)(0) ) { v = state.cgstate.curstpmax; vv = state.cgstate.x[i]-state.bndleffective[i]; if( (double)(vv)<(double)(0) ) { vv = 0; } state.cgstate.curstpmax = apserv.safeminposrv(vv, -state.cgstate.d[i], state.cgstate.curstpmax); if( (double)(state.cgstate.curstpmax)<(double)(v) ) { state.variabletofreeze = i; state.valuetofreeze = state.bndleffective[i]; } } if( state.hasbndu[i] && (double)(state.cgstate.d[i])>(double)(0) ) { v = state.cgstate.curstpmax; vv = state.bndueffective[i]-state.cgstate.x[i]; if( (double)(vv)<(double)(0) ) { vv = 0; } state.cgstate.curstpmax = apserv.safeminposrv(vv, state.cgstate.d[i], state.cgstate.curstpmax); if( (double)(state.cgstate.curstpmax)<(double)(v) ) { state.variabletofreeze = i; state.valuetofreeze = state.bndueffective[i]; } } } for(i=0; i<=nslack-1; i++) { if( (double)(state.cgstate.d[nmain+i])<(double)(0) ) { v = state.cgstate.curstpmax; vv = state.cgstate.x[nmain+i]; if( (double)(vv)<(double)(0) ) { vv = 0; } state.cgstate.curstpmax = apserv.safeminposrv(vv, -state.cgstate.d[nmain+i], state.cgstate.curstpmax); if( (double)(state.cgstate.curstpmax)<(double)(v) ) { state.variabletofreeze = nmain+i; state.valuetofreeze = 0; } } } if( (double)(state.cgstate.curstpmax)==(double)(0) ) { state.activeconstraints[state.variabletofreeze] = true; state.constrainedvalues[state.variabletofreeze] = state.valuetofreeze; state.cgstate.x[state.variabletofreeze] = state.valuetofreeze; state.cgstate.terminationneeded = true; } goto lbl_17; } if( state.cgstate.lsend ) { // // Line search just finished. // Maybe we should activate some constraints? // if( state.variabletofreeze>=0 ) { // // We have candidate constraint for activation, which can be // activated in two cases: // // 1. monotonic step with Stp=StpMax is performed (MCINFO=5), // we just have to post-process this step to ensure that // constraint is activated // // 2. MCINFO<>5 and MCINFO<>1, and BOTH of the following holds: // 2.a) ScaledNorm(State.CGState.G)>InnerEpsG (strict inequality to handle EpsG=0) // 2.b) ScaledNorm(StpMax*State.CGState.D)<=MaxNonMonotonicLen*MachineEpsilon; // // What do these conditions mean? // a) Having MCINFO which is not 5 or 1 means that it is // possible that our function is actually increasing in // the direction of D instead of decreasing. // b) clause (2.a) means that gradient value is large enough // to consider it non-zero (EpsG=0 means that any non-zero // gradient value will be large enough). // c) clause (2.b) means that step length is very small and // function non-monotonicity can be just a consequence of // numerical noise. // // Given all of the listed above, it may be worth trying to // perform step which increases function value. // // NOTE: some non-monotonic algorithms can be caught in an eternal // loop. However, this algorithm is guaranteed from having // such eternal loop because non-monotonic step always activates // at least one constraint, and in order to have eternal loop // we need to deactivate this constraint, which can be done // only at the beginning of the outer iteration. And we have // upper limit on the number of outer iterations, so we can // be sure that algorithm will stop eventually. // if( state.cgstate.mcinfo!=5 && state.cgstate.mcinfo!=1 ) { v = 0; vv = 0; for(i=0; i<=nmain+nslack-1; i++) { v = v+math.sqr(state.cgstate.g[i]*state.seffective[i]); vv = vv+math.sqr(state.cgstate.curstpmax*state.cgstate.d[i]/state.seffective[i]); } v = Math.Sqrt(v); vv = Math.Sqrt(vv); b = (double)(v)>(double)(state.innerepsg) && (double)(vv)<=(double)(maxnonmonotoniclen*math.machineepsilon); } else { b = state.cgstate.mcinfo==5; } if( b ) { state.activeconstraints[state.variabletofreeze] = true; state.constrainedvalues[state.variabletofreeze] = state.valuetofreeze; } } else { b = false; } // // Additional activation of constraints // b = b || additionalcheckforconstraints(state, state.cgstate.x); // // If at least one constraint was activated we have to rebuild constraint matrices // if( b ) { for(i_=0; i_<=nmain+nslack-1;i_++) { state.tmp0[i_] = state.cgstate.x[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { state.tmp1[i_] = state.lastg[i_]; } if( !prepareconstraintmatrix(state, state.tmp0, state.tmp1, ref state.cgstate.x, ref state.cgstate.g) ) { state.repterminationtype = -3; result = false; return result; } state.cgstate.innerresetneeded = true; } goto lbl_17; } if( !state.cgstate.needfg ) { goto lbl_19; } for(i_=0; i_<=nmain+nslack-1;i_++) { state.tmp1[i_] = state.cgstate.x[i_]; } projectpointandunscale(state, ref state.tmp1, ref state.x, ref state.r, ref vv); clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_21; } state.needfg = true; state.rstate.stage = 2; goto lbl_rcomm; lbl_2: state.needfg = false; goto lbl_22; lbl_21: state.needf = true; state.rstate.stage = 3; goto lbl_rcomm; lbl_3: state.fbase = state.f; i = 0; lbl_23: if( i>nmain-1 ) { goto lbl_25; } v = state.x[i]; b = false; if( state.hasbndl[i] ) { b = b || (double)(v-state.diffstep*state.soriginal[i])<(double)(state.bndloriginal[i]); } if( state.hasbndu[i] ) { b = b || (double)(v+state.diffstep*state.soriginal[i])>(double)(state.bnduoriginal[i]); } if( b ) { goto lbl_26; } state.x[i] = v-state.diffstep*state.soriginal[i]; state.rstate.stage = 4; goto lbl_rcomm; lbl_4: state.fm2 = state.f; state.x[i] = v-0.5*state.diffstep*state.soriginal[i]; state.rstate.stage = 5; goto lbl_rcomm; lbl_5: state.fm1 = state.f; state.x[i] = v+0.5*state.diffstep*state.soriginal[i]; state.rstate.stage = 6; goto lbl_rcomm; lbl_6: state.fp1 = state.f; state.x[i] = v+state.diffstep*state.soriginal[i]; state.rstate.stage = 7; goto lbl_rcomm; lbl_7: state.fp2 = state.f; state.g[i] = (8*(state.fp1-state.fm1)-(state.fp2-state.fm2))/(6*state.diffstep*state.soriginal[i]); goto lbl_27; lbl_26: state.xm1 = Math.Max(v-state.diffstep*state.soriginal[i], state.bndloriginal[i]); state.x[i] = state.xm1; state.rstate.stage = 8; goto lbl_rcomm; lbl_8: state.fm1 = state.f; state.xp1 = Math.Min(v+state.diffstep*state.soriginal[i], state.bnduoriginal[i]); state.x[i] = state.xp1; state.rstate.stage = 9; goto lbl_rcomm; lbl_9: state.fp1 = state.f; state.g[i] = (state.fp1-state.fm1)/(state.xp1-state.xm1); lbl_27: state.x[i] = v; i = i+1; goto lbl_23; lbl_25: state.f = state.fbase; state.needf = false; lbl_22: if( (double)(state.f)<(double)(state.trimthreshold) ) { // // normal processing // state.cgstate.f = state.f; scalegradientandexpand(state, state.g, ref state.cgstate.g); for(i_=0; i_<=nmain+nslack-1;i_++) { state.lastg[i_] = state.cgstate.g[i_]; } modifytargetfunction(state, state.tmp1, state.r, vv, ref state.cgstate.f, ref state.cgstate.g, ref state.gnorm, ref state.mpgnorm); } else { // // function value is too high, trim it // state.cgstate.f = state.trimthreshold; for(i=0; i<=nmain+nslack-1; i++) { state.cgstate.g[i] = 0.0; } } goto lbl_17; lbl_19: if( !state.cgstate.xupdated ) { goto lbl_28; } // // Report // unscalepoint(state, state.cgstate.x, ref state.x); state.f = state.cgstate.f; clearrequestfields(state); state.xupdated = true; state.rstate.stage = 10; goto lbl_rcomm; lbl_10: state.xupdated = false; goto lbl_17; lbl_28: goto lbl_17; lbl_18: mincg.mincgresults(state.cgstate, ref state.xcur, state.cgrep); unscalepoint(state, state.xcur, ref state.xend); state.repinneriterationscount = state.repinneriterationscount+state.cgrep.iterationscount; state.repouteriterationscount = state.repouteriterationscount+1; state.repnfev = state.repnfev+state.cgrep.nfev; // // Update RepDebugFF with function value at current point // unscalepoint(state, state.xcur, ref state.x); clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_30; } state.needfg = true; state.rstate.stage = 11; goto lbl_rcomm; lbl_11: state.needfg = false; goto lbl_31; lbl_30: state.needf = true; state.rstate.stage = 12; goto lbl_rcomm; lbl_12: state.needf = false; lbl_31: state.repnfev = state.repnfev+1; state.repdebugff = state.f; // // Check for stopping: // * "normal", outer step size is small enough, infeasibility is within bounds // * "inconsistent", if Lagrange multipliers increased beyond threshold given by MaxLagrangeMul // * "too stringent", in other cases // v = 0; for(i=0; i<=nmain-1; i++) { v = v+math.sqr((state.xcur[i]-state.xprev[i])/state.seffective[i]); } v = Math.Sqrt(v); if( (double)(v)<=(double)(state.outerepsx) ) { state.repterminationtype = 4; goto lbl_16; } if( state.maxits>0 ) { state.itsleft = state.itsleft-state.cgrep.iterationscount; if( state.itsleft<=0 ) { state.repterminationtype = 5; goto lbl_16; } } if( (double)(state.repouteriterationscount)>=(double)(maxouterits) ) { // // Stopped because of reaching limit on the number of outer iterations. // // IMPORTANT: DO NOT REMOVE THIS STOPPING CRITERIA because it // saves algorithm from eternal loop with non-monotonic // steps being made (first step increases function value, // next step moves us back). See comments on the code which // handles State.CGState.LSEnd for more info on this subject. // state.repterminationtype = 5; goto lbl_16; } // // Next iteration // for(i_=0; i_<=nmain+nslack-1;i_++) { state.xprev[i_] = state.xcur[i_]; } goto lbl_15; lbl_16: // // We've stopped, fill debug information // state.repdebugeqerr = 0.0; for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += state.ceeffective[i,i_]*state.xcur[i_]; } state.repdebugeqerr = state.repdebugeqerr+math.sqr(v-state.ceeffective[i,nmain+nslack]); } state.repdebugeqerr = Math.Sqrt(state.repdebugeqerr); state.repdebugdx = 0; for(i=0; i<=nmain-1; i++) { state.repdebugdx = state.repdebugdx+math.sqr(state.xcur[i]-state.xstart[i]); } state.repdebugdx = Math.Sqrt(state.repdebugdx); result = false; return result; // // Saving state // lbl_rcomm: result = true; state.rstate.ia[0] = nmain; state.rstate.ia[1] = nslack; state.rstate.ia[2] = m; state.rstate.ia[3] = i; state.rstate.ia[4] = j; state.rstate.ba[0] = b; state.rstate.ra[0] = v; state.rstate.ra[1] = vv; return result; } /************************************************************************* BLEIC results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one. More information about fields of this structure can be found in the comments on MinBLEICReport datatype. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicresults(minbleicstate state, ref double[] x, minbleicreport rep) { x = new double[0]; minbleicresultsbuf(state, ref x, rep); } /************************************************************************* BLEIC results Buffered implementation of MinBLEICResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicresultsbuf(minbleicstate state, ref double[] x, minbleicreport rep) { int i = 0; int i_ = 0; if( alglib.ap.len(x)0 ) { for(i_=0; i_<=state.nmain-1;i_++) { x[i_] = state.xend[i_]; } } else { for(i=0; i<=state.nmain-1; i++) { x[i] = Double.NaN; } } rep.debugeqerr = state.repdebugeqerr; rep.debugfs = state.repdebugfs; rep.debugff = state.repdebugff; rep.debugdx = state.repdebugdx; rep.debugfeasqpits = state.repdebugfeasqpits; rep.debugfeasgpaits = state.repdebugfeasgpaits; } /************************************************************************* This subroutine restarts algorithm from new point. All optimization parameters (including constraints) are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure previously allocated with MinBLEICCreate call. X - new starting point. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicrestartfrom(minbleicstate state, double[] x) { int n = 0; int i_ = 0; n = state.nmain; // // First, check for errors in the inputs // alglib.ap.assert(alglib.ap.len(x)>=n, "MinBLEICRestartFrom: Length(X)(double)(state.bnduoriginal[i]) ) { v = state.bnduoriginal[i]; } } xunscaled[i] = v; } } /************************************************************************* This function: 1. makes projection of XScaled into equality constrained subspace (X is modified in-place) 2. stores residual from the projection into R 3. unscales projected XScaled and stores result into XUnscaled with additional enforcement It calculates set of additional values which are used later for modification of the target function F. INPUT PARAMETERS: State - optimizer state (we use its fields to get information about constraints) X - vector being projected R - preallocated buffer, used to store residual from projection OUTPUT PARAMETERS: X - projection of input X R - residual RNorm - residual norm squared, used later to modify target function *************************************************************************/ private static void projectpointandunscale(minbleicstate state, ref double[] xscaled, ref double[] xunscaled, ref double[] rscaled, ref double rnorm2) { double v = 0; int i = 0; int nmain = 0; int nslack = 0; int i_ = 0; rnorm2 = 0; nmain = state.nmain; nslack = state.nslack; // // * subtract XE from XScaled // * project XScaled // * calculate norm of deviation from null space, store it in RNorm2 // * calculate residual from projection, store it in R // * add XE to XScaled // * unscale variables // for(i_=0; i_<=nmain+nslack-1;i_++) { xscaled[i_] = xscaled[i_] - state.xe[i_]; } rnorm2 = 0; for(i=0; i<=nmain+nslack-1; i++) { rscaled[i] = 0; } for(i=0; i<=nmain+nslack-1; i++) { if( state.activeconstraints[i] ) { v = xscaled[i]; xscaled[i] = 0; rscaled[i] = rscaled[i]+v; rnorm2 = rnorm2+math.sqr(v); } } for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += xscaled[i_]*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { xscaled[i_] = xscaled[i_] - v*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { rscaled[i_] = rscaled[i_] + v*state.cecurrent[i,i_]; } rnorm2 = rnorm2+math.sqr(v); } for(i_=0; i_<=nmain+nslack-1;i_++) { xscaled[i_] = xscaled[i_] + state.xe[i_]; } unscalepoint(state, xscaled, ref xunscaled); } /************************************************************************* This function scales and copies NMain elements of GUnscaled into GScaled. Other NSlack components of GScaled are set to zero. *************************************************************************/ private static void scalegradientandexpand(minbleicstate state, double[] gunscaled, ref double[] gscaled) { int i = 0; for(i=0; i<=state.nmain-1; i++) { gscaled[i] = gunscaled[i]*state.transforms[i]; } for(i=0; i<=state.nslack-1; i++) { gscaled[state.nmain+i] = 0; } } /************************************************************************* This subroutine applies modifications to the target function given by its value F and gradient G at the projected point X which lies in the equality constrained subspace. Following modifications are applied: * modified barrier functions to handle inequality constraints (both F and G are modified) * projection of gradient into equality constrained subspace (only G is modified) * quadratic penalty for deviations from equality constrained subspace (both F and G are modified) It also calculates gradient norm (three different norms for three different types of gradient), feasibility and complementary slackness errors. INPUT PARAMETERS: State - optimizer state (we use its fields to get information about constraints) X - point (projected into equality constrained subspace) R - residual from projection RNorm2 - residual norm squared F - function value at X G - function gradient at X OUTPUT PARAMETERS: F - modified function value at X G - modified function gradient at X GNorm - 2-norm of unmodified G MPGNorm - 2-norm of modified G MBA - minimum argument of barrier functions. If X is strictly feasible, it is greater than zero. If X lies on a boundary, it is zero. It is negative for infeasible X. FIErr - 2-norm of feasibility error with respect to inequality/bound constraints CSErr - 2-norm of complementarity slackness error *************************************************************************/ private static void