The States of Matter Search (SMS)

Erik Cuevas; Daniel Zaldívar; Marco Pérez-Cisneros

doi:10.1007/978-3-319-89309-9_6

Advances in Metaheuristics Algorithms: Methods and Applications pp 93-118 | Cite as

The States of Matter Search (SMS)

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Erik Cuevas
Daniel Zaldívar
Marco Pérez-Cisneros

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First Online: 11 April 2018

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Part of the Studies in Computational Intelligence book series (SCI, volume 775)

Abstract

The capacity of a metaheuristic method to attain the global optimal solution maintains an explicit dependency on its potential to find a good balance between exploitation and exploration of the search strategy.

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Appendix: List of Benchmark Functions

See Tables 6.8, 6.9, 6.10 and 6.11.

Table 6.8

Unimodal test functions

Test function	\( S \)	\( n \)
\( f_{1} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} } \)	\( \left[ { - 100{,}100} \right]^{n} \)	30
\( f_{2} \left( {\mathbf{x}} \right) = \hbox{max} \left\{ {\left\| {x_{i} } \right\|,1 \le i \le n} \right\} \)	\( \left[ { - 100{,}100} \right]^{n} \)	30
\( f_{3} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n - 1} {\left[ {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]} \)	\( \left[ { - 30{,}30} \right]^{n} \)	30
\( f_{4} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {ix_{i}^{4} + rand(0,1)} \)	\( \left[ { - 1.28{,}1.28} \right]^{n} \)	30

Table 6.9

Multimodal test functions

Test function	\( S \)	\( n \)
\( f_{5} \left( {\mathbf{x}} \right) = 418.9829n + \sum\nolimits_{i = 1}^{n} {\left( { - x_{i} \sin \left( {\sqrt {\left\| {x_{i} } \right\|} } \right)} \right)} \)	\( \left[ { - 500{,}500} \right]^{n} \)	30
\( f_{6} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{50} {\left( {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right)} \)	\( \left[ { - 5.12{,}5.12} \right]^{n} \)	30
\( f_{7} \left( {\mathbf{x}} \right) = \frac{1}{4000}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } - \prod\nolimits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1} \)	\( \left[ { - 600{,}600} \right]^{n} \)	30
\(\begin{aligned} f_{8} ({\mathbf{x}}) & = \frac{\pi }{n}\left\{ {10\,\sin \,(\pi y_{1} ) + \sum\nolimits_{i = 1}^{n - 1} {(y_{i} - 1)^{2} \left[ {1 + 10\,\sin^{2} (\pi y_{i + 1} )} \right] + (y_{n} - 1)^{2} } } \right\} \\ & \quad + \sum\nolimits_{i = 1}^{n} {u(x_{i} ,10,100,4)} \end{aligned}\) \( y_{i} = 1 + \frac{{\left( {x_{i} + 1} \right)}}{4} \) \( u\left( {x_{i} ,a,k,m} \right) = \left\{ {\begin{array}{*{20}c} {k\left( {x_{i} - a} \right)^{m} } & {x_{i} > a} \\ 0 & { - a \le x_{i} \le a} \\ {k\left( { - x_{i} - a} \right)^{m} } & {x_{i} < a} \\ \end{array} } \right. \)	\( \left[ { - 50{,}50} \right]^{n} \)	30
\( \begin{aligned} f_{9} ({\mathbf{x}})& = 0.1\left\{ {\sin^{2} (3\pi x_{1} ) + \sum\nolimits_{i = 1}^{n} {(x_{i} - 1)^{2} \left[ {1 + \sin^{2} (3\pi x_{i} + 1)} \right] + (x_{n} - 1)^{2} \left[ {1 + \sin^{2} (2\pi x_{n} )} \right]} } \right\} \\ & \quad + \sum\nolimits_{i = 1}^{n} {u(x_{i} } ,5,100,4) \end{aligned} \) where \( u\left( {x_{i} ,a,k,m} \right) \) is the same as \( f_{8} \)	\( \left[ { - 50{,}50} \right]^{n} \)	30
\( f_{10} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} } + \left( {\sum\nolimits_{i = 1}^{n} {0.5ix_{i} } } \right)^{2} + \left( {\sum\nolimits_{i = 1}^{n} {0.5ix_{i} } } \right)^{4} \)	\( \left[ { - 10{,}10} \right]^{n} \)	30
\( f_{11} \left( {\mathbf{x}} \right) = 1 - \cos \left( {2\pi \left\\| x \right\\|} \right) + 0.1\left\\| x \right\\| \) where \( \left\\| x \right\\| = \sqrt {\sum\nolimits_{i = 1}^{n} {x_{j}^{2} } } \)	\( \left[ { - 100{,}100} \right]^{n} \)	30

