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)

Authors
Authors and affiliations

Erik CuevasEmail author
Daniel Zaldívar
Marco Pérez-Cisneros

Chapter

First Online: 11 April 2018

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]

References

1.
Han, M.-F., Liao, S.-H., Chang, J.-Y., Lin, C.T.: Dynamic group-based differential evolution using a self-adaptive strategy for global optimization problems. Appl. Intell. https://doi.org/10.1007/s10489-012-0393-5
2.
Pardalos Panos, M., Romeijn Edwin H., Tuy, H.: Recent developments and trends in global optimization. J. Comput. Appl. Math. 124, 209–228 (2000)MathSciNetCrossRefGoogle Scholar
3.
Floudas, C., Akrotirianakis, I., Caratzoulas, S., Meyer, C., Kallrath, J.: Global optimization in the 21st century: advances and challenges. Comput. Chem. Eng. 29(6), 1185–1202 (2005)CrossRefGoogle Scholar
4.
Ying, J., Ke-Cun, Z., Shao-Jian, Q.: A deterministic global optimization algorithm. Appl. Math. Comput. 185(1), 382–387 (2007)MathSciNetMATHGoogle Scholar
5.
Georgieva, A., Jordanov, I.: Global optimization based on novel heuristics, low-discrepancy sequences and genetic algorithms. Eur. J. Oper. Res. 196, 413–422 (2009)CrossRefGoogle Scholar
6.
Lera, D., Sergeyev, Y.: Lipschitz and Hölder global optimization using space-filling curves. Appl. Numer. Math. 60(1–2), 115–129 (2010)MathSciNetCrossRefGoogle Scholar
7.
Fogel, L.J., Owens, A.J., Walsh, M.J.: Artificial Intelligence through Simulated Evolution. Wiley, Chichester, UK (1966)MATHGoogle Scholar
8.
De Jong, K.: Analysis of the behavior of a class of genetic adaptive systems. Ph.D. thesis, University of Michigan, Ann Arbor, MI (1975)Google Scholar
9.
Koza, J.R.: Genetic programming: a paradigm for genetically breeding populations of computer programs to solve problems. Rep. No. STAN-CS-90-1314, Stanford University, CA (1990)Google Scholar
10.
Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI (1975)Google Scholar
11.
Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley, Boston, MA (1989)MATHGoogle Scholar
12.
de Castro, L.N., Von Zuben, F.J.: Artificial immune systems: part I—basic theory and applications. Technical report, TR-DCA 01/99, December 1999Google Scholar
13.
Storn, R., Price, K.: Differential evolution—a simple and efficient adaptive scheme for global optimisation over continuous spaces. Technical Report TR-95–012, ICSI, Berkeley, Calif (1995)Google Scholar
14.
Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefGoogle Scholar
15.
İlker, B., Birbil, S., Shu-Cherng, F.: An electromagnetism-like mechanism for global optimization. J. Global Optim. 25, 263–282 (2003)MathSciNetCrossRefGoogle Scholar
16.
Rashedia, E., Nezamabadi-pour, H., Saryazdi, S.: Filter modeling using gravitational search algorithm. Eng. Appl. Artif. Intell. 24(1), 117–122 (2011)CrossRefGoogle Scholar
17.
Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the 1995 IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948, December 1995Google Scholar
18.
Dorigo, M., Maniezzo, V., Colorni, A.: Positive feedback as a search strategy. Technical Report No. 91-016, Politecnico di Milano (1991)Google Scholar
19.
Tan, K.C., Chiam, S.C., Mamun, A.A., Goh, C.K.: Balancing exploration and exploitation with adaptive variation for evolutionary multi-objective optimization. Eur. J. Oper. Res. 197, 701–713 (2009)CrossRefGoogle Scholar
20.
Chen, G., Low, C.P., Yang, Z.: Preserving and exploiting genetic diversity in evolutionary programming algorithms. IEEE Trans. Evol. Comput. 13(3), 661–673 (2009)CrossRefGoogle Scholar
21.
Liu, S.-H., Mernik, M., Bryant, B.: To explore or to exploit: an entropy-driven approach for evolutionary algorithms. Int. J. Knowl. Based Intell. Eng. Syst. 13(3), 185–206 (2009)Google Scholar
22.
Alba, E., Dorronsoro, B.: The exploration/exploitation tradeoff in dynamic cellular genetic algorithms. IEEE Trans. Evol. Comput. 9(3), 126–142 (2005)CrossRefGoogle Scholar
