Abstract
Multimodal function optimization, where the aim is to locate more than one solution, has attracted growing interest especially in the evolutionary computing research community. To evaluate experimentally the strengths and weaknesses of multimodal optimization algorithms, it is important to use test functions representing different characteristics and various levels of difficulty. The available selection of multimodal test problems is, however, rather limited and no general framework exists. This paper describes an attempt to construct a software framework which includes a variety of easily tunable test functions. The aim is to provide a general and easily expandable environment for testing different methods of multimodal optimization. Several function families with different characteristics are included. The framework implements new parameterizable function families for generating desired landscapes. Additionally the framework implements a selection of well known test functions from the literature, which can be rotated and stretched. The software module can easily be imported to any optimization algorithm implementation compatible with the C programming language. As an application example, 8 optimization approaches are compared by their ability to locate several global optima over a set of 16 functions with different properties generated by the proposed module. The effects of function regularity, dimensionality and number of local optima on the performance of different algorithms are studied.
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References
Beasley D, Bull D, Martin R (1993a) An overview of genetic algorithms: part 2, research topics. Univ Comput 15:170–181
Beasley D, Bull D, Martin R (1993b) A sequential niche technique for multimodal function optimization. Evol Comput 1(2):101–125
Bezier P (1968) How renault uses numerical control for car body design and tooling. In: SAE Paper 680010, Society of Automotive Engineers Congress, Detroit, MI, USA.
Bezier P (1986) The mathematical basis of UNISURF CAD System. Butterworth-Heinemann, London. ISBN 978-0408221757
Branke J (2002) Evolutionary optimization in dynamic environments. Kluwer, Norwell
Crutchley DA, Zwolinski M (2002) Using evolutionary and hybrid algorithms for DC operating point analysis of nonlinear circuits. In: Proceedings of 2002 IEEE world congress on computational intelligence, pp 753–758, Honolulu, USA, 12–15 May 2002. ISBN 0-7803-7282-4
De Jong K (1975) An analysis of the behavior of a class of genetic adaptive systems. PhD thesis, University of Michigan
Dong Z, Lu M, Lu Z, Wong K (2006) A differential evolution based method for power system planning. In: Proceedings of 2006 IEEE world congress on computational intelligence, pp 2699–2706, Vancouver, Canada, 16–21 July 2006. ISBN 0-7803-9489-5
Gallagher M, Yuan B (2006) A general-purpose tunable landscape generator. IEEE Trans Evol Comput 10:590–603
Gaviano M, Kvasov D, Lera D, Sergeyev Y (2003) Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans Math Softw 29(4):469–480
Ghosh A, Tsutsui S, Tanaka H, Corne D (2000) Genetic algorithms with substitution and re-entry of individuals. Int J Knowl Based Intell Eng Syst 4(1):64–71
Goldberg D, Richardson J (1987) Genetic algorithms with sharing for multimodal function optimization. In: Grefenstette J (ed) Proceedins of the second international conference on genetic algorithms, pp 41–49
Hansen N, Ostermeier A (2001) Completely derandomized self adaptation in evolution strategies. Evol Comput 9(2):159–195
Harik G (1995) Finding multimodal solutions using restricted tournament selection. In: Eshelman L (eds) Proceedings of the sixth international conference on genetic algorithms. Morgan Kaufmann, San Francisco, pp 24–31
Li J, Balazs M, Parks G, Clarkson P (2002) A species conserving genetic algorithm for multimodal function optimization. Evol Comput 10(3):207–234 (ISSN 1063-6560)
Li X (2005) Efficient differential evolution using speciation for multimodal function optimization. In: Proceedings of the conference on genetic and evolutionary computation (GECCO 2005). Washington DC, USA, pp 873–880
Liang J, Suganthan P, Deb K (2005) Novel composition test functions for numerical global optimization. In: Proceedings of the 2005 IEEE congress on evolutionary computation, pp 68–75
Lobo F, Lima C (2006) On the utility of the multimodal problem generator for assessing the performance of evolutionary algorithms. In: Proceedings of the ACM genetic and evolutionary computation conference (GECCO 2006), ACM Press
Macnish C (2007) Towards unbiased benchmarking of evolutionary and hybrid algorithms for real-valued optimisation. Connect Sci 19(4):361–385
Mahfoud S (1994) Genetic drift in sharing methods. In: Profeedings of the First IEEE conference on evolutionary computation, pp 67–72