modifytargetfunction(minbleicstate state, double[] x, double[] r, double rnorm2, ref double f, ref double[] g, ref double gnorm, ref double mpgnorm) { double v = 0; int i = 0; int nmain = 0; int nslack = 0; bool hasconstraints = new bool(); int i_ = 0; gnorm = 0; mpgnorm = 0; nmain = state.nmain; nslack = state.nslack; hasconstraints = false; // // GNorm // v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += g[i_]*g[i_]; } gnorm = Math.Sqrt(v); // // Process equality constraints: // * modify F to handle penalty term for equality constraints // * project gradient on null space of equality constraints // * add penalty term for equality constraints to gradient // f = f+rnorm2; for(i=0; i<=nmain+nslack-1; i++) { if( state.activeconstraints[i] ) { g[i] = 0; } } for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += g[i_]*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { g[i_] = g[i_] - v*state.cecurrent[i,i_]; } } for(i_=0; i_<=nmain+nslack-1;i_++) { g[i_] = g[i_] + 2*r[i_]; } // // MPGNorm // v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += g[i_]*g[i_]; } mpgnorm = Math.Sqrt(v); } /************************************************************************* This function makes additional check for constraints which can be activated. We try activate constraints one by one, but it is possible that several constraints should be activated during one iteration. It this case only one of them (probably last) will be activated. This function will fix it - it will pass through constraints and activate those which are at the boundary or beyond it. It will return True, if at least one constraint was activated by this function. *************************************************************************/ private static bool additionalcheckforconstraints(minbleicstate state, double[] x) { bool result = new bool(); int i = 0; int nmain = 0; int nslack = 0; result = false; nmain = state.nmain; nslack = state.nslack; for(i=0; i<=nmain-1; i++) { if( !state.activeconstraints[i] ) { if( state.hasbndl[i] ) { if( (double)(x[i])<=(double)(state.bndleffective[i]) ) { state.activeconstraints[i] = true; state.constrainedvalues[i] = state.bndleffective[i]; result = true; } } if( state.hasbndu[i] ) { if( (double)(x[i])>=(double)(state.bndueffective[i]) ) { state.activeconstraints[i] = true; state.constrainedvalues[i] = state.bndueffective[i]; result = true; } } } } for(i=0; i<=nslack-1; i++) { if( !state.activeconstraints[nmain+i] ) { if( (double)(x[nmain+i])<=(double)(0) ) { state.activeconstraints[nmain+i] = true; state.constrainedvalues[nmain+i] = 0; result = true; } } } return result; } /************************************************************************* This function rebuilds CECurrent and XE according to current set of active bound constraints. *************************************************************************/ private static void rebuildcexe(minbleicstate state) { int i = 0; int j = 0; int k = 0; int nmain = 0; int nslack = 0; double v = 0; int i_ = 0; nmain = state.nmain; nslack = state.nslack; ablas.rmatrixcopy(state.cecnt, nmain+nslack+1, state.ceeffective, 0, 0, ref state.cecurrent, 0, 0); for(i=0; i<=state.cecnt-1; i++) { // // "Subtract" active bound constraints from I-th linear constraint // for(j=0; j<=nmain+nslack-1; j++) { if( state.activeconstraints[j] ) { state.cecurrent[i,nmain+nslack] = state.cecurrent[i,nmain+nslack]-state.cecurrent[i,j]*state.constrainedvalues[j]; state.cecurrent[i,j] = 0.0; } } // // Reorthogonalize I-th constraint with respect to previous ones // NOTE: we also update right part, which is CECurrent[...,NMain+NSlack]. // for(k=0; k<=i-1; k++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += state.cecurrent[k,i_]*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack;i_++) { state.cecurrent[i,i_] = state.cecurrent[i,i_] - v*state.cecurrent[k,i_]; } } // // Calculate norm of I-th row of CECurrent. Fill by zeros, if it is // too small. Normalize otherwise. // // NOTE: we also scale last column of CECurrent (right part) // v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += state.cecurrent[i,i_]*state.cecurrent[i,i_]; } v = Math.Sqrt(v); if( (double)(v)>(double)(10000*math.machineepsilon) ) { v = 1/v; for(i_=0; i_<=nmain+nslack;i_++) { state.cecurrent[i,i_] = v*state.cecurrent[i,i_]; } } else { for(j=0; j<=nmain+nslack; j++) { state.cecurrent[i,j] = 0; } } } for(j=0; j<=nmain+nslack-1; j++) { state.xe[j] = 0; } for(i=0; i<=nmain+nslack-1; i++) { if( state.activeconstraints[i] ) { state.xe[i] = state.xe[i]+state.constrainedvalues[i]; } } for(i=0; i<=state.cecnt-1; i++) { v = state.cecurrent[i,nmain+nslack]; for(i_=0; i_<=nmain+nslack-1;i_++) { state.xe[i_] = state.xe[i_] + v*state.cecurrent[i,i_]; } } } /************************************************************************* This function projects gradient onto equality constrained subspace *************************************************************************/ private static void makegradientprojection(minbleicstate state, ref double[] pg) { int i = 0; int nmain = 0; int nslack = 0; double v = 0; int i_ = 0; nmain = state.nmain; nslack = state.nslack; for(i=0; i<=nmain+nslack-1; i++) { if( state.activeconstraints[i] ) { pg[i] = 0; } } for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += pg[i_]*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { pg[i_] = pg[i_] - v*state.cecurrent[i,i_]; } } } /************************************************************************* This function prepares equality constrained subproblem: 1. X is used to activate constraints (if there are ones which are still inactive, but should be activated). 2. constraints matrix CEOrt is copied to CECurrent and modified according to the list of active bound constraints (corresponding elements are filled by zeros and reorthogonalized). 3. XE - least squares solution of equality constraints - is recalculated 4. X is copied to PX and projected onto equality constrained subspace 5. inactive constraints are checked against PX - if there is at least one which should be activated, we activate it and move back to (2) 6. as result, PX is feasible with respect to bound constraints - step (5) guarantees it. But PX can be infeasible with respect to equality ones, because step (2) is done without checks for consistency. As the final step, we check that PX is feasible. If not, we return False. True is returned otherwise. If this algorithm returned True, then: * X is not changed * PX contains projection of X onto constrained subspace * G is not changed * PG contains projection of G onto constrained subspace * PX is feasible with respect to all constraints * all constraints which are active at PX, are activated *************************************************************************/ private static bool prepareconstraintmatrix(minbleicstate state, double[] x, double[] g, ref double[] px, ref double[] pg) { bool result = new bool(); int i = 0; int nmain = 0; int nslack = 0; double v = 0; double ferr = 0; int i_ = 0; nmain = state.nmain; nslack = state.nslack; result = true; // // Step 1 // additionalcheckforconstraints(state, x); // // Steps 2-5 // do { // // Steps 2-3 // rebuildcexe(state); // // Step 4 // // Calculate PX, PG // for(i_=0; i_<=nmain+nslack-1;i_++) { px[i_] = x[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { px[i_] = px[i_] - state.xe[i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { pg[i_] = g[i_]; } for(i=0; i<=nmain+nslack-1; i++) { if( state.activeconstraints[i] ) { px[i] = 0; pg[i] = 0; } } for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += px[i_]*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { px[i_] = px[i_] - v*state.cecurrent[i,i_]; } v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += pg[i_]*state.cecurrent[i,i_]; } for(i_=0; i_<=nmain+nslack-1;i_++) { pg[i_] = pg[i_] - v*state.cecurrent[i,i_]; } } for(i_=0; i_<=nmain+nslack-1;i_++) { px[i_] = px[i_] + state.xe[i_]; } // // Step 5 (loop condition below) // } while( additionalcheckforconstraints(state, px) ); // // Step 6 // ferr = 0; for(i=0; i<=state.cecnt-1; i++) { v = 0.0; for(i_=0; i_<=nmain+nslack-1;i_++) { v += px[i_]*state.ceeffective[i,i_]; } v = v-state.ceeffective[i,nmain+nslack]; ferr = Math.Max(ferr, Math.Abs(v)); } result = (double)(ferr)<=(double)(state.outerepsi); return result; } /************************************************************************* Internal initialization subroutine *************************************************************************/ private static void minbleicinitinternal(int n, double[] x, double diffstep, minbleicstate state) { int i = 0; double[,] c = new double[0,0]; int[] ct = new int[0]; state.nmain = n; state.optdim = 0; state.diffstep = diffstep; state.bndloriginal = new double[n]; state.bndleffective = new double[n]; state.hasbndl = new bool[n]; state.bnduoriginal = new double[n]; state.bndueffective = new double[n]; state.hasbndu = new bool[n]; state.xstart = new double[n]; state.soriginal = new double[n]; state.x = new double[n]; state.g = new double[n]; for(i=0; i<=n-1; i++) { state.bndloriginal[i] = Double.NegativeInfinity; state.hasbndl[i] = false; state.bnduoriginal[i] = Double.PositiveInfinity; state.hasbndu[i] = false; state.soriginal[i] = 1.0; } minbleicsetlc(state, c, ct, 0); minbleicsetinnercond(state, 0.0, 0.0, 0.0); minbleicsetoutercond(state, 1.0E-6, 1.0E-6); minbleicsetmaxits(state, 0); minbleicsetxrep(state, false); minbleicsetstpmax(state, 0.0); minbleicsetprecdefault(state); minbleicrestartfrom(state, x); } } public class minlbfgs { public class minlbfgsstate { public int n; public int m; public double epsg; public double epsf; public double epsx; public int maxits; public bool xrep; public double stpmax; public double[] s; public double diffstep; public int nfev; public int mcstage; public int k; public int q; public int p; public double[] rho; public double[,] yk; public double[,] sk; public double[] theta; public double[] d; public double stp; public double[] work; public double fold; public double trimthreshold; public int prectype; public double gammak; public double[,] denseh; public double[] diagh; public double fbase; public double fm2; public double fm1; public double fp1; public double fp2; public double[] autobuf; public double[] x; public double f; public double[] g; public bool needf; public bool needfg; public bool xupdated; public rcommstate rstate; public int repiterationscount; public int repnfev; public int repterminationtype; public linmin.linminstate lstate; public minlbfgsstate() { s = new double[0]; rho = new double[0]; yk = new double[0,0]; sk = new double[0,0]; theta = new double[0]; d = new double[0]; work = new double[0]; denseh = new double[0,0]; diagh = new double[0]; autobuf = new double[0]; x = new double[0]; g = new double[0]; rstate = new rcommstate(); lstate = new linmin.linminstate(); } }; public class minlbfgsreport { public int iterationscount; public int nfev; public int terminationtype; }; public const double gtol = 0.4; /************************************************************************* LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION DESCRIPTION: The subroutine minimizes function F(x) of N arguments by using a quasi- Newton method (LBFGS scheme) which is optimized to use a minimum amount of memory. The subroutine generates the approximation of an inverse Hessian matrix by using information about the last M steps of the algorithm (instead of N). It lessens a required amount of memory from a value of order N^2 to a value of order 2*N*M. REQUIREMENTS: Algorithm will request following information during its operation: * function value F and its gradient G (simultaneously) at given point X USAGE: 1. User initializes algorithm state with MinLBFGSCreate() call 2. User tunes solver parameters with MinLBFGSSetCond() MinLBFGSSetStpMax() and other functions 3. User calls MinLBFGSOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 4. User calls MinLBFGSResults() to get solution 5. Optionally user may call MinLBFGSRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLBFGSRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension. N>0 M - number of corrections in the BFGS scheme of Hessian approximation update. Recommended value: 3<=M<=7. The smaller value causes worse convergence, the bigger will not cause a considerably better convergence, but will cause a fall in the performance. M<=N. X - initial solution approximation, array[0..N-1]. OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLBFGSSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLBFGSSetStpMax() function to bound algorithm's steps. However, L-BFGS rarely needs such a tuning. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgscreate(int n, int m, double[] x, minlbfgsstate state) { alglib.ap.assert(n>=1, "MinLBFGSCreate: N<1!"); alglib.ap.assert(m>=1, "MinLBFGSCreate: M<1"); alglib.ap.assert(m<=n, "MinLBFGSCreate: M>N"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinLBFGSCreate: Length(X)0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of corrections in the BFGS scheme of Hessian approximation update. Recommended value: 3<=M<=7. The smaller value causes worse convergence, the bigger will not cause a considerably better convergence, but will cause a fall in the performance. M<=N. X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinLBFGSSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. LBFGS needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/ public static void minlbfgscreatef(int n, int m, double[] x, double diffstep, minlbfgsstate state) { alglib.ap.assert(n>=1, "MinLBFGSCreateF: N too small!"); alglib.ap.assert(m>=1, "MinLBFGSCreateF: M<1"); alglib.ap.assert(m<=n, "MinLBFGSCreateF: M>N"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinLBFGSCreateF: Length(X)(double)(0), "MinLBFGSCreateF: DiffStep is non-positive!"); minlbfgscreatex(n, m, x, 0, diffstep, state); } /************************************************************************* This function sets stopping conditions for L-BFGS optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinLBFGSSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetcond(minlbfgsstate state, double epsg, double epsf, double epsx, int maxits) { alglib.ap.assert(math.isfinite(epsg), "MinLBFGSSetCond: EpsG is not finite number!"); alglib.ap.assert((double)(epsg)>=(double)(0), "MinLBFGSSetCond: negative EpsG!"); alglib.ap.assert(math.isfinite(epsf), "MinLBFGSSetCond: EpsF is not finite number!"); alglib.ap.assert((double)(epsf)>=(double)(0), "MinLBFGSSetCond: negative EpsF!"); alglib.ap.assert(math.isfinite(epsx), "MinLBFGSSetCond: EpsX is not finite number!"); alglib.ap.assert((double)(epsx)>=(double)(0), "MinLBFGSSetCond: negative EpsX!"); alglib.ap.assert(maxits>=0, "MinLBFGSSetCond: negative MaxIts!"); if( (((double)(epsg)==(double)(0) && (double)(epsf)==(double)(0)) && (double)(epsx)==(double)(0)) && maxits==0 ) { epsx = 1.0E-6; } state.epsg = epsg; state.epsf = epsf; state.epsx = epsx; state.maxits = maxits; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinLBFGSOptimize(). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetxrep(minlbfgsstate state, bool needxrep) { state.xrep = needxrep; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0 (default), if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetstpmax(minlbfgsstate state, double stpmax) { alglib.ap.assert(math.isfinite(stpmax), "MinLBFGSSetStpMax: StpMax is not finite!"); alglib.ap.assert((double)(stpmax)>=(double)(0), "MinLBFGSSetStpMax: StpMax<0!"); state.stpmax = stpmax; } /************************************************************************* This function sets scaling coefficients for LBFGS optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the LBFGS too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinLBFGSSetPrec...() functions. There is special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetscale(minlbfgsstate state, double[] s) { int i = 0; alglib.ap.assert(alglib.ap.len(s)>=state.n, "MinLBFGSSetScale: Length(S)=1, "MinLBFGS: N too small!"); alglib.ap.assert(m>=1, "MinLBFGS: M too small!"); alglib.ap.assert(m<=n, "MinLBFGS: M too large!"); // // Initialize // state.diffstep = diffstep; state.n = n; state.m = m; allocatemem = flags%2==0; flags = flags/2; if( allocatemem ) { state.rho = new double[m]; state.theta = new double[m]; state.yk = new double[m, n]; state.sk = new double[m, n]; state.d = new double[n]; state.x = new double[n]; state.s = new double[n]; state.g = new double[n]; state.work = new double[n]; } minlbfgssetcond(state, 0, 0, 0, 0); minlbfgssetxrep(state, false); minlbfgssetstpmax(state, 0); minlbfgsrestartfrom(state, x); for(i=0; i<=n-1; i++) { state.s[i] = 1.0; } state.prectype = 0; } /************************************************************************* Modification of the preconditioner: default preconditioner (simple scaling, same for all elements of X) is used. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetprecdefault(minlbfgsstate state) { state.prectype = 0; } /************************************************************************* Modification of the preconditioner: Cholesky factorization of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state P - triangular preconditioner, Cholesky factorization of the approximate Hessian. array[0..N-1,0..N-1], (if larger, only leading N elements are used). IsUpper - whether upper or lower triangle of P is given (other triangle is not referenced) After call to this function preconditioner is changed to P (P is copied into the internal buffer). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: P should be nonsingular. Exception will be thrown otherwise. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetpreccholesky(minlbfgsstate state, double[,] p, bool isupper) { int i = 0; double mx = 0; alglib.ap.assert(apserv.isfinitertrmatrix(p, state.n, isupper), "MinLBFGSSetPrecCholesky: P contains infinite or NAN values!"); mx = 0; for(i=0; i<=state.n-1; i++) { mx = Math.Max(mx, Math.Abs(p[i,i])); } alglib.ap.assert((double)(mx)>(double)(0), "MinLBFGSSetPrecCholesky: P is strictly singular!"); if( alglib.ap.rows(state.denseh)=state.n, "MinLBFGSSetPrecDiag: D is too short"); for(i=0; i<=state.n-1; i++) { alglib.ap.assert(math.isfinite(d[i]), "MinLBFGSSetPrecDiag: D contains infinite or NAN elements"); alglib.ap.assert((double)(d[i])>(double)(0), "MinLBFGSSetPrecDiag: D contains non-positive elements"); } apserv.rvectorsetlengthatleast(ref state.diagh, state.n); state.prectype = 2; for(i=0; i<=state.n-1; i++) { state.diagh[i] = d[i]; } } /************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinLBFGSSetScale() call (before or after MinLBFGSSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetprecscale(minlbfgsstate state) { state.prectype = 3; } /************************************************************************* NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinLBFGSCreate() for analytical gradient or MinLBFGSCreateF() for numerical differentiation) you should choose appropriate variant of MinLBFGSOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinLBFGSOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinLBFGSOptimize() and specific function used to create optimizer. | USER PASSED TO MinLBFGSOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinLBFGSCreateF() | work FAIL MinLBFGSCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinLBFGSOptimize() version. Attemps to use such combination (for example, to create optimizer with MinLBFGSCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static bool minlbfgsiteration(minlbfgsstate state) { bool result = new bool(); int n = 0; int m = 0; int i = 0; int j = 0; int ic = 0; int mcinfo = 0; double v = 0; double vv = 0; int i_ = 0; // // Reverse communication preparations // I know it looks ugly, but it works the same way // anywhere from C++ to Python. // // This code initializes locals by: // * random values determined during code // generation - on first subroutine call // * values from previous call - on subsequent calls // if( state.rstate.stage>=0 ) { n = state.rstate.ia[0]; m = state.rstate.ia[1]; i = state.rstate.ia[2]; j = state.rstate.ia[3]; ic = state.rstate.ia[4]; mcinfo = state.rstate.ia[5]; v = state.rstate.ra[0]; vv = state.rstate.ra[1]; } else { n = -983; m = -989; i = -834; j = 900; ic = -287; mcinfo = 364; v = 214; vv = -338; } if( state.rstate.stage==0 ) { goto lbl_0; } if( state.rstate.stage==1 ) { goto lbl_1; } if( state.rstate.stage==2 ) { goto lbl_2; } if( state.rstate.stage==3 ) { goto lbl_3; } if( state.rstate.stage==4 ) { goto lbl_4; } if( state.rstate.stage==5 ) { goto lbl_5; } if( state.rstate.stage==6 ) { goto lbl_6; } if( state.rstate.stage==7 ) { goto lbl_7; } if( state.rstate.stage==8 ) { goto lbl_8; } if( state.rstate.stage==9 ) { goto lbl_9; } if( state.rstate.stage==10 ) { goto lbl_10; } if( state.rstate.stage==11 ) { goto lbl_11; } if( state.rstate.stage==12 ) { goto lbl_12; } if( state.rstate.stage==13 ) { goto lbl_13; } // // Routine body // // // Unload frequently used variables from State structure // (just for typing convinience) // n = state.n; m = state.m; state.repterminationtype = 0; state.repiterationscount = 0; state.repnfev = 0; // // Calculate F/G at the initial point // clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_14; } state.needfg = true; state.rstate.stage = 0; goto lbl_rcomm; lbl_0: state.needfg = false; goto lbl_15; lbl_14: state.needf = true; state.rstate.stage = 1; goto lbl_rcomm; lbl_1: state.fbase = state.f; i = 0; lbl_16: if( i>n-1 ) { goto lbl_18; } v = state.x[i]; state.x[i] = v-state.diffstep*state.s[i]; state.rstate.stage = 2; goto lbl_rcomm; lbl_2: state.fm2 = state.f; state.x[i] = v-0.5*state.diffstep*state.s[i]; state.rstate.stage = 3; goto lbl_rcomm; lbl_3: state.fm1 = state.f; state.x[i] = v+0.5*state.diffstep*state.s[i]; state.rstate.stage = 4; goto lbl_rcomm; lbl_4: state.fp1 = state.f; state.x[i] = v+state.diffstep*state.s[i]; state.rstate.stage = 5; goto lbl_rcomm; lbl_5: state.fp2 = state.f; state.x[i] = v; state.g[i] = (8*(state.fp1-state.fm1)-(state.fp2-state.fm2))/(6*state.diffstep*state.s[i]); i = i+1; goto lbl_16; lbl_18: state.f = state.fbase; state.needf = false; lbl_15: optserv.trimprepare(state.f, ref state.trimthreshold); if( !state.xrep ) { goto lbl_19; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 6; goto lbl_rcomm; lbl_6: state.xupdated = false; lbl_19: state.repnfev = 1; state.fold = state.f; v = 0; for(i=0; i<=n-1; i++) { v = v+math.sqr(state.g[i]*state.s[i]); } if( (double)(Math.Sqrt(v))<=(double)(state.epsg) ) { state.repterminationtype = 4; result = false; return result; } // // Choose initial step and direction. // Apply preconditioner, if we have something other than default. // for(i_=0; i_<=n-1;i_++) { state.d[i_] = -state.g[i_]; } if( state.prectype==0 ) { // // Default preconditioner is used, but we can't use it before iterations will start // v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.g[i_]*state.g[i_]; } v = Math.Sqrt(v); if( (double)(state.stpmax)==(double)(0) ) { state.stp = Math.Min(1.0/v, 1); } else { state.stp = Math.Min(1.0/v, state.stpmax); } } if( state.prectype==1 ) { // // Cholesky preconditioner is used // fbls.fblscholeskysolve(state.denseh, 1.0, n, true, ref state.d, ref state.autobuf); state.stp = 1; } if( state.prectype==2 ) { // // diagonal approximation is used // for(i=0; i<=n-1; i++) { state.d[i] = state.d[i]/state.diagh[i]; } state.stp = 1; } if( state.prectype==3 ) { // // scale-based preconditioner is used // for(i=0; i<=n-1; i++) { state.d[i] = state.d[i]*state.s[i]*state.s[i]; } state.stp = 1; } // // Main cycle // state.k = 0; lbl_21: if( false ) { goto lbl_22; } // // Main cycle: prepare to 1-D line search // state.p = state.k%m; state.q = Math.Min(state.k, m-1); // // Store X[k], G[k] // for(i_=0; i_<=n-1;i_++) { state.sk[state.p,i_] = -state.x[i_]; } for(i_=0; i_<=n-1;i_++) { state.yk[state.p,i_] = -state.g[i_]; } // // Minimize F(x+alpha*d) // Calculate S[k], Y[k] // state.mcstage = 0; if( state.k!=0 ) { state.stp = 1.0; } linmin.linminnormalized(ref state.d, ref state.stp, n); linmin.mcsrch(n, ref state.x, ref state.f, ref state.g, state.d, ref state.stp, state.stpmax, gtol, ref mcinfo, ref state.nfev, ref state.work, state.lstate, ref state.mcstage); lbl_23: if( state.mcstage==0 ) { goto lbl_24; } clearrequestfields(state); if( (double)(state.diffstep)!=(double)(0) ) { goto lbl_25; } state.needfg = true; state.rstate.stage = 7; goto lbl_rcomm; lbl_7: state.needfg = false; goto lbl_26; lbl_25: state.needf = true; state.rstate.stage = 8; goto lbl_rcomm; lbl_8: state.fbase = state.f; i = 0; lbl_27: if( i>n-1 ) { goto lbl_29; } v = state.x[i]; state.x[i] = v-state.diffstep*state.s[i]; state.rstate.stage = 9; goto lbl_rcomm; lbl_9: state.fm2 = state.f; state.x[i] = v-0.5*state.diffstep*state.s[i]; state.rstate.stage = 10; goto lbl_rcomm; lbl_10: state.fm1 = state.f; state.x[i] = v+0.5*state.diffstep*state.s[i]; state.rstate.stage = 11; goto lbl_rcomm; lbl_11: state.fp1 = state.f; state.x[i] = v+state.diffstep*state.s[i]; state.rstate.stage = 12; goto lbl_rcomm; lbl_12: state.fp2 = state.f; state.x[i] = v; state.g[i] = (8*(state.fp1-state.fm1)-(state.fp2-state.fm2))/(6*state.diffstep*state.s[i]); i = i+1; goto lbl_27; lbl_29: state.f = state.fbase; state.needf = false; lbl_26: optserv.trimfunction(ref state.f, ref state.g, n, state.trimthreshold); linmin.mcsrch(n, ref state.x, ref state.f, ref state.g, state.d, ref state.stp, state.stpmax, gtol, ref mcinfo, ref state.nfev, ref state.work, state.lstate, ref state.mcstage); goto lbl_23; lbl_24: if( !state.xrep ) { goto lbl_30; } // // report // clearrequestfields(state); state.xupdated = true; state.rstate.stage = 13; goto lbl_rcomm; lbl_13: state.xupdated = false; lbl_30: state.repnfev = state.repnfev+state.nfev; state.repiterationscount = state.repiterationscount+1; for(i_=0; i_<=n-1;i_++) { state.sk[state.p,i_] = state.sk[state.p,i_] + state.x[i_]; } for(i_=0; i_<=n-1;i_++) { state.yk[state.p,i_] = state.yk[state.p,i_] + state.g[i_]; } // // Stopping conditions // if( state.repiterationscount>=state.maxits && state.maxits>0 ) { // // Too many iterations // state.repterminationtype = 5; result = false; return result; } v = 0; for(i=0; i<=n-1; i++) { v = v+math.sqr(state.g[i]*state.s[i]); } if( (double)(Math.Sqrt(v))<=(double)(state.epsg) ) { // // Gradient is small enough // state.repterminationtype = 4; result = false; return result; } if( (double)(state.fold-state.f)<=(double)(state.epsf*Math.Max(Math.Abs(state.fold), Math.Max(Math.Abs(state.f), 1.0))) ) { // // F(k+1)-F(k) is small enough // state.repterminationtype = 1; result = false; return result; } v = 0; for(i=0; i<=n-1; i++) { v = v+math.sqr(state.sk[state.p,i]/state.s[i]); } if( (double)(Math.Sqrt(v))<=(double)(state.epsx) ) { // // X(k+1)-X(k) is small enough // state.repterminationtype = 2; result = false; return result; } // // If Wolfe conditions are satisfied, we can update // limited memory model. // // However, if conditions are not satisfied (NFEV limit is met, // function is too wild, ...), we'll skip L-BFGS update // if( mcinfo!=1 ) { // // Skip update. // // In such cases we'll initialize search direction by // antigradient vector, because it leads to more // transparent code with less number of special cases // state.fold = state.f; for(i_=0; i_<=n-1;i_++) { state.d[i_] = -state.g[i_]; } } else { // // Calculate Rho[k], GammaK // v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.yk[state.p,i_]*state.sk[state.p,i_]; } vv = 0.0; for(i_=0; i_<=n-1;i_++) { vv += state.yk[state.p,i_]*state.yk[state.p,i_]; } if( (double)(v)==(double)(0) || (double)(vv)==(double)(0) ) { // // Rounding errors make further iterations impossible. // state.repterminationtype = -2; result = false; return result; } state.rho[state.p] = 1/v; state.gammak = v/vv; // // Calculate d(k+1) = -H(k+1)*g(k+1) // // for I:=K downto K-Q do // V = s(i)^T * work(iteration:I) // theta(i) = V // work(iteration:I+1) = work(iteration:I) - V*Rho(i)*y(i) // work(last iteration) = H0*work(last iteration) - preconditioner // for I:=K-Q to K do // V = y(i)^T*work(iteration:I) // work(iteration:I+1) = work(iteration:I) +(-V+theta(i))*Rho(i)*s(i) // // NOW WORK CONTAINS d(k+1) // for(i_=0; i_<=n-1;i_++) { state.work[i_] = state.g[i_]; } for(i=state.k; i>=state.k-state.q; i--) { ic = i%m; v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.sk[ic,i_]*state.work[i_]; } state.theta[ic] = v; vv = v*state.rho[ic]; for(i_=0; i_<=n-1;i_++) { state.work[i_] = state.work[i_] - vv*state.yk[ic,i_]; } } if( state.prectype==0 ) { // // Simple preconditioner is used // v = state.gammak; for(i_=0; i_<=n-1;i_++) { state.work[i_] = v*state.work[i_]; } } if( state.prectype==1 ) { // // Cholesky preconditioner is used // fbls.fblscholeskysolve(state.denseh, 1, n, true, ref state.work, ref state.autobuf); } if( state.prectype==2 ) { // // diagonal approximation is used // for(i=0; i<=n-1; i++) { state.work[i] = state.work[i]/state.diagh[i]; } } if( state.prectype==3 ) { // // scale-based preconditioner is used // for(i=0; i<=n-1; i++) { state.work[i] = state.work[i]*state.s[i]*state.s[i]; } } for(i=state.k-state.q; i<=state.k; i++) { ic = i%m; v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.yk[ic,i_]*state.work[i_]; } vv = state.rho[ic]*(-v+state.theta[ic]); for(i_=0; i_<=n-1;i_++) { state.work[i_] = state.work[i_] + vv*state.sk[ic,i_]; } } for(i_=0; i_<=n-1;i_++) { state.d[i_] = -state.work[i_]; } // // Next step // state.fold = state.f; state.k = state.k+1; } goto lbl_21; lbl_22: result = false; return result; // // Saving state // lbl_rcomm: result = true; state.rstate.ia[0] = n; state.rstate.ia[1] = m; state.rstate.ia[2] = i; state.rstate.ia[3] = j; state.rstate.ia[4] = ic; state.rstate.ia[5] = mcinfo; state.rstate.ra[0] = v; state.rstate.ra[1] = vv; return result; } /************************************************************************* L-BFGS algorithm results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * -2 rounding errors prevent further improvement. X contains best point found. * -1 incorrect parameters were specified * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgsresults(minlbfgsstate state, ref double[] x, minlbfgsreport rep) { x = new double[0]; minlbfgsresultsbuf(state, ref x, rep); } /************************************************************************* L-BFGS algorithm results Buffered implementation of MinLBFGSResults which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.08.