23.
Fister, I., Mernik, M., Filipič, B.: A hybrid self-adaptive evolutionary algorithm for marker optimization in the clothing industry. Appl. Soft Comput. 10(2), 409–422 (2010)CrossRefGoogle Scholar
24.
Gong, W., Cai, Z., Jiang, L.: Enhancing the performance of differential evolution using orthogonal design method. Appl. Math. Comput. 206(1), 56–69 (2008)MATHGoogle Scholar
25.
Joan-Arinyo, R., Luzon, M.V., Yeguas, E.: Parameter tuning of PBIL and CHC evolutionary algorithms applied to solve the root identification problem. Appl. Soft Comput. 11(1), 754–767 (2011)CrossRefGoogle Scholar
26.
Mallipeddi, R., Suganthan, P.N., Pan, Q.K., Tasgetiren, M.F.: Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl. Soft Comput. 11(2), 1679–1696 (2011)CrossRefGoogle Scholar
27.
Sadegh, M., Reza, M., Palhang, M.: LADPSO: using fuzzy logic to conduct PSO algorithm. Appl. Intell. 37(2), 290–304 (1012)Google Scholar
28.
Yadav, P., Kumar, R., Panda, S.K., Chang, C.S.: An intelligent tuned harmony search algorithm for optimization. Inf. Sci. 196(1), 47–72 (2012)CrossRefGoogle Scholar
29.
Khajehzadeh, M., Taha, M.R., El-Shafie, A., Eslami, M.: A modified gravitational search algorithm for slope stability analysis. Eng. Appl. Artif. Intell. 25(8), 1589–1597 (2012)CrossRefGoogle Scholar
30.
Koumousis, V., Katsaras, C.P.: A saw-tooth genetic algorithm combining the effects of variable population size and reinitialization to enhance performance. IEEE Trans. Evol. Comput. 10(1), 19–28 (2006)CrossRefGoogle Scholar
31.
Han, M.-F., Liao, S.-H., Chang, J.-Y., Lin, C.-T.: Dynamic group-based differential evolution using a self-adaptive strategy for global optimization problems. Appl. Intell. (2012). https://doi.org/10.1007/s10489-012-0393-5CrossRefGoogle Scholar
32.
Brest, J., Maučec, M.S.: Population size reduction for the differential evolution algorithm. Appl. Intell. 29(3), 228–247 (2008)CrossRefGoogle Scholar
33.
Li, Y., Zeng, X.: Multi-population co-genetic algorithm with double chain-like agents structure for parallel global numerical optimization. Appl. Intell. 32(3), 292–310 (2010)MathSciNetCrossRefGoogle Scholar
34.
Paenke, I., Jin, Y., Branke, J.: Balancing population- and individual-level adaptation in changing environments. Adapt. Behav. 17(2), 153–174 (2009)CrossRefGoogle Scholar
35.
Araujo, L., Merelo, J.J.: Diversity through multiculturality: assessing migrant choice policies in an island model. IEEE Trans. Evol. Comput. 15(4), 456–468 (2011)CrossRefGoogle Scholar
36.
Gao, H., Xu, W.: Particle swarm algorithm with hybrid mutation strategy. Appl. Soft Comput. 11(8), 5129–5142 (2011)CrossRefGoogle Scholar
37.
Jia, D., Zheng, G., Khan, M.K. (2011). An effective memetic differential evolution algorithm based on chaotic local search. Inf. Sci. 181(15), 3175–3187CrossRefGoogle Scholar
38.
Lozano, M., Herrera, F., Cano, J.R.: Replacement strategies to preserve useful diversity in steady-state genetic algorithms. Inf. Sci. 178(23), 4421–4433 (2008)CrossRefGoogle Scholar
39.
Ostadmohammadi, B., Mirzabeygi, P., Panahi, M.: An improved PSO algorithm with a territorial diversity-preserving scheme and enhanced exploration–exploitation balance. Swarm Evol. Comput. (In Press)Google Scholar
40.
Yang, G.-P., Liu, S.-Y., Zhang, J.-K., Feng, Q.-X.: Control and synchronization of chaotic systems by an improved biogeography-based optimization algorithm. Appl. Intell. https://doi.org/10.1007/s10489-012-0398-0
41.
Hasanzadeh, M., Meybodi, M.R., Ebadzadeh, M.M.: Adaptive cooperative particle swarm optimizer. Appl. Intell. https://doi.org/10.1007/s10489-012-0420-6
42.
Aribarg, T., Supratid, S., Lursinsap, C.: Optimizing the modified fuzzy ant-miner for efficient medical diagnosis. Appl. Intell. 37(3), 357–376 (2012)CrossRefGoogle Scholar
43.