Table 6.10

Multimodal test functions with fixed dimensions

Test function

\( S \)

\( f_{opt} \)

\( n \)

\( f_{12} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{11} {\left[ {a_{i} - \frac{{x_{i} \left( {b_{i}^{2} + b_{i} x_{2} } \right)}}{{b_{i}^{2} + b_{i} x_{3} + x_{4} }}} \right]}^{2} \)

\( {\mathbf{a}} = \left[ {0.1957,0.1947,0.1735,0.1600,0.0844,0.0627,0.456,0.0342,0.0323,0.0235,0.0246} \right] \)

\( {\mathbf{b}} = \left[ {0.25,0.5,1,2,4,6,8,10,12,14,16} \right] \)

\( \left[ { - 5{,}5} \right]^{n} \)

0.00030

4

\( f_{13} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{4} {c_{i} } \,\exp \,\left( { - \sum\nolimits_{j = 1}^{3} {A_{ij} \left( {x_{j} - P_{ij} } \right)^{2} } } \right) \)

\( {\mathbf{A}} = \left[ {\begin{array}{*{20}c} {10} & 3 & {17} & {3.5} & {1.7} & 8 \\ {0.05} & {10} & {17} & {0.1} & 8 & {14} \\ 3 & {3.5} & {17} & {10} & {17} & 8 \\ {17} & 8 & {0.05} & {10} & {0.1} & {14} \\ \end{array} } \right] \)

\( {\mathbf{c}} = \left[ {1,1.2,3,3.2} \right] \)

\( {\mathbf{P}} = \left[ {\begin{array}{*{20}c} {0.131} & {0.169} & {0.556} & {0.012} & {0.828} & {0.588} \\ {0.232} & {0.413} & {0.830} & {0.373} & {0.100} & {0.999} \\ {0.234} & {0.141} & {0.352} & {0.288} & {0.304} & {0.665} \\ {0.404} & {0.882} & {0.873} & {0.574} & {0.109} & {0.038} \\ \end{array} } \right] \)

\( \left[ {0{,}1} \right]^{n} \)

−3.32

6

\( f_{14} \left( {\mathbf{x}} \right) = \left( {1.5 - x_{1} \left( {1 - x_{2} } \right)} \right)^{2} + \left( {2.25 - x_{1} \left( {1 - x_{2} } \right)} \right)^{2} + \left( {2.625 - x_{1} \left( {1 - x_{2} } \right)} \right)^{2} \)

\( \left[ { - 4.5{,}4.5} \right]^{n} \)