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgsresultsbuf(minlbfgsstate state, ref double[] x, minlbfgsreport rep) { int i_ = 0; if( alglib.ap.len(x)=state.n, "MinLBFGSRestartFrom: Length(X)=1, "MinQPCreate: N<1"); // // initialize QP solver // state.n = n; state.akind = -1; state.repterminationtype = 0; state.b = new double[n]; state.bndl = new double[n]; state.bndu = new double[n]; state.workbndl = new double[n]; state.workbndu = new double[n]; state.havebndl = new bool[n]; state.havebndu = new bool[n]; state.startx = new double[n]; state.xorigin = new double[n]; state.xcand0 = new double[n]; state.xcand1 = new double[n]; state.xc = new double[n]; state.xn = new double[n]; state.gc = new double[n]; state.pg = new double[n]; state.dc = new double[n]; state.activeset = new bool[n]; for(i=0; i<=n-1; i++) { state.b[i] = 0.0; state.bndl[i] = Double.NegativeInfinity; state.bndu[i] = Double.PositiveInfinity; state.havebndl[i] = false; state.havebndu[i] = false; state.startx[i] = 0.0; state.xorigin[i] = 0.0; } state.havex = false; minqpsetalgocholesky(state); } /************************************************************************* This function sets linear term for QP solver. By default, linear term is zero. INPUT PARAMETERS: State - structure which stores algorithm state B - linear term, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetlinearterm(minqpstate state, double[] b) { int n = 0; n = state.n; alglib.ap.assert(alglib.ap.len(b)>=n, "MinQPSetLinearTerm: Length(B)=n, "MinQPSetQuadraticTerm: Rows(A)=n, "MinQPSetQuadraticTerm: Cols(A)=n, "MinQPSetStartingPoint: Length(B)=n, "MinQPSetOrigin: Length(B)=n, "MinQPSetBC: Length(BndL)=n, "MinQPSetBC: Length(BndU)(double)(state.bndu[i]) ) { state.repterminationtype = -3; return; } } } // // count number of bound and linear constraints // nbc = 0; nlc = 0; for(i=0; i<=n-1; i++) { if( state.havebndl[i] ) { nbc = nbc+1; } if( state.havebndu[i] ) { nbc = nbc+1; } } // // Our formulation of quadratic problem includes origin point, // i.e. we have F(x-x_origin) which is minimized subject to // constraints on x, instead of having simply F(x). // // Here we make transition from non-zero origin to zero one. // In order to make such transition we have to: // 1. subtract x_origin from x_start // 2. modify constraints // 3. solve problem // 4. add x_origin to solution // // There is alternate solution - to modify quadratic function // by expansion of multipliers containing (x-x_origin), but // we prefer to modify constraints, because it is a) more precise // and b) easier to to. // // Parts (1)-(2) are done here. After this block is over, // we have: // * XC, which stores shifted XStart (if we don't have XStart, // value of XC will be ignored later) // * WorkBndL, WorkBndU, which store modified boundary constraints. // for(i=0; i<=n-1; i++) { if( state.havebndl[i] ) { state.workbndl[i] = state.bndl[i]-state.xorigin[i]; } else { state.workbndl[i] = Double.NegativeInfinity; } if( state.havebndu[i] ) { state.workbndu[i] = state.bndu[i]-state.xorigin[i]; } else { state.workbndu[i] = Double.PositiveInfinity; } } // // modify starting point XC according to boundary constraints // if( state.havex ) { // // We have starting point in StartX, so we just have to shift and bound it // for(i=0; i<=n-1; i++) { state.xc[i] = state.startx[i]-state.xorigin[i]; if( state.havebndl[i] ) { if( (double)(state.xc[i])<(double)(state.workbndl[i]) ) { state.xc[i] = state.workbndl[i]; } } if( state.havebndu[i] ) { if( (double)(state.xc[i])>(double)(state.workbndu[i]) ) { state.xc[i] = state.workbndu[i]; } } } } else { // // We don't have starting point, so we deduce it from // constraints (if they are present). // // NOTE: XC contains some meaningless values from previous block // which are ignored by code below. // for(i=0; i<=n-1; i++) { if( state.havebndl[i] && state.havebndu[i] ) { state.xc[i] = 0.5*(state.workbndl[i]+state.workbndu[i]); if( (double)(state.xc[i])<(double)(state.workbndl[i]) ) { state.xc[i] = state.workbndl[i]; } if( (double)(state.xc[i])>(double)(state.workbndu[i]) ) { state.xc[i] = state.workbndu[i]; } continue; } if( state.havebndl[i] ) { state.xc[i] = state.workbndl[i]; continue; } if( state.havebndu[i] ) { state.xc[i] = state.workbndu[i]; continue; } state.xc[i] = 0; } } // // Select algo // if( state.algokind==1 && state.akind==0 ) { // // Cholesky-based algorithm for dense bound constrained problems. // // This algorithm exists in two variants: // * unconstrained one, which can solve problem using only one NxN // double matrix // * bound constrained one, which needs two NxN matrices // // We will try to solve problem using unconstrained algorithm, // and will use bound constrained version only when constraints // are actually present // if( nbc==0 && nlc==0 ) { // // "Simple" unconstrained version // apserv.rvectorsetlengthatleast(ref state.tmp0, n); apserv.rvectorsetlengthatleast(ref state.bufb, n); apserv.rmatrixsetlengthatleast(ref state.bufa, n, n); for(i=0; i<=n-1; i++) { for(j=i; j<=n-1; j++) { state.bufa[i,j] = state.densea[i,j]; } } for(i_=0; i_<=n-1;i_++) { state.bufb[i_] = state.b[i_]; } state.repncholesky = 1; if( !trfac.spdmatrixcholeskyrec(ref state.bufa, 0, n, true, ref state.tmp0) ) { state.repterminationtype = -5; return; } fbls.fblscholeskysolve(state.bufa, 1.0, n, true, ref state.bufb, ref state.tmp0); for(i_=0; i_<=n-1;i_++) { state.xc[i_] = -state.bufb[i_]; } for(i_=0; i_<=n-1;i_++) { state.xc[i_] = state.xc[i_] + state.xorigin[i_]; } state.repouteriterationscount = 1; state.repterminationtype = 4; return; } alglib.ap.assert(nlc==0, "MinQPOptimize: internal error"); // // BoxQP algorithm // apserv.rvectorsetlengthatleast(ref state.tmp0, n); apserv.rmatrixsetlengthatleast(ref state.bufa, n, n); while( true ) { // // Phase 1. // // Determine active set using only current point location // (we ignore antigradient direction, constraint is activated // when current point is located at the bound of the area). // // This phase allows us to reduce function value without // leaving current face. // for(i=0; i<=n-1; i++) { state.activeset[i] = (state.havebndl[i] && (double)(state.xc[i])==(double)(state.workbndl[i])) || (state.havebndu[i] && (double)(state.xc[i])==(double)(state.workbndu[i])); } state.repncholesky = state.repncholesky+1; if( !minqppreparenewtonstep(state, state.xc, state.activeset, ref state.xn, n, ref state.bufa, ref state.tmp0, ref state.tmp1) ) { state.repterminationtype = -5; return; } minqpstepto(state, state.xc, ref state.xn, state.workbndl, state.havebndl, state.workbndu, state.havebndu, n, ref state.tmp0, ref state.tmp1, ref constraintsactivated, ref constraintsdeactivated, ref noisystep); for(i_=0; i_<=n-1;i_++) { state.xc[i_] = state.xn[i_]; } // // Phase 2. // // Determine active set using current point location and // antigradient direction. Then perform steepest descent step. // // 1. we calculate projected gradient PG and scale using // diagonal of Hessian matrix // 2. we calculate Cauchy point XN // 3. in case difference between f(XN) and f(XC) is dominated by // numerical noise, we terminate algorithm. // 4. otherwise, we perform bounded step to XN // // This is the only phase were we can terminate algorithm // // NOTE: Cauchy point is calculated as minimizer of // g(t) = f(xc-t*pg) = t^2*(0.5*pg'*A*pg) + t*(-pg*g) + const // where pg is a projected gradient, t is a step length, g is // a non-projected gradient. // // NOTE 2: here we assume that our problem is convex, hence pg'*A*pg>=0, // and pg'*A*pg=0 if and only if pg=0. // minqpgrad(state, ref state.xc, ref state.gc); for(i_=0; i_<=n-1;i_++) { state.pg[i_] = state.gc[i_]; } minqpscalegrad(state, ref state.pg); optserv.projectgradientintobc(state.xc, ref state.pg, state.workbndl, state.havebndl, state.workbndu, state.havebndu, n, 0); minqptargetfunction(state, state.xc, ref v, ref vn0); v0 = minqpxtax2(state, state.pg); if( (double)(v0)>(double)(0) ) { // // Function curvature is positive. // Calculate predicted function decrease, which is equal to (pg'*g)^2/(4*pg'*A*pg). // In case step is dominated by noise, we exit. // v1 = 0.0; for(i_=0; i_<=n-1;i_++) { v1 += state.pg[i_]*state.gc[i_]; } if( (double)(math.sqr(v1)/(4*v0))<(double)(noisetolerance*vn0) ) { break; } v = v1/(2*v0); for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.xc[i_]; } for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.xn[i_] - v*state.pg[i_]; } minqpstepto(state, state.xc, ref state.xn, state.workbndl, state.havebndl, state.workbndu, state.havebndu, n, ref state.tmp0, ref state.tmp1, ref constraintsactivated, ref constraintsdeactivated, ref noisystep); for(i_=0; i_<=n-1;i_++) { state.xc[i_] = state.xn[i_]; } } else { // // non-positive curvature detected. // because it is assumed that function is positive definite, // it means that projected gradient is zero (termination is needed). // break; } // // Phase 3. // // Determine active set using current point location and // antigradient direction. Then perform Newton step. // // This phase allows us to leave current face. // minqpgrad(state, ref state.xc, ref state.gc); for(i=0; i<=n-1; i++) { state.activeset[i] = false; state.activeset[i] = state.activeset[i] || ((state.havebndl[i] && (double)(state.xc[i])==(double)(state.workbndl[i])) && (double)(state.gc[i])>(double)(0)); state.activeset[i] = state.activeset[i] || ((state.havebndu[i] && (double)(state.xc[i])==(double)(state.workbndu[i])) && (double)(state.gc[i])<(double)(0)); } if( !minqppreparenewtonstep(state, state.xc, state.activeset, ref state.xn, n, ref state.bufa, ref state.tmp0, ref state.tmp1) ) { state.repterminationtype = -5; return; } state.repncholesky = state.repncholesky+1; minqpstepto(state, state.xc, ref state.xn, state.workbndl, state.havebndl, state.workbndu, state.havebndu, n, ref state.tmp0, ref state.tmp1, ref constraintsactivated, ref constraintsdeactivated, ref noisystep); for(i_=0; i_<=n-1;i_++) { state.xc[i_] = state.xn[i_]; } } // // Post-process: add XOrigin to XC // for(i=0; i<=n-1; i++) { if( state.havebndl[i] && (double)(state.xc[i])==(double)(state.workbndl[i]) ) { state.xc[i] = state.bndl[i]; continue; } if( state.havebndu[i] && (double)(state.xc[i])==(double)(state.workbndu[i]) ) { state.xc[i] = state.bndu[i]; continue; } state.xc[i] = apserv.boundval(state.xc[i]+state.xorigin[i], state.bndl[i], state.bndu[i]); } state.repterminationtype = 4; return; } } /************************************************************************* QP solver results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType, which contains completion code, and you may check another fields which contain another information about algorithm functioning. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpresults(minqpstate state, ref double[] x, minqpreport rep) { x = new double[0]; minqpresultsbuf(state, ref x, rep); } /************************************************************************* QP results Buffered implementation of MinQPResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpresultsbuf(minqpstate state, ref double[] x, minqpreport rep) { int i_ = 0; if( alglib.ap.len(x)0)"); n = state.n; for(k=0; k<=n-1; k++) { state.densea[k,k] = s[k]; } } /************************************************************************* Fast version of MinQPSetStartingPoint(), which doesn't check its arguments. For internal use only. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetstartingpointfast(minqpstate state, double[] x) { int n = 0; int i_ = 0; n = state.n; for(i_=0; i_<=n-1;i_++) { state.startx[i_] = x[i_]; } state.havex = true; } /************************************************************************* Fast version of MinQPSetOrigin(), which doesn't check its arguments. For internal use only. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minqpsetoriginfast(minqpstate state, double[] xorigin) { int n = 0; int i_ = 0; n = state.n; for(i_=0; i_<=n-1;i_++) { state.xorigin[i_] = xorigin[i_]; } } /************************************************************************* This function calculates gradient of quadratic function at XC and stores it in the GC. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ private static void minqpgrad(minqpstate state, ref double[] xc, ref double[] gc) { int n = 0; int i = 0; double v = 0; int i_ = 0; n = state.n; alglib.ap.assert(state.akind==-1 || state.akind==0, "MinQPGrad: internal error"); // // zero A // if( state.akind==-1 ) { for(i_=0; i_<=n-1;i_++) { gc[i_] = state.b[i_]; } return; } // // dense A // if( state.akind==0 ) { for(i_=0; i_<=n-1;i_++) { gc[i_] = state.b[i_]; } gc[0] = gc[0]+state.densea[0,0]*xc[0]; for(i=1; i<=n-1; i++) { v = 0.0; for(i_=0; i_<=i-1;i_++) { v += state.densea[i,i_]*xc[i_]; } gc[i] = gc[i]+v+state.densea[i,i]*xc[i]; v = xc[i]; for(i_=0; i_<=i-1;i_++) { gc[i_] = gc[i_] + v*state.densea[i,i_]; } } return; } } /************************************************************************* This function scales gradient by multiplying it by inverse of the diagonal of the Hessian matrix. It should be used to accelerate steepest descent phase of the QP solver. Although it is called "scale-grad", it can be called for any vector, whether it is gradient, anti-gradient, or just some vector. It MUST be called only for problems with SPD matrices. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ private static void minqpscalegrad(minqpstate state, ref double[] g) { int n = 0; int i = 0; n = state.n; alglib.ap.assert(state.akind==0, "MinQPGrad: internal error"); // // dense A // if( state.akind==0 ) { for(i=0; i<=n-1; i++) { g[i] = g[i]/state.densea[i,i]; } return; } } /************************************************************************* This function calculates 0.5*x'*A*x for given X. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ private static double minqpxtax2(minqpstate state, double[] x) { double result = 0; int n = 0; int i = 0; int j = 0; n = state.n; alglib.ap.assert(state.akind==-1 || state.akind==0, "MinQPXTAX: internal error"); result = 0; // // zero A // if( state.akind==-1 ) { result = 0.0; return result; } // // dense A // if( state.akind==0 ) { result = 0; for(i=0; i<=n-1; i++) { for(j=0; j<=i-1; j++) { result = result+state.densea[i,j]*x[i]*x[j]; } result = result+0.5*state.densea[i,i]*math.sqr(x[i]); } return result; } return result; } /************************************************************************* This function calculates 0.5*x'*A*x+b*x for given X. INPUT PARAMETERS: State - quadratic model X - point of calculation OUTPUT PARAMETERS: F - function value Noise - estimate of the numerical noise level -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/ private static void minqptargetfunction(minqpstate state, double[] x, ref double f, ref double noise) { int n = 0; int i = 0; int j = 0; double v = 0; f = 0; noise = 0; n = state.n; alglib.ap.assert(state.akind==-1 || state.akind==0, "MinQPXTAX: internal error"); // // zero A // if( state.akind==-1 ) { f = 0.0; noise = 0.0; // // Linear term // for(i=0; i<=n-1; i++) { v = state.b[i]*x[i]; f = f+v; noise = Math.Max(noise, Math.Abs(v)); } // // End // noise = math.machineepsilon*n*noise; return; } // // dense A // if( state.akind==0 ) { f = 0; noise = 0; // // Quadratic term // for(i=0; i<=n-1; i++) { for(j=0; j<=i-1; j++) { v = state.densea[i,j]*x[i]*x[j]; f = f+v; noise = Math.Max(noise, Math.Abs(v)); } v = 0.5*state.densea[i,i]*math.sqr(x[i]); f = f+v; noise = Math.Max(noise, Math.Abs(v)); } // // Linear term // for(i=0; i<=n-1; i++) { v = state.b[i]*x[i]; f = f+v; noise = Math.Max(noise, Math.Abs(v)); } // // End // noise = math.machineepsilon*n*noise; return; } } /************************************************************************* This function calculates Newton step subject to the set of the active equality constraints. INPUT PARAMETERS: State - MinQP state. This structure is used ONLY to get quadratic model. This function does NOT use other fields of the State structure (like temporary buffers or bounds). XC - current point ActiveSet- set of active equality constraints XN - preallocated buffer N - problem size BufA - temporary buffer (automatically reallocated, if needed) BufB - temporary buffer (automatically reallocated, if needed) OUTPUT PARAMETERS: XN - minimizer of the quadratic model subject to equality constraints. This point can be infeasible because full Newton step may lead us to infeasible area. RESULT: True, when Newton step was correctly calculated False, when this function was called for non-convex problem (matrix is not positive definite, Cholesky factorization failed) NOTE: this function correctly works when all constraints are active. -- ALGLIB -- Copyright 29.02.2012 by Bochkanov Sergey *************************************************************************/ private static bool minqppreparenewtonstep(minqpstate state, double[] xc, bool[] activeset, ref double[] xn, int n, ref double[,] bufa, ref double[] bufb, ref double[] bufc) { bool result = new bool(); int i = 0; int j = 0; int k = 0; int nfree = 0; alglib.ap.assert(state.akind==0, "MinQPNewtonStep: unexpected model type"); // // Determine XN - minimizer of the quadratic function subject to active constraints. // It is done in two substeps: // * first, we determine values of the constrained components // * then we determine unconstrained components by splitting x'*A*x+b'*x into // ( A0 A1 ) ( x0 ) ( x0 ) // ( x0' x1' )*( )*( ) + (b0' b1')* ( ), with fixed x1 and "free" x0, // ( A1' A2 ) ( x1 ) ( x1 ) // and seeking for minimizer x0_min = inv(A0)*(-A1*x1-b0). We store A0 into BufA, // and -A1*x1-b0 is stored into BufB. We make implicit permutation of variables // when storing these matrices/vectors and retreiving result into XN. // apserv.rmatrixsetlengthatleast(ref bufa, n, n); apserv.rvectorsetlengthatleast(ref bufb, n); for(i=0; i<=n-1; i++) { if( activeset[i] ) { xn[i] = xc[i]; } } nfree = 0; for(i=0; i<=n-1; i++) { if( !activeset[i] ) { bufb[nfree] = -state.b[i]; k = 0; for(j=0; j<=n-1; j++) { if( !activeset[j] ) { bufa[nfree,k] = state.densea[i,j]; k = k+1; } else { bufb[nfree] = bufb[nfree]-state.densea[i,j]*xn[j]; } } nfree = nfree+1; } } if( nfree>0 ) { if( !trfac.spdmatrixcholeskyrec(ref bufa, 0, nfree, true, ref bufc) ) { result = false; return result; } fbls.fblscholeskysolve(bufa, 1.0, nfree, true, ref bufb, ref bufc); k = 0; for(i=0; i<=n-1; i++) { if( !activeset[i] ) { xn[i] = bufb[k]; k = k+1; } } } result = true; return result; } /************************************************************************* This function performs step from XC to XN subject to bound constraints and on condition that function value decreases. User prepares two points: * XC, current point, feasible w.r.t. bound constraints * XN, candidate point, which minimizes function value in some sense (either it is Newton minimizer, or minimizer along steepest descent direction), subject to equality constraints active at XC. It can be infeasible w.r.t. other constraints - ones which were not active at XC. It is important that function MUST monotonically decrease all the way from XC to XN. This function: 1. compares quadratic model values at XN and XC; if f(XN)>=f(XC), it sets NoisyStep, sets XN to XC and returns 2. performs longest step from XC to XN until we activate constraints 3. handles the rest of the step by performing Armijo line search for f()=PROJ(XC_new+alpha*(XN-XC_new)) where PROJ() is a projection on bound constrained area, alpha is a search parameter (we start from alpha=1 and decrease it), and XC_new is a new point obtained at step 2. 4. New location is saved into XN. In case some constraints were activated or deactivated, we set corresponding flag variable. In case function value decrease is dominated by the numerical noise, we set NoisyStep flag. INPUT PARAMETERS: State - MinQP state. This structure is used ONLY to get quadratic model. This function does NOT use other fields of the State structure (like temporary buffers or bounds). XC - current point, must be feasible w.r.t. constraints XN - candidate point, can be infeasible BndL - lower bounds, array[N] (may contain -INF, when bound is not present) HaveBndL- array[N], if HaveBndL[i] is False, then i-th bound is not present BndU - array[NMain], upper bounds (may contain +INF, when bound is not present) HaveBndU- array[NMain], if HaveBndU[i] is False, then i-th bound is not present N - problem size Tmp0 - temporary buffer (automatically reallocated if needed) OUTPUT PARAMETERS: XN - new point, feasible w.r.t. bound constraints NOTE 1: it is important that XN is constrained only by constraints, which are active at XC. Any attempt to introduce additional constraints may lead to algorithm stagnation -- ALGLIB -- Copyright 29.02.2012 by Bochkanov Sergey *************************************************************************/ private static void minqpstepto(minqpstate state, double[] xc, ref double[] xn, double[] bndl, bool[] havebndl, double[] bndu, bool[] havebndu, int n, ref double[] tmp0, ref double[] tmp1, ref bool constraintsactivated, ref bool constraintsdeactivated, ref bool noisystep) { int variabletofreeze = 0; double valuetofreeze = 0; double maxsteplen = 0; double armijostep = 0; int armijocnt = 0; double v = 0; double v0 = 0; double v1 = 0; double vn0 = 0; double vn1 = 0; bool b0 = new bool(); bool b1 = new bool(); int i = 0; int i_ = 0; constraintsactivated = new bool(); constraintsdeactivated = new bool(); noisystep = new bool(); apserv.rvectorsetlengthatleast(ref tmp0, n); apserv.rvectorsetlengthatleast(ref tmp1, n); // // Compare function values at XC and XN // minqptargetfunction(state, xc, ref v0, ref v); minqptargetfunction(state, xn, ref v1, ref v); if( (double)(v1)>=(double)(v0) ) { constraintsactivated = false; constraintsdeactivated = false; noisystep = true; for(i_=0; i_<=n-1;i_++) { xn[i_] = xc[i_]; } return; } // // Perform bounded step (we activate at most one constraint during this phase) // for(i_=0; i_<=n-1;i_++) { tmp0[i_] = xn[i_]; } for(i_=0; i_<=n-1;i_++) { tmp0[i_] = tmp0[i_] - xc[i_]; } optserv.calculatestepbound(xc, tmp0, 1.0, bndl, havebndl, bndu, havebndu, n, 0, ref variabletofreeze, ref valuetofreeze, ref maxsteplen); if( variabletofreeze>=0 && (double)(maxsteplen)<=(double)(1) ) { // // Perform bounded step // for(i_=0; i_<=n-1;i_++) { tmp0[i_] = xn[i_]; } for(i_=0; i_<=n-1;i_++) { tmp0[i_] = tmp0[i_] - xc[i_]; } for(i_=0; i_<=n-1;i_++) { tmp0[i_] = maxsteplen*tmp0[i_]; } for(i_=0; i_<=n-1;i_++) { tmp0[i_] = tmp0[i_] + xc[i_]; } optserv.postprocessboundedstep(ref tmp0, xc, bndl, havebndl, bndu, havebndu, n, 0, variabletofreeze, valuetofreeze, maxsteplen, maxsteplen); } else { // // Perform full step // for(i_=0; i_<=n-1;i_++) { tmp0[i_] = xn[i_]; } } // // Tmp0 contains either XN - or intermediate point halfway between XC and XN. // In case it contains intermediate point, handle the rest using Armijo search and projections. // if( variabletofreeze>=0 && (double)(maxsteplen)<=(double)(1) ) { // // Armijo search. // After this block Tmp1 contains best point found so far (can be equal to Tmp0). // minqptargetfunction(state, tmp0, ref v0, ref v); armijostep = 1.0; armijocnt = 5; for(i_=0; i_<=n-1;i_++) { tmp1[i_] = tmp0[i_]; } while( armijocnt>0 ) { // // Store candidate point in Tmp1 // for(i_=0; i_<=n-1;i_++) { tmp1[i_] = xn[i_]; } for(i_=0; i_<=n-1;i_++) { tmp1[i_] = tmp1[i_] - tmp0[i_]; } for(i_=0; i_<=n-1;i_++) { tmp1[i_] = armijostep*tmp1[i_]; } for(i_=0; i_<=n-1;i_++) { tmp1[i_] = tmp1[i_] + tmp0[i_]; } for(i=0; i<=n-1; i++) { if( havebndl[i] && (double)(tmp1[i])<(double)(bndl[i]) ) { tmp1[i] = bndl[i]; } if( havebndu[i] && (double)(tmp1[i])>(double)(bndu[i]) ) { tmp1[i] = bndu[i]; } } // // Evaluate function // minqptargetfunction(state, tmp1, ref v1, ref v); if( (double)(v1)>=(double)(v0) ) { for(i_=0; i_<=n-1;i_++) { tmp1[i_] = tmp0[i_]; } } else { break; } // // Next iteration // armijocnt = armijocnt-1; armijostep = 0.5*armijostep; } for(i_=0; i_<=n-1;i_++) { xn[i_] = tmp1[i_]; } } else { // // No Armijo search // for(i_=0; i_<=n-1;i_++) { xn[i_] = tmp0[i_]; } } // // // Check constraints status // constraintsactivated = false; constraintsdeactivated = false; for(i=0; i<=n-1; i++) { b0 = (havebndl[i] && (double)(bndl[i])==(double)(xc[i])) || (havebndu[i] && (double)(bndu[i])==(double)(xc[i])); b1 = (havebndl[i] && (double)(bndl[i])==(double)(xn[i])) || (havebndu[i] && (double)(bndu[i])==(double)(xn[i])); constraintsactivated = constraintsactivated || (b1 && !b0); constraintsdeactivated = constraintsdeactivated || (b0 && !b1); } minqptargetfunction(state, xc, ref v0, ref vn0); minqptargetfunction(state, xn, ref v1, ref vn1); noisystep = (double)(v0-v1)<(double)(Math.Max(vn0, vn1)); } } public class minlm { /************************************************************************* Levenberg-Marquardt optimizer. This structure should be created using one of the MinLMCreate???() functions. You should not access its fields directly; use ALGLIB functions to work with it. *************************************************************************/ public class minlmstate { public int n; public int m; public double diffstep; public double epsg; public double epsf; public double epsx; public int maxits; public bool xrep; public double stpmax; public int maxmodelage; public bool makeadditers; public double[] x; public double f; public double[] fi; public double[,] j; public double[,] h; public double[] g; public bool needf; public bool needfg; public bool needfgh; public bool needfij; public bool needfi; public bool xupdated; public int algomode; public bool hasf; public bool hasfi; public bool hasg; public double[] xbase; public double fbase; public double[] fibase; public double[] gbase; public double[,] quadraticmodel; public double[] bndl; public double[] bndu; public bool[] havebndl; public bool[] havebndu; public double[] s; public double lambdav; public double nu; public int modelage; public double[] xdir; public double[] deltax; public double[] deltaf; public bool deltaxready; public bool deltafready; public int repiterationscount; public int repterminationtype; public int repnfunc; public int repnjac; public int repngrad; public int repnhess; public int repncholesky; public rcommstate rstate; public double[] choleskybuf; public double[] tmp0; public double actualdecrease; public double predicteddecrease; public double xm1; public double xp1; public double[] fm1; public double[] fp1; public minlbfgs.minlbfgsstate internalstate; public minlbfgs.minlbfgsreport internalrep; public minqp.minqpstate qpstate; public minqp.minqpreport qprep; public minlmstate() { x = new double[0]; fi = new double[0]; j = new double[0,0]; h = new double[0,0]; g = new double[0]; xbase = new double[0]; fibase = new double[0]; gbase = new double[0]; quadraticmodel = new double[0,0]; bndl = new double[0]; bndu = new double[0]; havebndl = new bool[0]; havebndu = new bool[0]; s = new double[0]; xdir = new double[0]; deltax = new double[0]; deltaf = new double[0]; rstate = new rcommstate(); choleskybuf = new double[0]; tmp0 = new double[0]; fm1 = new double[0]; fp1 = new double[0]; internalstate = new minlbfgs.minlbfgsstate(); internalrep = new minlbfgs.minlbfgsreport(); qpstate = new minqp.minqpstate(); qprep = new minqp.minqpreport(); } }; /************************************************************************* Optimization report, filled by MinLMResults() function FIELDS: * TerminationType, completetion code: * -9 derivative correctness check failed; see Rep.WrongNum, Rep.WrongI, Rep.WrongJ for more information. * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient is no more than EpsG. * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * IterationsCount, contains iterations count * NFunc, number of function calculations * NJac, number of Jacobi matrix calculations * NGrad, number of gradient calculations * NHess, number of Hessian calculations * NCholesky, number of Cholesky decomposition calculations *************************************************************************/ public class minlmreport { public int iterationscount; public int terminationtype; public int nfunc; public int njac; public int ngrad; public int nhess; public int ncholesky; }; public const int lmmodefj = 0; public const int lmmodefgj = 1; public const int lmmodefgh = 2; public const int lmflagnoprelbfgs = 1; public const int lmflagnointlbfgs = 2; public const int lmprelbfgsm = 5; public const int lmintlbfgsits = 5; public const int lbfgsnorealloc = 1; public const double lambdaup = 2.0; public const double lambdadown = 0.33; public const double suspiciousnu = 16; public const int smallmodelage = 3; public const int additers = 5; /************************************************************************* IMPROVED LEVENBERG-MARQUARDT METHOD FOR NON-LINEAR LEAST SQUARES OPTIMIZATION DESCRIPTION: This function is used to find minimum of function which is represented as sum of squares: F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1]) using value of function vector f[] and Jacobian of f[]. REQUIREMENTS: This algorithm will request following information during its operation: * function vector f[] at given point X * function vector f[] and Jacobian of f[] (simultaneously) at given point There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts fvec() and jac() callbacks. First one is used to calculate f[] at given point, second one calculates f[] and Jacobian df[i]/dx[j]. You can try to initialize MinLMState structure with VJ function and then use incorrect version of MinLMOptimize() (for example, version which works with general form function and does not provide Jacobian), but it will lead to exception being thrown after first attempt to calculate Jacobian. USAGE: 1. User initializes algorithm state with MinLMCreateVJ() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of functions f[i] X - initial solution, array[0..N-1] OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatevj(int n, int m, double[] x, minlmstate state) { alglib.ap.assert(n>=1, "MinLMCreateVJ: N<1!"); alglib.ap.assert(m>=1, "MinLMCreateVJ: M<1!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinLMCreateVJ: Length(X)1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of functions f[i] X - initial solution, array[0..N-1] DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state See also MinLMIteration, MinLMResults. NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatev(int n, int m, double[] x, double diffstep, minlmstate state) { alglib.ap.assert(math.isfinite(diffstep), "MinLMCreateV: DiffStep is not finite!"); alglib.ap.assert((double)(diffstep)>(double)(0), "MinLMCreateV: DiffStep<=0!"); alglib.ap.assert(n>=1, "MinLMCreateV: N<1!"); alglib.ap.assert(m>=1, "MinLMCreateV: M<1!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinLMCreateV: Length(X)1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - initial solution, array[0..N-1] OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmcreatefgh(int n, double[] x, minlmstate state) { alglib.ap.assert(n>=1, "MinLMCreateFGH: N<1!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinLMCreateFGH: Length(X)=0 The subroutine finishes its work if the condition |v|=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinLMSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Only Levenberg-Marquardt iterations are counted (L-BFGS/CG iterations are NOT counted because their cost is very low compared to that of LM). Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetcond(minlmstate state, double epsg, double epsf, double epsx, int maxits) { alglib.ap.assert(math.isfinite(epsg), "MinLMSetCond: EpsG is not finite number!"); alglib.ap.assert((double)(epsg)>=(double)(0), "MinLMSetCond: negative EpsG!"); alglib.ap.assert(math.isfinite(epsf), "MinLMSetCond: EpsF is not finite number!"); alglib.ap.assert((double)(epsf)>=(double)(0), "MinLMSetCond: negative EpsF!"); alglib.ap.assert(math.isfinite(epsx), "MinLMSetCond: EpsX is not finite number!"); alglib.ap.assert((double)(epsx)>=(double)(0), "MinLMSetCond: negative EpsX!"); alglib.ap.assert(maxits>=0, "MinLMSetCond: negative MaxIts!"); if( (((double)(epsg)==(double)(0) && (double)(epsf)==(double)(0)) && (double)(epsx)==(double)(0)) && maxits==0 ) { epsx = 1.0E-6; } state.epsg = epsg; state.epsf = epsf; state.epsx = epsx; state.maxits = maxits; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinLMOptimize(). Both Levenberg-Marquardt and internal L-BFGS iterations are reported. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetxrep(minlmstate state, bool needxrep) { state.xrep = needxrep; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. NOTE: non-zero StpMax leads to moderate performance degradation because intermediate step of preconditioned L-BFGS optimization is incompatible with limits on step size. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minlmsetstpmax(minlmstate state, double stpmax) { alglib.ap.assert(math.isfinite(stpmax), "MinLMSetStpMax: StpMax is not finite!"); alglib.ap.assert((double)(stpmax)>=(double)(0), "MinLMSetStpMax: StpMax<0!"); state.stpmax = stpmax; } /************************************************************************* This function sets scaling coefficients for LM optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Generally, scale is NOT considered to be a form of preconditioner. But LM optimizer is unique in that it uses scaling matrix both in the stopping condition tests and as Marquardt damping factor. Proper scaling is very important for the algorithm performance. It is less important for the quality of results, but still has some influence (it is easier to converge when variables are properly scaled, so premature stopping is possible when very badly scalled variables are combined with relaxed stopping conditions). INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void minlmsetscale(minlmstate state, double[] s) { int i = 0; alglib.ap.assert(alglib.ap.len(s)>=state.n, "MinLMSetScale: Length(S)=n, "MinLMSetBC: Length(BndL)=n, "MinLMSetBC: Length(BndU)=0 ) { n = state.rstate.ia[0]; m = state.rstate.ia[1]; iflag = state.rstate.ia[2]; i = state.rstate.ia[3]; k = state.rstate.ia[4]; bflag = state.rstate.ba[0]; v = state.rstate.ra[0]; s = state.rstate.ra[1]; t = state.rstate.ra[2]; } else { n = -983; m = -989; iflag = -834; i = 900; k = -287; bflag = false; v = 214; s = -338; t = -686; } if( state.rstate.stage==0 ) { goto lbl_0; } if( state.rstate.stage==1 ) { goto lbl_1; } if( state.rstate.stage==2 ) { goto lbl_2; } if( state.rstate.stage==3 ) { goto lbl_3; } if( state.rstate.stage==4 ) { goto lbl_4; } if( state.rstate.stage==5 ) { goto lbl_5; } if( state.rstate.stage==6 ) { goto lbl_6; } if( state.rstate.stage==7 ) { goto lbl_7; } if( state.rstate.stage==8 ) { goto lbl_8; } if( state.rstate.stage==9 ) { goto lbl_9; } if( state.rstate.stage==10 ) { goto lbl_10; } if( state.rstate.stage==11 ) { goto lbl_11; } if( state.rstate.stage==12 ) { goto lbl_12; } if( state.rstate.stage==13 ) { goto lbl_13; } if( state.rstate.stage==14 ) { goto lbl_14; } if( state.rstate.stage==15 ) { goto lbl_15; } // // Routine body // // // prepare // n = state.n; m = state.m; state.repiterationscount = 0; state.repterminationtype = 0; state.repnfunc = 0; state.repnjac = 0; state.repngrad = 0; state.repnhess = 0; state.repncholesky = 0; // // check consistency of constraints, // enforce feasibility of the solution // set constraints // if( !optserv.enforceboundaryconstraints(ref state.xbase, state.bndl, state.havebndl, state.bndu, state.havebndu, n, 0) ) { state.repterminationtype = -3; result = false; return result; } minqp.minqpsetbc(state.qpstate, state.bndl, state.bndu); // // Initial report of current point // // Note 1: we rewrite State.X twice because // user may accidentally change it after first call. // // Note 2: we set NeedF or NeedFI depending on what // information about function we have. // if( !state.xrep ) { goto lbl_16; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); if( !state.hasf ) { goto lbl_18; } state.needf = true; state.rstate.stage = 0; goto lbl_rcomm; lbl_0: state.needf = false; goto lbl_19; lbl_18: alglib.ap.assert(state.hasfi, "MinLM: internal error 2!"); state.needfi = true; state.rstate.stage = 1; goto lbl_rcomm; lbl_1: state.needfi = false; v = 0.0; for(i_=0; i_<=m-1;i_++) { v += state.fi[i_]*state.fi[i_]; } state.f = v; lbl_19: state.repnfunc = state.repnfunc+1; for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 2; goto lbl_rcomm; lbl_2: state.xupdated = false; lbl_16: // // Prepare control variables // state.nu = 1; state.lambdav = -math.maxrealnumber; state.modelage = state.maxmodelage+1; state.deltaxready = false; state.deltafready = false; // // Main cycle. // // We move through it until either: // * one of the stopping conditions is met // * we decide that stopping conditions are too stringent // and break from cycle // // lbl_20: if( false ) { goto lbl_21; } // // First, we have to prepare quadratic model for our function. // We use BFlag to ensure that model is prepared; // if it is false at the end of this block, something went wrong. // // We may either calculate brand new model or update old one. // // Before this block we have: // * State.XBase - current position. // * State.DeltaX - if DeltaXReady is True // * State.DeltaF - if DeltaFReady is True // // After this block is over, we will have: // * State.XBase - base point (unchanged) // * State.FBase - F(XBase) // * State.GBase - linear term // * State.QuadraticModel - quadratic term // * State.LambdaV - current estimate for lambda // // We also clear DeltaXReady/DeltaFReady flags // after initialization is done. // bflag = false; if( !(state.algomode==0 || state.algomode==1) ) { goto lbl_22; } // // Calculate f[] and Jacobian // if( !(state.modelage>state.maxmodelage || !(state.deltaxready && state.deltafready)) ) { goto lbl_24; } // // Refresh model (using either finite differences or analytic Jacobian) // if( state.algomode!=0 ) { goto lbl_26; } // // Optimization using F values only. // Use finite differences to estimate Jacobian. // alglib.ap.assert(state.hasfi, "MinLMIteration: internal error when estimating Jacobian (no f[])"); k = 0; lbl_28: if( k>n-1 ) { goto lbl_30; } // // We guard X[k] from leaving [BndL,BndU]. // In case BndL=BndU, we assume that derivative in this direction is zero. // for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } state.x[k] = state.x[k]-state.s[k]*state.diffstep; if( state.havebndl[k] ) { state.x[k] = Math.Max(state.x[k], state.bndl[k]); } if( state.havebndu[k] ) { state.x[k] = Math.Min(state.x[k], state.bndu[k]); } state.xm1 = state.x[k]; clearrequestfields(state); state.needfi = true; state.rstate.stage = 3; goto lbl_rcomm; lbl_3: state.repnfunc = state.repnfunc+1; for(i_=0; i_<=m-1;i_++) { state.fm1[i_] = state.fi[i_]; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } state.x[k] = state.x[k]+state.s[k]*state.diffstep; if( state.havebndl[k] ) { state.x[k] = Math.Max(state.x[k], state.bndl[k]); } if( state.havebndu[k] ) { state.x[k] = Math.Min(state.x[k], state.bndu[k]); } state.xp1 = state.x[k]; clearrequestfields(state); state.needfi = true; state.rstate.stage = 4; goto lbl_rcomm; lbl_4: state.repnfunc = state.repnfunc+1; for(i_=0; i_<=m-1;i_++) { state.fp1[i_] = state.fi[i_]; } v = state.xp1-state.xm1; if( (double)(v)!=(double)(0) ) { v = 1/v; for(i_=0; i_<=m-1;i_++) { state.j[i_,k] = v*state.fp1[i_]; } for(i_=0; i_<=m-1;i_++) { state.j[i_,k] = state.j[i_,k] - v*state.fm1[i_]; } } else { for(i=0; i<=m-1; i++) { state.j[i,k] = 0; } } k = k+1; goto lbl_28; lbl_30: // // Calculate F(XBase) // for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); state.needfi = true; state.rstate.stage = 5; goto lbl_rcomm; lbl_5: state.needfi = false; state.repnfunc = state.repnfunc+1; state.repnjac = state.repnjac+1; // // New model // state.modelage = 0; goto lbl_27; lbl_26: // // Obtain f[] and Jacobian // for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); state.needfij = true; state.rstate.stage = 6; goto lbl_rcomm; lbl_6: state.needfij = false; state.repnfunc = state.repnfunc+1; state.repnjac = state.repnjac+1; // // New model // state.modelage = 0; lbl_27: goto lbl_25; lbl_24: // // State.J contains Jacobian or its current approximation; // refresh it using secant updates: // // f(x0+dx) = f(x0) + J*dx, // J_new = J_old + u*h' // h = x_new-x_old // u = (f_new - f_old - J_old*h)/(h'h) // // We can explicitly generate h and u, but it is // preferential to do in-place calculations. Only // I-th row of J_old is needed to calculate u[I], // so we can update J row by row in one pass. // // NOTE: we expect that State.XBase contains new point, // State.FBase contains old point, State.DeltaX and // State.DeltaY contain updates from last step. // alglib.ap.assert(state.deltaxready && state.deltafready, "MinLMIteration: uninitialized DeltaX/DeltaF"); t = 0.0; for(i_=0; i_<=n-1;i_++) { t += state.deltax[i_]*state.deltax[i_]; } alglib.ap.assert((double)(t)!=(double)(0), "MinLM: internal error (T=0)"); for(i=0; i<=m-1; i++) { v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.j[i,i_]*state.deltax[i_]; } v = (state.deltaf[i]-v)/t; for(i_=0; i_<=n-1;i_++) { state.j[i,i_] = state.j[i,i_] + v*state.deltax[i_]; } } for(i_=0; i_<=m-1;i_++) { state.fi[i_] = state.fibase[i_]; } for(i_=0; i_<=m-1;i_++) { state.fi[i_] = state.fi[i_] + state.deltaf[i_]; } // // Increase model age // state.modelage = state.modelage+1; lbl_25: // // Generate quadratic model: // f(xbase+dx) = // = (f0 + J*dx)'(f0 + J*dx) // = f0^2 + dx'J'f0 + f0*J*dx + dx'J'J*dx // = f0^2 + 2*f0*J*dx + dx'J'J*dx // // Note that we calculate 2*(J'J) instead of J'J because // our quadratic model is based on Tailor decomposition, // i.e. it has 0.5 before quadratic term. // ablas.rmatrixgemm(n, n, m, 2.0, state.j, 0, 0, 1, state.j, 0, 0, 0, 0.0, ref state.quadraticmodel, 0, 0); ablas.rmatrixmv(n, m, state.j, 0, 0, 1, state.fi, 0, ref state.gbase, 0); for(i_=0; i_<=n-1;i_++) { state.gbase[i_] = 2*state.gbase[i_]; } v = 0.0; for(i_=0; i_<=m-1;i_++) { v += state.fi[i_]*state.fi[i_]; } state.fbase = v; for(i_=0; i_<=m-1;i_++) { state.fibase[i_] = state.fi[i_]; } // // set control variables // bflag = true; lbl_22: if( state.algomode!=2 ) { goto lbl_31; } alglib.ap.assert(!state.hasfi, "MinLMIteration: internal error (HasFI is True in Hessian-based mode)"); // // Obtain F, G, H // for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); state.needfgh = true; state.rstate.stage = 7; goto lbl_rcomm; lbl_7: state.needfgh = false; state.repnfunc = state.repnfunc+1; state.repngrad = state.repngrad+1; state.repnhess = state.repnhess+1; ablas.rmatrixcopy(n, n, state.h, 0, 0, ref state.quadraticmodel, 0, 0); for(i_=0; i_<=n-1;i_++) { state.gbase[i_] = state.g[i_]; } state.fbase = state.f; // // set control variables // bflag = true; state.modelage = 0; lbl_31: alglib.ap.assert(bflag, "MinLM: internal integrity check failed!"); state.deltaxready = false; state.deltafready = false; // // If Lambda is not initialized, initialize it using quadratic model // if( (double)(state.lambdav)<(double)(0) ) { state.lambdav = 0; for(i=0; i<=n-1; i++) { state.lambdav = Math.Max(state.lambdav, Math.Abs(state.quadraticmodel[i,i])*math.sqr(state.s[i])); } state.lambdav = 0.001*state.lambdav; if( (double)(state.lambdav)==(double)(0) ) { state.lambdav = 1; } } // // Test stopping conditions for function gradient // if( (double)(boundedscaledantigradnorm(state, state.xbase, state.gbase))>(double)(state.epsg) ) { goto lbl_33; } if( state.modelage!=0 ) { goto lbl_35; } // // Model is fresh, we can rely on it and terminate algorithm // state.repterminationtype = 4; if( !state.xrep ) { goto lbl_37; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } state.f = state.fbase; clearrequestfields(state); state.xupdated = true; state.rstate.stage = 8; goto lbl_rcomm; lbl_8: state.xupdated = false; lbl_37: result = false; return result; goto lbl_36; lbl_35: // // Model is not fresh, we should refresh it and test // conditions once more // state.modelage = state.maxmodelage+1; goto lbl_20; lbl_36: lbl_33: // // Find value of Levenberg-Marquardt damping parameter which: // * leads to positive definite damped model // * within bounds specified by StpMax // * generates step which decreases function value // // After this block IFlag is set to: // * -3, if constraints are infeasible // * -2, if model update is needed (either Lambda growth is too large // or step is too short, but we can't rely on model and stop iterations) // * -1, if model is fresh, Lambda have grown too large, termination is needed // * 0, if everything is OK, continue iterations // // State.Nu can have any value on enter, but after exit it is set to 1.0 // iflag = -99; lbl_39: if( false ) { goto lbl_40; } // // Do we need model update? // if( state.modelage>0 && (double)(state.nu)>=(double)(suspiciousnu) ) { iflag = -2; goto lbl_40; } // // Setup quadratic solver and solve quadratic programming problem. // After problem is solved we'll try to bound step by StpMax // (Lambda will be increased if step size is too large). // // We use BFlag variable to indicate that we have to increase Lambda. // If it is False, we will try to increase Lambda and move to new iteration. // bflag = true; minqp.minqpsetstartingpointfast(state.qpstate, state.xbase); minqp.minqpsetoriginfast(state.qpstate, state.xbase); minqp.minqpsetlineartermfast(state.qpstate, state.gbase); minqp.minqpsetquadratictermfast(state.qpstate, state.quadraticmodel, true, 0.0); for(i=0; i<=n-1; i++) { state.tmp0[i] = state.quadraticmodel[i,i]+state.lambdav/math.sqr(state.s[i]); } minqp.minqprewritediagonal(state.qpstate, state.tmp0); minqp.minqpoptimize(state.qpstate); minqp.minqpresultsbuf(state.qpstate, ref state.xdir, state.qprep); if( state.qprep.terminationtype>0 ) { // // successful solution of QP problem // for(i_=0; i_<=n-1;i_++) { state.xdir[i_] = state.xdir[i_] - state.xbase[i_]; } v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.xdir[i_]*state.xdir[i_]; } if( math.isfinite(v) ) { v = Math.Sqrt(v); if( (double)(state.stpmax)>(double)(0) && (double)(v)>(double)(state.stpmax) ) { bflag = false; } } else { bflag = false; } } else { // // Either problem is non-convex (increase LambdaV) or constraints are inconsistent // alglib.ap.assert(state.qprep.terminationtype==-3 || state.qprep.terminationtype==-5, "MinLM: unexpected completion code from QP solver"); if( state.qprep.terminationtype==-3 ) { iflag = -3; goto lbl_40; } bflag = false; } if( !bflag ) { // // Solution failed: // try to increase lambda to make matrix positive definite and continue. // if( !increaselambda(ref state.lambdav, ref state.nu) ) { iflag = -1; goto lbl_40; } goto lbl_39; } // // Step in State.XDir and it is bounded by StpMax. // // We should check stopping conditions on step size here. // DeltaX, which is used for secant updates, is initialized here. // // This code is a bit tricky because sometimes XDir<>0, but // it is so small that XDir+XBase==XBase (in finite precision // arithmetics). So we set DeltaX to XBase, then // add XDir, and then subtract XBase to get exact value of // DeltaX. // // Step length is estimated using DeltaX. // // NOTE: stopping conditions are tested // for fresh models only (ModelAge=0) // for(i_=0; i_<=n-1;i_++) { state.deltax[i_] = state.xbase[i_]; } for(i_=0; i_<=n-1;i_++) { state.deltax[i_] = state.deltax[i_] + state.xdir[i_]; } for(i_=0; i_<=n-1;i_++) { state.deltax[i_] = state.deltax[i_] - state.xbase[i_]; } state.deltaxready = true; v = 0.0; for(i=0; i<=n-1; i++) { v = v+math.sqr(state.deltax[i]/state.s[i]); } v = Math.Sqrt(v); if( (double)(v)>(double)(state.epsx) ) { goto lbl_41; } if( state.modelage!