Fernandes, C.M., Laredo, J.L.J., Rosa, A.C., Merelo, J.J.: The sandpile mutation genetic algorithm: an investigation on the working mechanisms of a diversity-oriented and self-organized mutation operator for non-stationary functions. Appl. Intell. https://doi.org/10.1007/s10489-012-0413-5
44.
Gwak, J., Sim, K.M.: A novel method for coevolving PS-optimizing negotiation strategies using improved diversity controlling EDAs. Appl. Intell. 38(3), 384–417 (2013)CrossRefGoogle Scholar
45.
Cheshmehgaz, H.R., Desa, M.I., Wibowo, A.: Effective local evolutionary searches distributed on an island model solving bi-objective optimization problems. Appl. Intell. 38(3), 331–356 (2013)Google Scholar
46.
Cuevas, E., González, M.: Multi-circle detection on images inspired by collective animal behavior. Appl. Intell. https://doi.org/10.1007/s10489-012-0396-2
47.
Adra, S.F., Fleming, P.J.: Diversity management in evolutionary many-objective optimization. IEEE Trans. Evol. Comput. 15(2), 183–195 (2011)CrossRefGoogle Scholar
48.
Črepineš, M., Liu, S.H., Mernik, M.: Exploration and exploitation in evolutionary algorithms: a survey. ACM Comput. Surv. 1(1), 1–33 (2011)Google Scholar
49.
Ceruti, G., Rubin, H.: Infodynamics: analogical analysis of states of matter and information. Inf. Sci. 177, 969–987 (2007)CrossRefGoogle Scholar
50.
Chowdhury, D., Stauffer, D.: Principles of Equilibrium Statistical Mechanics, 1st edn. Wiley-VCH, Germany (2000)CrossRefGoogle Scholar
51.
Betts, D.S., Turner, R.E.: Introductory Statistical Mechanics, 1st edn. Addison Wesley, Boston (1992)Google Scholar
52.
Cengel, Y.A., Boles, M.A.: Thermodynamics: An Engineering Approach, 5th edn. McGraw-Hill, USA (2005)Google Scholar
53.
Bueche, F., Hecht, E.: Schaum’s Outline of College Physics, 11th edn. McGraw-Hill, USA (2012)Google Scholar
54.
Piotrowski, A.P., Napiorkowski, J.J., Kiczko, A.: Differential evolution algorithm with separated groups for multi-dimensional optimization problems. Eur. J. Oper. Res. 216(1), 33–46 (2012)MathSciNetCrossRefGoogle Scholar
55.
Mariani, V.C., Luvizotto, L.G.J., Guerra, F.A., dos Santos Coelho, L.: A hybrid shuffled complex evolution approach based on differential evolution for unconstrained optimization. Appl. Math. Comput. 217(12), 5822–5829 (2011)MathSciNetCrossRefGoogle Scholar
56.
Yao, X., Liu, Y., Lin, G.: Evolutionary programming made faster. IEEE Trans. Evol. Comput. 3(2), 82–102 (1999)CrossRefGoogle Scholar
57.
Moré, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7(1), 17–41 (1981)MathSciNetCrossRefGoogle Scholar
58.
Tsoulos, I.G.: Modifications of real code genetic algorithm for global optimization. Appl. Math. Comput. 203(2), 598–607 (2008)MathSciNetMATHGoogle Scholar
59.
Black-Box Optimization Benchmarking (BBOB) 2010. In: 2nd GECCO Workshop for Real-Parameter Optimization. http://coco.gforge.inria.fr/doku.php?id=bbob-2010
60.
Hedar, A.-R., Ali, A.F.: Tabu search with multi-level neighborhood structures for high dimensional problems. Appl. Intell. 37(2), 189–206 (2012)CrossRefGoogle Scholar
61.
Vafashoar, R., Meybodi, M.R., Momeni Azandaryani, A.H.: CLA-DE: a hybrid model based on cellular learning automata for numerical optimization. Appl. Intell. 36(3), 735–748 (2012)CrossRefGoogle Scholar
62.
Garcia, S., Molina, D., Lozano, M., Herrera, F.: A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC ’2005, Special session on real parameter optimization. J. Heurist (2008). https://doi.org/10.1007/s10732-008-9080-4CrossRefMATHGoogle Scholar
63.
Shilane, D., Martikainen, J., Dudoit, S., Ovaska, S.: A general framework for statistical performance comparison of evolutionary computation algorithms. Inf. Sci. 178, 2870–2879 (2008)CrossRefGoogle Scholar

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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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