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source: trunk/documentation/Test Functions/TestFunctions.tex @ 13397

Last change on this file since 13397 was 9887, checked in by jkarder, 9 years ago

updated template document for description of real valued test functions

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1\documentclass[12pt, a4paper]{article}
2
3%include packages
4\usepackage{a4}
5\usepackage[dvips]{graphicx}
6\usepackage[ansinew]{inputenc}
7\usepackage{epsfig}
8\usepackage{subcaption}
9\usepackage{amsmath}
10\usepackage{amssymb}
11
12\pagestyle{plain}
13\pagenumbering{arabic}
14
15\title{Real Valued Test Functions}
16\author{Heuristic and Evolutionary Algorithms Laboratory (HEAL)}
17\date{\today}
18
19\begin{document}
20  \maketitle
21
22  \section*{Ackley Function}
23    \begin{equation*}
24      f(x) = 20 + e - 20e^{-\frac{1}{5} \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2}} - e^{\frac{1}{n} \sum_{i=1}^n \cos(2 \pi x_i)}
25    \end{equation*}
26
27    \begin{tabbing}
28      \hspace{5cm}\=\kill
29      \textbf{Dimensions:}     \> $n$ \\
30      \textbf{Domain:}         \> $-32.768 \leq x_i \leq 32.768$ \\
31      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
32      \textbf{Operator:}       \> AckleyEvaluator \\
33      \textbf{Charts:}         \> \\
34    \end{tabbing}
35
36    \begin{figure}[ht]
37      \begin{subfigure}{0.49\textwidth}
38        \includegraphics[width=\linewidth]{Images/Ackley_large}
39        \caption{[-32.768, 32.768]}
40      \end{subfigure}
41      \begin{subfigure}{0.49\textwidth}
42        \includegraphics[width=\linewidth]{Images/Ackley_small}
43        \caption{[-6.0, 6.0]}
44      \end{subfigure}
45      \caption{Ackley function plots.}
46    \end{figure}
47
48  \newpage
49
50  \section*{Beale Function}
51    \begin{equation*}
52      f(x)=(1.5-x_1+x_1x_2)^2+(2.25-x_1+x_1x_2^2)^2+(2.625 - x_1 + x_1x_2^3)^2
53    \end{equation*}
54
55    \begin{tabbing}
56      \hspace{5cm}\=\kill
57      \textbf{Dimensions:}     \> $2$ \\
58      \textbf{Domain:}         \> $-4.5 \leq x_i \leq 4.5$ \\
59      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (3.0, 0.5)$ \\
60      \textbf{Operator:}       \> BealeEvaluator \\
61      \textbf{Charts:}         \> \\
62    \end{tabbing}
63
64    \begin{figure}[ht]
65      \includegraphics[width=\textwidth]{Images/Beale}
66      \caption{Beale function [-4.5, 4.5].}
67    \end{figure}
68
69  \newpage
70
71  \section*{Booth Function}
72    \begin{equation*}
73      f(x)=(x_1+2x_2-7)^2+(2x_1+x_2-5)^2
74    \end{equation*}
75
76    \begin{tabbing}
77      \hspace{5cm}\=\kill
78      \textbf{Dimensions:}     \> $2$ \\
79      \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\
80      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (1.0, 3.0)$ \\
81      \textbf{Operator:}       \> BoothEvaluator \\
82      \textbf{Charts:}         \> \\
83    \end{tabbing}
84
85    \begin{figure}[ht]
86      \includegraphics[width=\textwidth]{Images/Booth}
87      \caption{Booth function [-10.0, 10.0].}
88    \end{figure}
89
90  \newpage
91
92  \section*{Griewank Function}
93    \begin{equation*}
94      f(x) = 1 + \sum_{i=1}^n \frac{x_i^2}{4000} - \prod_{i=1}^n \cos(\frac{x_i}{\sqrt i})
95    \end{equation*}
96
97    \begin{tabbing}
98      \hspace{5cm}\=\kill
99      \textbf{Dimensions:}     \> $n$ \\