Mahfoud S (1995a) A comparison of parallel and sequential niching methods. In: Proceedings of 6th international conference on genetic algorithms, pp 136–143
Mahfoud S (1995b) Niching methods for genetic algorithms. PhD thesis, University of Illinois, Urbana, IL, USA
Michalewicz Z (1996) Genetic algorithms + data structures = evolution programs. Springer, Berlin
Michalewicz Z, Deb K, Schmidt M, Stidsen T (2000) Test-case generator for nonlinear continuous parameter optimization techniques. IEEE Trans Evol Comput 4:197–215
Morrison R (2004) Designing evolutionary algorithms for dynamic environments. Springer, Berlin
Morrison R, De Jong K (1999) A test problem generator for nonstationary evironments. In: Proceedings of the congress of evolutionary computation. IEEE Press, Piscataway, pp 1843–1850
Pétrowski A (1996) A clearing procedure as a niching method for genetic algorithms. In: Proceedings of the 3rd IEEE international conference on evolutionary computation, pp 798–803
Press W, Flannery B, Teukolsky S, Vetterling W (1992) Numerical recipes in C, 2nd edn. Cambridge University Press, Cambridge. ISBN 0-521-43108-5
Price K, Rönkkönen J (2006) Comparing the uni-modal scaling performance of global and local selection in mutation-only differential evolution algorithm. In: Proceedings of 2006 IEEE world congress on computational intelligence, Vancouver, Canada, pp 7387–7394, 16–21 July 2006. ISBN 0-7803-9489-5
Price K, Storn R, Lampinen J (2005) Differential evolution: a practical approach to global optimization. Springer, Berlin. ISBN 3-540-20950-6
Rönkkönen J, Lampinen J (2007a) An extended mutation concept for the local selection based differential evolution algorithm. In: Proceedings of genetic and evolutionary computation conference (GECCO 2007), London, England, pp 689–696. ISBN 978-1-59593-697-4
Rönkkönen J, Lampinen J (2007b) On determining multiple global optima by differential evolution. In: Evolutionary and deterministic methods for design, optimization and control, proceedings of eurogen 2007. Jyväskylä, Finland, pp 146–151. ISBN 978-84-96736-45-0
Rönkkönen J, Li X, Kyrki V (2009) The role of local and global search in solving problems with multiple global optima. Technical Report 110, Department of Information Technology, Lappeenranta University of Technology. ISBN 978-952-214-730-1
Salomon R (1996) Reevaluating genetic algorithms performance under coordinate rotation of benchmark functions. Biosystems 39:263–278
Shir O, Bäck T (2006) Niche radius adaptation in the cma-es niching algorithm. In: Parallel problem solving from nature (PPSN IX). Springer, Berlin, pp 142–151. ISBN 978-3-540-38990-3
Singh G, Deb K (2006) Comparison of multi-modal optimization algorithms based on evolutionary algorithms. In: Proceedings of the genetic and evolutionary computation conference. ACM Press, Seattle, WA, pp 1305–1312
Storn R, Price K (1995) Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95-012, International Computer Science Institute (ICSI)
Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11:341–359
Thomsen R (2004) Multimodal optimization using crowding-based differential evolution. In: Proceedings of the 2004 congress on evolutionary computation, vol 2. Portland, pp 1382–1389
Törn A, Zilinskas A (1989) Global optimization, Lecture Notes in Computer Science. Springer, Berlin. ISBN 9783540508717
Ursem R (1999) Multinational evolutionary algorithms. In: Proceedings of congress of evolutionary computation (CEC 1999), vol 3. IEEE Press
Weise T, Niemczyk S, Skubch H, Reichle R, Geihs K (2008) A tunable model for multi-objective, epistatic, rugged, and neutral fitness landscapes. In: Proceedings of the 10th annual conference on genetic and evolutionary computation. Atlanta, GA, USA, pp 795–802
Whitley D, Lunacek M, Sokolov A (2006) Comparing the niches of cma-es, chc and pattern search using diverse benchmarks. In: Parallel problem solving from nature (PPSN IX). Springer, pp 988–997. ISBN 978-3-540-38990-3
Whitley D, Mathias K, Rana S, Dzubera J (1996) Evaluating evolutionary algorithms. Artif Intell 85:245–276
Wilcoxon F (1945) Individual comparisons by ranking methods. Biometrics 1:80–83
Wolpert D, Macready W (1995) No free lunch theorems for search. Technical report SFI-TR-95-02-010, The Santa Fe Institute
Wolpert D, Macready W (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82
Acknowledgments
The authors want to thank Harri Lattu and Jarmo Ilonen for their help with implementing the software, Jouni Sampo for mathematical consultation, and the anonymous reviewers for their useful comments.
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Rönkkönen, J., Li, X., Kyrki, V. et al. A framework for generating tunable test functions for multimodal optimization. Soft Comput 15, 1689–1706 (2011). https://doi.org/10.1007/s00500-010-0611-1
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DOI: https://doi.org/10.1007/s00500-010-0611-1