0

2

Table 6.11

Set of representative GECCO functions

Test function	S	n	GECCO classification
\( f_{15} ({\mathbf{x}}) = 10^{6} \cdot z_{1}^{2} + \sum\nolimits_{i = 2}^{n} {z_{i} } + f_{opt} \) \( {\mathbf{z}} = T_{osz} \left( {{\mathbf{x}} - {\mathbf{x}}^{opt} } \right) \) \( T_{osz} :{\mathbb{R}}^{n} \to {\mathbb{R}}^{n} , \) for any positive integer n, it maps element-wise \( {\mathbf{a}} = T_{osz} ({\mathbf{h}}),\,{\mathbf{a}} = \left\{ {a_{1} ,a_{2} , \ldots ,a_{n} } \right\} \), \( {\mathbf{h}} = \left\{ {h_{1} ,h_{2} , \ldots ,h_{n} } \right\} \) \( a_{i} {\text{ = sign}}\left( {h_{i} } \right)\exp \left( {K + 0.049\left( {\sin \left( {c_{1} K} \right) + \sin \left( {c_{2} K} \right)} \right)} \right), \) where \( K = \left\{ {\begin{array}{{20}c} {\log (h_{i} )} & {{\text{if}}\,h_{i} \ne 0} \\ 0 & {\text{otherwise}} \\ \end{array} ,} \right.\,{\text{sign}}(h_{i} ) = \left\{ {\begin{array}{{20}c} { - 1} & {{\text{if}}\,h_{i} < 0} \\ { \, 0} & {{\text{if}}\,h_{i} = 0} \\ { \, 1} & {{\text{if}}\,h_{i} > 0} \\ \end{array} } \right., \) \( c_{1} = \left\{ {\begin{array}{{20}c} {10} & {{\text{if}}\,h_{i} > 0} \\ {5.5} & {\text{otherwise}} \\ \end{array} } \right. \) and \( c_{2} = \left\{ {\begin{array}{{20}c} {7.9} & {{\text{if}}\,h_{i} > 0} \\ {3.1} & {\text{otherwise}} \\ \end{array} } \right. \)	\( \left[ { - 5{,}5} \right]^{n} \)	30	GECCO2010 Discus function \( f_{11} ({\mathbf{x}}) \)
\( f_{16} ({\mathbf{x}}) = \sqrt {\sum\nolimits_{i = 1}^{n} {\left\| {z_{i} } \right\|^{{2 + 4\frac{i - 1}{n - 1}}} } } + f_{opt} \) \( {\mathbf{z}} = {\mathbf{x}} - {\mathbf{x}}^{opt} \)	\( \left[ { - 5{,}5} \right]^{n} \)	30	GECCO2010 Different powers function \( f_{14} ({\mathbf{x}}) \)
\( f_{17} ({\mathbf{x}}) = - \frac{1}{n}\sum\nolimits_{i = 1}^{n} {z_{i} \sin \left( {\sqrt {\left\| {z_{i} } \right\|} } \right)} + 4.189828872724339 + 100f_{pen} \left( {\frac{{\mathbf{z}}}{100}} \right) + f_{opt} \) \( {\hat{{\mathbf{x}}}} = 2 \times {\mathbf{1}}_{ - }^{ + } \otimes {\mathbf{x}} \) \( \hat{z}_{1} = \hat{x}_{1} ,\hat{z}_{i + 1} = \hat{x}_{i + 1} + 0.25\left( {\hat{x}_{i} - x_{i}^{opt} } \right) \) \( {\text{for}}\,i = 1, \ldots ,n - 1 \) \( {\mathbf{z}} = 100\left( {\Lambda ^{10} \left( {{\hat{\mathbf{z}}} - {\mathbf{x}}^{opt} } \right) + {\mathbf{x}}^{opt} } \right) \) \( f_{pen} :{\mathbb{R}}^{n} \to {\mathbb{R}}, \) \( a = f_{pen} ({\mathbf{h}}) \), \( {\mathbf{h}} = \left\{ {h_{1} ,h_{2} , \ldots ,h_{n} } \right\} \) \( a = 100\sum\nolimits_{i = 1}^{n} {\hbox{max} \left( {0,\left\| {h_{i} } \right\| - 5} \right)^{2} } \) \( {\mathbf{1}}_{ - }^{ + } \) is a n-dimensional vector with elements of −1 or 1 computed considering equal probability	\( \left[ { - 5{,}5} \right]^{n} \)	30	GECCO2010 Schwefel function \( f_{20} ({\mathbf{x}}) \)
\( f_{18} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {z_{i}^{2} } - 450 \) \( {\mathbf{z}} = {\mathbf{x}} - {\mathbf{x}}^{opt} \)	\( \left[ { - 100{,}100} \right]^{n} \)	30	GECCO2005 Shifted sphere function \( f_{1} ({\mathbf{x}}) \)
\( f_{19} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left( {\sum\nolimits_{j = 1}^{i} {z_{j} } } \right)}^{2} - 450 \) \( {\mathbf{z}} = {\mathbf{x}} - {\mathbf{x}}^{opt} \)	\( \left[ { - 100{,}100} \right]^{n} \)	30	GECCO2005 Shifted Schwefel’s problem \( f_{2} ({\mathbf{x}}) \)
\( f_{20} ({\mathbf{x}}) = \left( {\sum\nolimits_{i = 1}^{n} {\left( {\sum\nolimits_{j = 1}^{i} {z_{j} } } \right)}^{2} } \right) \cdot \left( {1 + 0.4\left\| {N\left( {0,1} \right)} \right\|} \right) - 450 \) \( {\mathbf{z}} = {\mathbf{x}} - {\mathbf{x}}^{opt} \)	\( \left[ { - 100{,}100} \right]^{n} \)	30	GECCO2005 Shifted Schwefel’s problem 1.2 with noise in fitness \( f_{4} ({\mathbf{x}}) \)