100      \textbf{Domain:}         \> $-600.0 \leq x_i \leq 600.0$ \\
101      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
102      \textbf{Operator:}       \> GriewankEvaluator \\
103      \textbf{Charts:}         \> \\
104    \end{tabbing}
105
106    \begin{figure}[ht]
107      \begin{subfigure}{0.49\textwidth}
108        \includegraphics[width=\linewidth]{Images/Griewank_large}
109        \caption{[-600.0, 600.0]}
110      \end{subfigure}
111      \begin{subfigure}{0.49\textwidth}
112        \includegraphics[width=\linewidth]{Images/Griewank_small}
113        \caption{[-10.0, 10.0]}
114      \end{subfigure}
115      \caption{Griewank function plots.}
116    \end{figure}
117
118  \newpage
119
120  \section*{Levy Function}
121    \begin{equation*}
122      f(x)=\sin^2(\pi w_1)+\sum\limits_{i=1}^{n-1}(w_i-1)^2[1+10\sin^2(\pi w_i+1)]+(w_n-1)^2[1+\sin^2(2\pi w_n)]
123    \end{equation*}
124    \begin{equation*}
125      w_i=1+\frac{x_i - 1}{4}, i=1,\dots,n
126    \end{equation*}
127
128    \begin{tabbing}
129      \hspace{5cm}\=\kill
130      \textbf{Dimensions:}     \> $n$ \\
131      \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\
132      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (1.0, 1.0)$ \\
133      \textbf{Operator:}       \> LevyEvaluator \\
134      \textbf{Charts:}         \> \\
135    \end{tabbing}
136
137    \begin{figure}[ht]
138      \includegraphics[width=\textwidth]{Images/Levy}
139      \caption{Levy function [-10.0, 10.0].}
140    \end{figure}
141
142  \newpage
143
144  \section*{Matyas Function}
145    \begin{equation*}
146      f(x)=0.26(x_1^2+x_2^2)-0.48x_1x_2
147    \end{equation*}
148
149    \begin{tabbing}
150      \hspace{5cm}\=\kill
151      \textbf{Dimensions:}     \> $2$ \\
152      \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\
153      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0)$ \\
154      \textbf{Operator:}       \> MatyasEvaluator \\
155      \textbf{Charts:}         \> \\
156    \end{tabbing}
157
158    \begin{figure}[ht]
159      \includegraphics[width=\textwidth]{Images/Matyas}
160      \caption{Matyas function [-10.0, 10.0].}
161    \end{figure}
162
163  \newpage
164
165  \section*{Rastrigin Function}
166    \begin{equation*}
167      f(x)=10n+\sum\limits_{i=1}^n[x_i^2-10\cos(2\pi x_i)]
168    \end{equation*}
169
170    \begin{tabbing}
171      \hspace{5cm}\=\kill
172      \textbf{Dimensions:}     \> $n$ \\
173      \textbf{Domain:}         \> $-5.12 \leq x_i \leq 5.12$ \\
174      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
175      \textbf{Operator:}       \> RastriginEvaluator \\
176      \textbf{Charts:}         \> \\
177    \end{tabbing}
178
179    \begin{figure}[ht]
180      \begin{subfigure}{0.49\textwidth}
181        \includegraphics[width=\linewidth]{Images/Rastrigin_large}
182        \caption{[-5.12, 5.12]}
183      \end{subfigure}
184      \begin{subfigure}{0.49\textwidth}
185        \includegraphics[width=\linewidth]{Images/Rastrigin_small}
186        \caption{[-2.0, 2.0]}
187      \end{subfigure}
188      \caption{Rastrigin function plots.}
189    \end{figure}
190
191  \newpage
192
193  \section*{Rosenbrock Function}
194    \begin{equation*}
195      f(x)=\sum\limits_{i=1}^{n-1}[100(x_i^2-x_{i+1})^2+(x_i-1)^2]
196    \end{equation*}
197
198    \begin{tabbing}
199      \hspace{5cm}\=\kill