=0 ) { goto lbl_43; } // // Step is too short, model is fresh and we can rely on it. // Terminating. // state.repterminationtype = 2; if( !state.xrep ) { goto lbl_45; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } state.f = state.fbase; clearrequestfields(state); state.xupdated = true; state.rstate.stage = 9; goto lbl_rcomm; lbl_9: state.xupdated = false; lbl_45: result = false; return result; goto lbl_44; lbl_43: // // Step is suspiciously short, but model is not fresh // and we can't rely on it. // iflag = -2; goto lbl_40; lbl_44: lbl_41: // // Let's evaluate new step: // a) if we have Fi vector, we evaluate it using rcomm, and // then we manually calculate State.F as sum of squares of Fi[] // b) if we have F value, we just evaluate it through rcomm interface // // We prefer (a) because we may need Fi vector for additional // iterations // alglib.ap.assert(state.hasfi || state.hasf, "MinLM: internal error 2!"); for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.x[i_] + state.xdir[i_]; } clearrequestfields(state); if( !state.hasfi ) { goto lbl_47; } state.needfi = true; state.rstate.stage = 10; goto lbl_rcomm; lbl_10: state.needfi = false; v = 0.0; for(i_=0; i_<=m-1;i_++) { v += state.fi[i_]*state.fi[i_]; } state.f = v; for(i_=0; i_<=m-1;i_++) { state.deltaf[i_] = state.fi[i_]; } for(i_=0; i_<=m-1;i_++) { state.deltaf[i_] = state.deltaf[i_] - state.fibase[i_]; } state.deltafready = true; goto lbl_48; lbl_47: state.needf = true; state.rstate.stage = 11; goto lbl_rcomm; lbl_11: state.needf = false; lbl_48: state.repnfunc = state.repnfunc+1; if( (double)(state.f)>=(double)(state.fbase) ) { // // Increase lambda and continue // if( !increaselambda(ref state.lambdav, ref state.nu) ) { iflag = -1; goto lbl_40; } goto lbl_39; } // // We've found our step! // iflag = 0; goto lbl_40; goto lbl_39; lbl_40: state.nu = 1; alglib.ap.assert(iflag>=-3 && iflag<=0, "MinLM: internal integrity check failed!"); if( iflag==-3 ) { state.repterminationtype = -3; result = false; return result; } if( iflag==-2 ) { state.modelage = state.maxmodelage+1; goto lbl_20; } if( iflag==-1 ) { goto lbl_21; } // // Levenberg-Marquardt step is ready. // Compare predicted vs. actual decrease and decide what to do with lambda. // // NOTE: we expect that State.DeltaX contains direction of step, // State.F contains function value at new point. // alglib.ap.assert(state.deltaxready, "MinLM: deltaX is not ready"); t = 0; for(i=0; i<=n-1; i++) { v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.quadraticmodel[i,i_]*state.deltax[i_]; } t = t+state.deltax[i]*state.gbase[i]+0.5*state.deltax[i]*v; } state.predicteddecrease = -t; state.actualdecrease = -(state.f-state.fbase); if( (double)(state.predicteddecrease)<=(double)(0) ) { goto lbl_21; } v = state.actualdecrease/state.predicteddecrease; if( (double)(v)>=(double)(0.1) ) { goto lbl_49; } if( increaselambda(ref state.lambdav, ref state.nu) ) { goto lbl_51; } // // Lambda is too large, we have to break iterations. // state.repterminationtype = 7; if( !state.xrep ) { goto lbl_53; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } state.f = state.fbase; clearrequestfields(state); state.xupdated = true; state.rstate.stage = 12; goto lbl_rcomm; lbl_12: state.xupdated = false; lbl_53: result = false; return result; lbl_51: lbl_49: if( (double)(v)>(double)(0.5) ) { decreaselambda(ref state.lambdav, ref state.nu); } // // Accept step, report it and // test stopping conditions on iterations count and function decrease. // // NOTE: we expect that State.DeltaX contains direction of step, // State.F contains function value at new point. // // NOTE2: we should update XBase ONLY. In the beginning of the next // iteration we expect that State.FIBase is NOT updated and // contains old value of a function vector. // for(i_=0; i_<=n-1;i_++) { state.xbase[i_] = state.xbase[i_] + state.deltax[i_]; } if( !state.xrep ) { goto lbl_55; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 13; goto lbl_rcomm; lbl_13: state.xupdated = false; lbl_55: state.repiterationscount = state.repiterationscount+1; if( state.repiterationscount>=state.maxits && state.maxits>0 ) { state.repterminationtype = 5; } if( state.modelage==0 ) { if( (double)(Math.Abs(state.f-state.fbase))<=(double)(state.epsf*Math.Max(1, Math.Max(Math.Abs(state.f), Math.Abs(state.fbase)))) ) { state.repterminationtype = 1; } } if( state.repterminationtype<=0 ) { goto lbl_57; } if( !state.xrep ) { goto lbl_59; } // // Report: XBase contains new point, F contains function value at new point // for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 14; goto lbl_rcomm; lbl_14: state.xupdated = false; lbl_59: result = false; return result; lbl_57: state.modelage = state.modelage+1; goto lbl_20; lbl_21: // // Lambda is too large, we have to break iterations. // state.repterminationtype = 7; if( !state.xrep ) { goto lbl_61; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xbase[i_]; } state.f = state.fbase; clearrequestfields(state); state.xupdated = true; state.rstate.stage = 15; goto lbl_rcomm; lbl_15: state.xupdated = false; lbl_61: result = false; return result; // // Saving state // lbl_rcomm: result = true; state.rstate.ia[0] = n; state.rstate.ia[1] = m; state.rstate.ia[2] = iflag; state.rstate.ia[3] = i; state.rstate.ia[4] = k; state.rstate.ba[0] = bflag; state.rstate.ra[0] = v; state.rstate.ra[1] = s; state.rstate.ra[2] = t; return result; } /************************************************************************* Levenberg-Marquardt algorithm results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report; see comments for this structure for more info. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmresults(minlmstate state, ref double[] x, minlmreport rep) { x = new double[0]; minlmresultsbuf(state, ref x, rep); } /************************************************************************* Levenberg-Marquardt algorithm results Buffered implementation of MinLMResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minlmresultsbuf(minlmstate state, ref double[] x, minlmreport rep) { int i_ = 0; if( alglib.ap.len(x)=state.n, "MinLMRestartFrom: Length(X)=1, "MinLMCreateFJ: N<1!"); alglib.ap.assert(m>=1, "MinLMCreateFJ: M<1!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinLMCreateFJ: Length(X)(double)(0.25*lnmax) ) { return result; } if( (double)(lnnu+Math.Log(2))>(double)(lnmax) ) { return result; } lambdav = lambdav*lambdaup*nu; nu = nu*2; result = true; return result; } /************************************************************************* Decreases lambda, but leaves it unchanged when there is danger of underflow. *************************************************************************/ private static void decreaselambda(ref double lambdav, ref double nu) { nu = 1; if( (double)(Math.Log(lambdav)+Math.Log(lambdadown))<(double)(Math.Log(math.minrealnumber)) ) { lambdav = math.minrealnumber; } else { lambdav = lambdav*lambdadown; } } /************************************************************************* Returns norm of bounded scaled anti-gradient. Bounded antigradient is a vector obtained from anti-gradient by zeroing components which point outwards: result = norm(v) v[i]=0 if ((-g[i]<0)and(x[i]=bndl[i])) or ((-g[i]>0)and(x[i]=bndu[i])) v[i]=-g[i]*s[i] otherwise, where s[i] is a scale for I-th variable This function may be used to check a stopping criterion. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ private static double boundedscaledantigradnorm(minlmstate state, double[] x, double[] g) { double result = 0; int n = 0; int i = 0; double v = 0; result = 0; n = state.n; for(i=0; i<=n-1; i++) { v = -(g[i]*state.s[i]); if( state.havebndl[i] ) { if( (double)(x[i])<=(double)(state.bndl[i]) && (double)(-g[i])<(double)(0) ) { v = 0; } } if( state.havebndu[i] ) { if( (double)(x[i])>=(double)(state.bndu[i]) && (double)(-g[i])>(double)(0) ) { v = 0; } } result = result+math.sqr(v); } result = Math.Sqrt(result); return result; } } public class mincomp { public class minasastate { public int n; public double epsg; public double epsf; public double epsx; public int maxits; public bool xrep; public double stpmax; public int cgtype; public int k; public int nfev; public int mcstage; public double[] bndl; public double[] bndu; public int curalgo; public int acount; public double mu; public double finit; public double dginit; public double[] ak; public double[] xk; public double[] dk; public double[] an; public double[] xn; public double[] dn; public double[] d; public double fold; public double stp; public double[] work; public double[] yk; public double[] gc; public double laststep; public double[] x; public double f; public double[] g; public bool needfg; public bool xupdated; public rcommstate rstate; public int repiterationscount; public int repnfev; public int repterminationtype; public int debugrestartscount; public linmin.linminstate lstate; public double betahs; public double betady; public minasastate() { bndl = new double[0]; bndu = new double[0]; ak = new double[0]; xk = new double[0]; dk = new double[0]; an = new double[0]; xn = new double[0]; dn = new double[0]; d = new double[0]; work = new double[0]; yk = new double[0]; gc = new double[0]; x = new double[0]; g = new double[0]; rstate = new rcommstate(); lstate = new linmin.linminstate(); } }; public class minasareport { public int iterationscount; public int nfev; public int terminationtype; public int activeconstraints; }; public const int n1 = 2; public const int n2 = 2; public const double stpmin = 1.0E-300; public const double gtol = 0.3; public const double gpaftol = 0.0001; public const double gpadecay = 0.5; public const double asarho = 0.5; /************************************************************************* Obsolete function, use MinLBFGSSetPrecDefault() instead. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetdefaultpreconditioner(minlbfgs.minlbfgsstate state) { minlbfgs.minlbfgssetprecdefault(state); } /************************************************************************* Obsolete function, use MinLBFGSSetCholeskyPreconditioner() instead. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/ public static void minlbfgssetcholeskypreconditioner(minlbfgs.minlbfgsstate state, double[,] p, bool isupper) { minlbfgs.minlbfgssetpreccholesky(state, p, isupper); } /************************************************************************* This is obsolete function which was used by previous version of the BLEIC optimizer. It does nothing in the current version of BLEIC. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetbarrierwidth(minbleic.minbleicstate state, double mu) { } /************************************************************************* This is obsolete function which was used by previous version of the BLEIC optimizer. It does nothing in the current version of BLEIC. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/ public static void minbleicsetbarrierdecay(minbleic.minbleicstate state, double mudecay) { } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 25.03.2010 by Bochkanov Sergey *************************************************************************/ public static void minasacreate(int n, double[] x, double[] bndl, double[] bndu, minasastate state) { int i = 0; alglib.ap.assert(n>=1, "MinASA: N too small!"); alglib.ap.assert(alglib.ap.len(x)>=n, "MinCGCreate: Length(X)=n, "MinCGCreate: Length(BndL)=n, "MinCGCreate: Length(BndU)=(double)(0), "MinASASetCond: negative EpsG!"); alglib.ap.assert(math.isfinite(epsf), "MinASASetCond: EpsF is not finite number!"); alglib.ap.assert((double)(epsf)>=(double)(0), "MinASASetCond: negative EpsF!"); alglib.ap.assert(math.isfinite(epsx), "MinASASetCond: EpsX is not finite number!"); alglib.ap.assert((double)(epsx)>=(double)(0), "MinASASetCond: negative EpsX!"); alglib.ap.assert(maxits>=0, "MinASASetCond: negative MaxIts!"); if( (((double)(epsg)==(double)(0) && (double)(epsf)==(double)(0)) && (double)(epsx)==(double)(0)) && maxits==0 ) { epsx = 1.0E-6; } state.epsg = epsg; state.epsf = epsf; state.epsx = epsx; state.maxits = maxits; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetxrep(minasastate state, bool needxrep) { state.xrep = needxrep; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetalgorithm(minasastate state, int algotype) { alglib.ap.assert(algotype>=-1 && algotype<=1, "MinASASetAlgorithm: incorrect AlgoType!"); if( algotype==-1 ) { algotype = 1; } state.cgtype = algotype; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void minasasetstpmax(minasastate state, double stpmax) { alglib.ap.assert(math.isfinite(stpmax), "MinASASetStpMax: StpMax is not finite!"); alglib.ap.assert((double)(stpmax)>=(double)(0), "MinASASetStpMax: StpMax<0!"); state.stpmax = stpmax; } /************************************************************************* -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static bool minasaiteration(minasastate state) { bool result = new bool(); int n = 0; int i = 0; double betak = 0; double v = 0; double vv = 0; int mcinfo = 0; bool b = new bool(); bool stepfound = new bool(); int diffcnt = 0; int i_ = 0; // // Reverse communication preparations // I know it looks ugly, but it works the same way // anywhere from C++ to Python. // // This code initializes locals by: // * random values determined during code // generation - on first subroutine call // * values from previous call - on subsequent calls // if( state.rstate.stage>=0 ) { n = state.rstate.ia[0]; i = state.rstate.ia[1]; mcinfo = state.rstate.ia[2]; diffcnt = state.rstate.ia[3]; b = state.rstate.ba[0]; stepfound = state.rstate.ba[1]; betak = state.rstate.ra[0]; v = state.rstate.ra[1]; vv = state.rstate.ra[2]; } else { n = -983; i = -989; mcinfo = -834; diffcnt = 900; b = true; stepfound = false; betak = 214; v = -338; vv = -686; } if( state.rstate.stage==0 ) { goto lbl_0; } if( state.rstate.stage==1 ) { goto lbl_1; } if( state.rstate.stage==2 ) { goto lbl_2; } if( state.rstate.stage==3 ) { goto lbl_3; } if( state.rstate.stage==4 ) { goto lbl_4; } if( state.rstate.stage==5 ) { goto lbl_5; } if( state.rstate.stage==6 ) { goto lbl_6; } if( state.rstate.stage==7 ) { goto lbl_7; } if( state.rstate.stage==8 ) { goto lbl_8; } if( state.rstate.stage==9 ) { goto lbl_9; } if( state.rstate.stage==10 ) { goto lbl_10; } if( state.rstate.stage==11 ) { goto lbl_11; } if( state.rstate.stage==12 ) { goto lbl_12; } if( state.rstate.stage==13 ) { goto lbl_13; } if( state.rstate.stage==14 ) { goto lbl_14; } // // Routine body // // // Prepare // n = state.n; state.repterminationtype = 0; state.repiterationscount = 0; state.repnfev = 0; state.debugrestartscount = 0; state.cgtype = 1; for(i_=0; i_<=n-1;i_++) { state.xk[i_] = state.x[i_]; } for(i=0; i<=n-1; i++) { if( (double)(state.xk[i])==(double)(state.bndl[i]) || (double)(state.xk[i])==(double)(state.bndu[i]) ) { state.ak[i] = 0; } else { state.ak[i] = 1; } } state.mu = 0.1; state.curalgo = 0; // // Calculate F/G, initialize algorithm // clearrequestfields(state); state.needfg = true; state.rstate.stage = 0; goto lbl_rcomm; lbl_0: state.needfg = false; if( !state.xrep ) { goto lbl_15; } // // progress report // clearrequestfields(state); state.xupdated = true; state.rstate.stage = 1; goto lbl_rcomm; lbl_1: state.xupdated = false; lbl_15: if( (double)(asaboundedantigradnorm(state))<=(double)(state.epsg) ) { state.repterminationtype = 4; result = false; return result; } state.repnfev = state.repnfev+1; // // Main cycle // // At the beginning of new iteration: // * CurAlgo stores current algorithm selector // * State.XK, State.F and State.G store current X/F/G // * State.AK stores current set of active constraints // lbl_17: if( false ) { goto lbl_18; } // // GPA algorithm // if( state.curalgo!=0 ) { goto lbl_19; } state.k = 0; state.acount = 0; lbl_21: if( false ) { goto lbl_22; } // // Determine Dk = proj(xk - gk)-xk // for(i=0; i<=n-1; i++) { state.d[i] = apserv.boundval(state.xk[i]-state.g[i], state.bndl[i], state.bndu[i])-state.xk[i]; } // // Armijo line search. // * exact search with alpha=1 is tried first, // 'exact' means that we evaluate f() EXACTLY at // bound(x-g,bndl,bndu), without intermediate floating // point operations. // * alpha<1 are tried if explicit search wasn't successful // Result is placed into XN. // // Two types of search are needed because we can't // just use second type with alpha=1 because in finite // precision arithmetics (x1-x0)+x0 may differ from x1. // So while x1 is correctly bounded (it lie EXACTLY on // boundary, if it is active), (x1-x0)+x0 may be // not bounded. // v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.d[i_]*state.g[i_]; } state.dginit = v; state.finit = state.f; if( !