\( f_{21} ({\mathbf{x}}) = \hbox{max} \left\{ {\left\| {{\mathbf{Ax}} - {\mathbf{b}}_{i} } \right\|} \right\} - 310 \) A is a \( n \times n \) matrix, \( a_{i,j} \) are integer random numbers in the range \( [ - 500{,}500] \), \( \det ({\mathbf{A}}) \ne 0 \) \( {\mathbf{b}}_{i} = {\mathbf{A}}_{i} \cdot {\mathbf{o}} \) \( {\mathbf{A}}_{i} \) is the ith row of A whereas \( {\mathbf{o}} \) is a \( n \times 1 \) vector whose elements are random numbers in the range [−100,100]	\( \left[ { - 100{,}100} \right]^{n} \)	30	GECCO2005 Schwefel’s problem 2.6 with global optimum on bounds \( f_{5} ({\mathbf{x}}) \)
\( f_{22} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left( {100\left( {z_{i}^{2} - z_{i + 1} } \right)^{2} + \left( {z_{i} - 1} \right)^{2} } \right)} + 390 \) \( {\mathbf{z}} = {\mathbf{x}} - {\mathbf{x}}^{opt} \)	\( \left[ { - 100{,}100} \right]^{n} \)	30	GECCO2005 Shifted Rosenbrock’s function \( f_{6} ({\mathbf{x}}) \)
\( f_{23} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{n} {\left( {A_{i} - B_{i} \left( {\mathbf{x}} \right)} \right)}^{2} - 460 \) \( A_{i} = \sum\nolimits_{j = 1}^{n} {\left( {a_{i,j} \sin \,\alpha_{j} + b_{i,j} \cos \,\alpha_{j} } \right)} \) \( B_{i} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {\left( {a_{i,j} \sin \,x_{j} + b_{i,j} \cos \,x_{j} } \right)} \) For \( i = 1, \ldots ,n \) \( a_{i,j} \) and \( b_{i,j} \) are integer random numbers in the range [−100,100], \( {\varvec{\upalpha}} = \left[ {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right] \), \( \alpha_{j} \) are random numbers in the range \( \left[ { - \pi {,}\pi } \right] \)	\( \left[ { - \pi ,\pi } \right]^{n} \)	30	GECCO2005 Schwefel’s problem 2.13 \( f_{12} ({\mathbf{x}}) \)
\( f_{24} ({\mathbf{x}}) = \sum\nolimits_{i = 1}^{10} {\hat{F}_{i} ({\mathbf{x}} - {\mathbf{x}}_{i}^{opt} )} /\lambda_{i} \) \( F_{1 - 2} \left( {\mathbf{x}} \right) = \) Ackley’s function \( F_{i} \left( {\mathbf{x}} \right) = - 20\,\exp \,\left( { - 0.2\sqrt {\frac{1}{D}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } } } \right) - \exp \,\left( {\frac{1}{D}\sum\nolimits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right) + 20 \) \( F_{3 - 4} \left( {\mathbf{x}} \right) = \) Rastringin’s function \( F_{i} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {\left( {x_{i}^{2} - 10\cos \,\left( {2\pi x_{i} } \right) + 10} \right)} \) \( F_{5 - 6} \left( {\mathbf{x}} \right) = \) Sphere function \( F_{i} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} } \) \( F_{7 - 8} \left( {\mathbf{x}} \right) = \) Weierstrass function \( \begin{aligned}F_{i} \left( x \right) & = \sum\nolimits_{i = 1}^{n} {\left( {\sum\nolimits_{k = 0}^{k{\text{max}} } {\left[ {a^{k} \cos \,\left( {2\pi b^{k} \left( {x_{i} + 0.5} \right)} \right)} \right]} } \right)}\\ & \quad - n\sum\nolimits_{k = 0}^{k{\text{max}} } {\left[ {a^{k} \cos \,\left( {2\pi b^{k} \left( {x_{i} \cdot 0.5} \right)} \right)} \right]}\end{aligned} \) \( F_{9 - 10} \left( {\mathbf{x}} \right) = \) Griewank’s function \( F_{i} \left( {\mathbf{x}} \right) = \sum\nolimits_{i = 1}^{n} {\frac{{x_{i}^{2} }}{4000} - \prod\nolimits_{i = 1}^{n} {\cos \,\left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1} } \) \( \hat{F}_{i} ({\mathbf{z}}) = F_{i} ({\mathbf{z}})/F_{i}^{\hbox{max} } \). \( F_{i}^{\hbox{max} } \) is the maximum value of the particular function i \( {\varvec{\uplambda}} = \left[ {\frac{10}{32},\frac{5}{32},2,1,\frac{10}{100},\frac{5}{100},20,10,\frac{10}{60},\frac{5}{60}} \right] \)	\( \left[ { - 5{,}5} \right]^{n} \)	30	GECCO2005 Rotated version of hybrid composition function \( f_{16} ({\mathbf{x}}) \)
The \( {\mathbf{x}}^{opt} \) and \( f_{opt} \) values have been set to default values which have been obtained from the Matlab© implementation for GECCO competitions, as it is provided in [51]

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Erik Cuevas
- 1
Email author
Daniel Zaldívar
- 1
Marco Pérez-Cisneros
- 1

1.CUCEIUniversidad de GuadalajaraGuadalajaraMexico

The States of Matter Search (SMS)

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