200      \textbf{Dimensions:}     \> $n$ \\
201      \textbf{Domain:}         \> $-2.048 \leq x_i \leq 2.048$ \\
202      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (1.0, 1.0, \dots, 1.0)$ \\
203      \textbf{Operator:}       \> RosenbrockEvaluator \\
204      \textbf{Charts:}         \> \\
205    \end{tabbing}
206
207    \begin{figure}[ht]
208      \includegraphics[width=\textwidth]{Images/Rosenbrock}
209      \caption{Rosenbrock function [-2.048, 2.048].}
210    \end{figure}
211
212  \newpage
213
214  \section*{Schwefel Function}
215    \begin{equation*}
216      f(x)=418.982887272433n - \sum\limits_{i=1}^n x_i\sin(\sqrt{|x_i|})
217    \end{equation*}
218
219    \begin{tabbing}
220      \hspace{5cm}\=\kill
221      \textbf{Dimensions:}     \> $n$ \\
222      \textbf{Domain:}         \> $-500.0 \leq x_i \leq 500.0$ \\
223      \textbf{Global Optimum:} \> $f(x) \approx 0.0$ at $x = (420.9687, 420.9687, \dots, 420.9687)$ \\
224      \textbf{Operator:}       \> SchwefelEvaluator \\
225      \textbf{Charts:}         \> \\
226    \end{tabbing}
227
228    \begin{figure}[ht]
229      \includegraphics[width=\textwidth]{Images/Schwefel}
230      \caption{Schwefel function [-500.0, 500.0].}
231    \end{figure}
232
233  \newpage
234
235  \section*{Sphere Function}
236    \begin{equation*}
237      f(x)=\sum\limits_{i=1}^n x_i^2
238    \end{equation*}
239
240    \begin{tabbing}
241      \hspace{5cm}\=\kill
242      \textbf{Dimensions:}     \> $n$ \\
243      \textbf{Domain:}         \> $-5.12 \leq x_i \leq 5.12$ \\
244      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
245      \textbf{Operator:}       \> SphereEvaluator \\
246      \textbf{Charts:}         \> \\
247    \end{tabbing}
248
249    \begin{figure}[ht]
250      \includegraphics[width=\textwidth]{Images/Sphere}
251      \caption{Sphere function [-5.12, 5.12].}
252    \end{figure}
253
254  \newpage
255
256  \section*{Sum Squares Function}
257    \begin{equation*}
258      f(x)=\sum\limits_{i=1}^n ix_i^2
259    \end{equation*}
260
261    \begin{tabbing}
262      \hspace{5cm}\=\kill
263      \textbf{Dimensions:}     \> $n$ \\
264      \textbf{Domain:}         \> $-10.0 \leq x_i \leq 10.0$ \\
265      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
266      \textbf{Operator:}       \> SumSquaresEvaluator \\
267      \textbf{Charts:}         \> \\
268    \end{tabbing}
269
270    \begin{figure}[ht]
271      \includegraphics[width=\textwidth]{Images/SumSquares}
272      \caption{Sum squares function [-10.0, 10.0].}
273    \end{figure}
274
275  \newpage
276
277  \section*{Zakharov Function}
278    \begin{equation*}
279      f(x)=\sum\limits_{i=1}^n x_i^2+\left(\sum\limits_{i=1}^n 0.5ix_i\right)^2+\left(\sum\limits_{i=1}^n 0.5ix_i\right)^4
280    \end{equation*}
281
282    \begin{tabbing}
283      \hspace{5cm}\=\kill
284      \textbf{Dimensions:}     \> $n$ \\
285      \textbf{Domain:}         \> $-5.0 \leq x_i \leq 10.0$ \\
286      \textbf{Global Optimum:} \> $f(x) = 0.0$ at $x = (0.0, 0.0, \dots, 0.0)$ \\
287      \textbf{Operator:}       \> ZakharovEvaluator \\
288      \textbf{Charts:}         \> \\
289    \end{tabbing}
290
291    \begin{figure}[ht]
292      \includegraphics[width=\textwidth]{Images/Zakharov}
293      \caption{Zakharov function [-5.0, 10.0].}
294    \end{figure}
295
296\end{document}
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