((double)(asad1norm(state))<=(double)(state.stpmax) || (double)(state.stpmax)==(double)(0)) ) { goto lbl_23; } // // Try alpha=1 step first // for(i=0; i<=n-1; i++) { state.x[i] = apserv.boundval(state.xk[i]-state.g[i], state.bndl[i], state.bndu[i]); } clearrequestfields(state); state.needfg = true; state.rstate.stage = 2; goto lbl_rcomm; lbl_2: state.needfg = false; state.repnfev = state.repnfev+1; stepfound = (double)(state.f)<=(double)(state.finit+gpaftol*state.dginit); goto lbl_24; lbl_23: stepfound = false; lbl_24: if( !stepfound ) { goto lbl_25; } // // we are at the boundary(ies) // for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.x[i_]; } state.stp = 1; goto lbl_26; lbl_25: // // alpha=1 is too large, try smaller values // state.stp = 1; linmin.linminnormalized(ref state.d, ref state.stp, n); state.dginit = state.dginit/state.stp; state.stp = gpadecay*state.stp; if( (double)(state.stpmax)>(double)(0) ) { state.stp = Math.Min(state.stp, state.stpmax); } lbl_27: if( false ) { goto lbl_28; } v = state.stp; for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.xk[i_]; } for(i_=0; i_<=n-1;i_++) { state.x[i_] = state.x[i_] + v*state.d[i_]; } clearrequestfields(state); state.needfg = true; state.rstate.stage = 3; goto lbl_rcomm; lbl_3: state.needfg = false; state.repnfev = state.repnfev+1; if( (double)(state.stp)<=(double)(stpmin) ) { goto lbl_28; } if( (double)(state.f)<=(double)(state.finit+state.stp*gpaftol*state.dginit) ) { goto lbl_28; } state.stp = state.stp*gpadecay; goto lbl_27; lbl_28: for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.x[i_]; } lbl_26: state.repiterationscount = state.repiterationscount+1; if( !state.xrep ) { goto lbl_29; } // // progress report // clearrequestfields(state); state.xupdated = true; state.rstate.stage = 4; goto lbl_rcomm; lbl_4: state.xupdated = false; lbl_29: // // Calculate new set of active constraints. // Reset counter if active set was changed. // Prepare for the new iteration // for(i=0; i<=n-1; i++) { if( (double)(state.xn[i])==(double)(state.bndl[i]) || (double)(state.xn[i])==(double)(state.bndu[i]) ) { state.an[i] = 0; } else { state.an[i] = 1; } } for(i=0; i<=n-1; i++) { if( (double)(state.ak[i])!=(double)(state.an[i]) ) { state.acount = -1; break; } } state.acount = state.acount+1; for(i_=0; i_<=n-1;i_++) { state.xk[i_] = state.xn[i_]; } for(i_=0; i_<=n-1;i_++) { state.ak[i_] = state.an[i_]; } // // Stopping conditions // if( !(state.repiterationscount>=state.maxits && state.maxits>0) ) { goto lbl_31; } // // Too many iterations // state.repterminationtype = 5; if( !state.xrep ) { goto lbl_33; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 5; goto lbl_rcomm; lbl_5: state.xupdated = false; lbl_33: result = false; return result; lbl_31: if( (double)(asaboundedantigradnorm(state))>(double)(state.epsg) ) { goto lbl_35; } // // Gradient is small enough // state.repterminationtype = 4; if( !state.xrep ) { goto lbl_37; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 6; goto lbl_rcomm; lbl_6: state.xupdated = false; lbl_37: result = false; return result; lbl_35: v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.d[i_]*state.d[i_]; } if( (double)(Math.Sqrt(v)*state.stp)>(double)(state.epsx) ) { goto lbl_39; } // // Step size is too small, no further improvement is // possible // state.repterminationtype = 2; if( !state.xrep ) { goto lbl_41; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 7; goto lbl_rcomm; lbl_7: state.xupdated = false; lbl_41: result = false; return result; lbl_39: if( (double)(state.finit-state.f)>(double)(state.epsf*Math.Max(Math.Abs(state.finit), Math.Max(Math.Abs(state.f), 1.0))) ) { goto lbl_43; } // // F(k+1)-F(k) is small enough // state.repterminationtype = 1; if( !state.xrep ) { goto lbl_45; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 8; goto lbl_rcomm; lbl_8: state.xupdated = false; lbl_45: result = false; return result; lbl_43: // // Decide - should we switch algorithm or not // if( asauisempty(state) ) { if( (double)(asaginorm(state))>=(double)(state.mu*asad1norm(state)) ) { state.curalgo = 1; goto lbl_22; } else { state.mu = state.mu*asarho; } } else { if( state.acount==n1 ) { if( (double)(asaginorm(state))>=(double)(state.mu*asad1norm(state)) ) { state.curalgo = 1; goto lbl_22; } } } // // Next iteration // state.k = state.k+1; goto lbl_21; lbl_22: lbl_19: // // CG algorithm // if( state.curalgo!=1 ) { goto lbl_47; } // // first, check that there are non-active constraints. // move to GPA algorithm, if all constraints are active // b = true; for(i=0; i<=n-1; i++) { if( (double)(state.ak[i])!=(double)(0) ) { b = false; break; } } if( b ) { state.curalgo = 0; goto lbl_17; } // // CG iterations // state.fold = state.f; for(i_=0; i_<=n-1;i_++) { state.xk[i_] = state.x[i_]; } for(i=0; i<=n-1; i++) { state.dk[i] = -(state.g[i]*state.ak[i]); state.gc[i] = state.g[i]*state.ak[i]; } lbl_49: if( false ) { goto lbl_50; } // // Store G[k] for later calculation of Y[k] // for(i=0; i<=n-1; i++) { state.yk[i] = -state.gc[i]; } // // Make a CG step in direction given by DK[]: // * calculate step. Step projection into feasible set // is used. It has several benefits: a) step may be // found with usual line search, b) multiple constraints // may be activated with one step, c) activated constraints // are detected in a natural way - just compare x[i] with // bounds // * update active set, set B to True, if there // were changes in the set. // for(i_=0; i_<=n-1;i_++) { state.d[i_] = state.dk[i_]; } for(i_=0; i_<=n-1;i_++) { state.xn[i_] = state.xk[i_]; } state.mcstage = 0; state.stp = 1; linmin.linminnormalized(ref state.d, ref state.stp, n); if( (double)(state.laststep)!=(double)(0) ) { state.stp = state.laststep; } linmin.mcsrch(n, ref state.xn, ref state.f, ref state.gc, state.d, ref state.stp, state.stpmax, gtol, ref mcinfo, ref state.nfev, ref state.work, state.lstate, ref state.mcstage); lbl_51: if( state.mcstage==0 ) { goto lbl_52; } // // preprocess data: bound State.XN so it belongs to the // feasible set and store it in the State.X // for(i=0; i<=n-1; i++) { state.x[i] = apserv.boundval(state.xn[i], state.bndl[i], state.bndu[i]); } // // RComm // clearrequestfields(state); state.needfg = true; state.rstate.stage = 9; goto lbl_rcomm; lbl_9: state.needfg = false; // // postprocess data: zero components of G corresponding to // the active constraints // for(i=0; i<=n-1; i++) { if( (double)(state.x[i])==(double)(state.bndl[i]) || (double)(state.x[i])==(double)(state.bndu[i]) ) { state.gc[i] = 0; } else { state.gc[i] = state.g[i]; } } linmin.mcsrch(n, ref state.xn, ref state.f, ref state.gc, state.d, ref state.stp, state.stpmax, gtol, ref mcinfo, ref state.nfev, ref state.work, state.lstate, ref state.mcstage); goto lbl_51; lbl_52: diffcnt = 0; for(i=0; i<=n-1; i++) { // // XN contains unprojected result, project it, // save copy to X (will be used for progress reporting) // state.xn[i] = apserv.boundval(state.xn[i], state.bndl[i], state.bndu[i]); // // update active set // if( (double)(state.xn[i])==(double)(state.bndl[i]) || (double)(state.xn[i])==(double)(state.bndu[i]) ) { state.an[i] = 0; } else { state.an[i] = 1; } if( (double)(state.an[i])!=(double)(state.ak[i]) ) { diffcnt = diffcnt+1; } state.ak[i] = state.an[i]; } for(i_=0; i_<=n-1;i_++) { state.xk[i_] = state.xn[i_]; } state.repnfev = state.repnfev+state.nfev; state.repiterationscount = state.repiterationscount+1; if( !state.xrep ) { goto lbl_53; } // // progress report // clearrequestfields(state); state.xupdated = true; state.rstate.stage = 10; goto lbl_rcomm; lbl_10: state.xupdated = false; lbl_53: // // Update info about step length // v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.d[i_]*state.d[i_]; } state.laststep = Math.Sqrt(v)*state.stp; // // Check stopping conditions. // if( (double)(asaboundedantigradnorm(state))>(double)(state.epsg) ) { goto lbl_55; } // // Gradient is small enough // state.repterminationtype = 4; if( !state.xrep ) { goto lbl_57; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 11; goto lbl_rcomm; lbl_11: state.xupdated = false; lbl_57: result = false; return result; lbl_55: if( !(state.repiterationscount>=state.maxits && state.maxits>0) ) { goto lbl_59; } // // Too many iterations // state.repterminationtype = 5; if( !state.xrep ) { goto lbl_61; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 12; goto lbl_rcomm; lbl_12: state.xupdated = false; lbl_61: result = false; return result; lbl_59: if( !((double)(asaginorm(state))>=(double)(state.mu*asad1norm(state)) && diffcnt==0) ) { goto lbl_63; } // // These conditions (EpsF/EpsX) are explicitly or implicitly // related to the current step size and influenced // by changes in the active constraints. // // For these reasons they are checked only when we don't // want to 'unstick' at the end of the iteration and there // were no changes in the active set. // // NOTE: consition |G|>=Mu*|D1| must be exactly opposite // to the condition used to switch back to GPA. At least // one inequality must be strict, otherwise infinite cycle // may occur when |G|=Mu*|D1| (we DON'T test stopping // conditions and we DON'T switch to GPA, so we cycle // indefinitely). // if( (double)(state.fold-state.f)>(double)(state.epsf*Math.Max(Math.Abs(state.fold), Math.Max(Math.Abs(state.f), 1.0))) ) { goto lbl_65; } // // F(k+1)-F(k) is small enough // state.repterminationtype = 1; if( !state.xrep ) { goto lbl_67; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 13; goto lbl_rcomm; lbl_13: state.xupdated = false; lbl_67: result = false; return result; lbl_65: if( (double)(state.laststep)>(double)(state.epsx) ) { goto lbl_69; } // // X(k+1)-X(k) is small enough // state.repterminationtype = 2; if( !state.xrep ) { goto lbl_71; } clearrequestfields(state); state.xupdated = true; state.rstate.stage = 14; goto lbl_rcomm; lbl_14: state.xupdated = false; lbl_71: result = false; return result; lbl_69: lbl_63: // // Check conditions for switching // if( (double)(asaginorm(state))<(double)(state.mu*asad1norm(state)) ) { state.curalgo = 0; goto lbl_50; } if( diffcnt>0 ) { if( asauisempty(state) || diffcnt>=n2 ) { state.curalgo = 1; } else { state.curalgo = 0; } goto lbl_50; } // // Calculate D(k+1) // // Line search may result in: // * maximum feasible step being taken (already processed) // * point satisfying Wolfe conditions // * some kind of error (CG is restarted by assigning 0.0 to Beta) // if( mcinfo==1 ) { // // Standard Wolfe conditions are satisfied: // * calculate Y[K] and BetaK // for(i_=0; i_<=n-1;i_++) { state.yk[i_] = state.yk[i_] + state.gc[i_]; } vv = 0.0; for(i_=0; i_<=n-1;i_++) { vv += state.yk[i_]*state.dk[i_]; } v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.gc[i_]*state.gc[i_]; } state.betady = v/vv; v = 0.0; for(i_=0; i_<=n-1;i_++) { v += state.gc[i_]*state.yk[i_]; } state.betahs = v/vv; if( state.cgtype==0 ) { betak = state.betady; } if( state.cgtype==1 ) { betak = Math.Max(0, Math.Min(state.betady, state.betahs)); } } else { // // Something is wrong (may be function is too wild or too flat). // // We'll set BetaK=0, which will restart CG algorithm. // We can stop later (during normal checks) if stopping conditions are met. // betak = 0; state.debugrestartscount = state.debugrestartscount+1; } for(i_=0; i_<=n-1;i_++) { state.dn[i_] = -state.gc[i_]; } for(i_=0; i_<=n-1;i_++) { state.dn[i_] = state.dn[i_] + betak*state.dk[i_]; } for(i_=0; i_<=n-1;i_++) { state.dk[i_] = state.dn[i_]; } // // update other information // state.fold = state.f; state.k = state.k+1; goto lbl_49; lbl_50: lbl_47: goto lbl_17; lbl_18: result = false; return result; // // Saving state // lbl_rcomm: result = true; state.rstate.ia[0] = n; state.rstate.ia[1] = i; state.rstate.ia[2] = mcinfo; state.rstate.ia[3] = diffcnt; state.rstate.ba[0] = b; state.rstate.ba[1] = stepfound; state.rstate.ra[0] = betak; state.rstate.ra[1] = v; state.rstate.ra[2] = vv; return result; } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minasaresults(minasastate state, ref double[] x, minasareport rep) { x = new double[0]; minasaresultsbuf(state, ref x, rep); } /************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ public static void minasaresultsbuf(minasastate state, ref double[] x, minasareport rep) { int i = 0; int i_ = 0; if( alglib.ap.len(x)=state.n, "MinASARestartFrom: Length(X)=state.n, "MinASARestartFrom: Length(BndL)=state.n, "MinASARestartFrom: Length(BndU)0)and(x[i]=bndu[i])) v[i]=-g[i] otherwise This function may be used to check a stopping criterion. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ private static double asaboundedantigradnorm(minasastate state) { double result = 0; int i = 0; double v = 0; result = 0; for(i=0; i<=state.n-1; i++) { v = -state.g[i]; if( (double)(state.x[i])==(double)(state.bndl[i]) && (double)(-state.g[i])<(double)(0) ) { v = 0; } if( (double)(state.x[i])==(double)(state.bndu[i]) && (double)(-state.g[i])>(double)(0) ) { v = 0; } result = result+math.sqr(v); } result = Math.Sqrt(result); return result; } /************************************************************************* Returns norm of GI(x). GI(x) is a gradient vector whose components associated with active constraints are zeroed. It differs from bounded anti-gradient because components of GI(x) are zeroed independently of sign(g[i]), and anti-gradient's components are zeroed with respect to both constraint and sign. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ private static double asaginorm(minasastate state) { double result = 0; int i = 0; result = 0; for(i=0; i<=state.n-1; i++) { if( (double)(state.x[i])!=(double)(state.bndl[i]) && (double)(state.x[i])!=(double)(state.bndu[i]) ) { result = result+math.sqr(state.g[i]); } } result = Math.Sqrt(result); return result; } /************************************************************************* Returns norm(D1(State.X)) For a meaning of D1 see 'NEW ACTIVE SET ALGORITHM FOR BOX CONSTRAINED OPTIMIZATION' by WILLIAM W. HAGER AND HONGCHAO ZHANG. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ private static double asad1norm(minasastate state) { double result = 0; int i = 0; result = 0; for(i=0; i<=state.n-1; i++) { result = result+math.sqr(apserv.boundval(state.x[i]-state.g[i], state.bndl[i], state.bndu[i])-state.x[i]); } result = Math.Sqrt(result); return result; } /************************************************************************* Returns True, if U set is empty. * State.X is used as point, * State.G - as gradient, * D is calculated within function (because State.D may have different meaning depending on current optimization algorithm) For a meaning of U see 'NEW ACTIVE SET ALGORITHM FOR BOX CONSTRAINED OPTIMIZATION' by WILLIAM W. HAGER AND HONGCHAO ZHANG. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/ private static bool asauisempty(minasastate state) { bool result = new bool(); int i = 0; double d = 0; double d2 = 0; double d32 = 0; d = asad1norm(state); d2 = Math.Sqrt(d); d32 = d*d2; result = true; for(i=0; i<=state.n-1; i++) { if( (double)(Math.Abs(state.g[i]))>=(double)(d2) && (double)(Math.Min(state.x[i]-state.bndl[i], state.bndu[i]-state.x[i]))>=(double)(d32) ) { result = false; return result; } } return result; } /************************************************************************* Clears request fileds (to be sure that we don't forgot to clear something) *************************************************************************/ private static void clearrequestfields(minasastate state) { state.needfg = false; state.xupdated = false; } } }