A framework for generating tunable test functions for multimodal optimization

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.

[1]  Alain Pétrowski,et al.  A clearing procedure as a niching method for genetic algorithms , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

[2]  L. Darrell Whitley,et al.  Evaluating Evolutionary Algorithms , 1996, Artif. Intell..

[3]  D. Wolpert,et al.  No Free Lunch Theorems for Search , 1995 .

[4]  Mark Zwolinski,et al.  Using evolutionary and hybrid algorithms for DC operating point analysis of nonlinear circuits , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[5]  William H. Press,et al.  Numerical Recipes in Fortran 77: The Art of Scientific Computing 2nd Editionn - Volume 1 of Fortran Numerical Recipes , 1992 .

[6]  Cláudio F. Lima,et al.  On the utility of the multimodal problem generator for assessing the performance of evolutionary algorithms , 2006, GECCO '06.

[7]  Kalyanmoy Deb,et al.  Comparison of multi-modal optimization algorithms based on evolutionary algorithms , 2006, GECCO.

[8]  P. John Clarkson,et al.  A Species Conserving Genetic Algorithm for Multimodal Function Optimization , 2002, Evolutionary Computation.

[9]  René Thomsen,et al.  Multimodal optimization using crowding-based differential evolution , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[10]  Yaroslav D. Sergeyev,et al.  Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization , 2003, TOMS.

[11]  Kurt Geihs,et al.  A tunable model for multi-objective, epistatic, rugged, and neutral fitness landscapes , 2008, GECCO '08.

[12]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[13]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[14]  Jürgen Branke,et al.  Evolutionary Optimization in Dynamic Environments , 2001, Genetic Algorithms and Evolutionary Computation.

[15]  Ralph R. Martin,et al.  A Sequential Niche Technique for Multimodal Function Optimization , 1993, Evolutionary Computation.

[16]  William H. Press,et al.  Numerical Recipes in C, 2nd Edition , 1992 .

[17]  Samir W. Mahfoud.  Niching methods for genetic algorithms , 1996 .

[18]  R. K. Ursem.  Multinational evolutionary algorithms , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[19]  Pierre Bezier,et al.  The Mathematical Basis of the Unisurf CAD System , 1986 .

[20]  Georges R. Harik,et al.  Finding Multimodal Solutions Using Restricted Tournament Selection , 1995, ICGA.

[21]  R. Storn,et al.  Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces , 2004 .

[22]  Xiaodong Li,et al.  Efficient differential evolution using speciation for multimodal function optimization , 2005, GECCO '05.

[23]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[24]  Ofer M. Shir,et al.  Niche Radius Adaptation in the CMA-ES Niching Algorithm , 2006, PPSN.

[25]  R.W. Morrison,et al.  A test problem generator for non-stationary environments , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[26]  K. Dejong,et al.  An analysis of the behavior of a class of genetic adaptive systems , 1975 .

[27]  Jouni Lampinen,et al.  An extended mutation concept for the local selection based differential evolution algorithm , 2007, GECCO '07.

[28]  Bull,et al.  An Overview of Genetic Algorithms: Part 2, Research Topics , 1993 .

[29]  Marcus Gallagher,et al.  A general-purpose tunable landscape generator , 2006, IEEE Transactions on Evolutionary Computation.

[30]  Zbigniew Michalewicz,et al.  Test-case generator for nonlinear continuous parameter optimization techniques , 2000, IEEE Trans. Evol. Comput..

[31]  L. Darrell Whitley,et al.  Comparing the Niches of CMA-ES, CHC and Pattern Search Using Diverse Benchmarks , 2006, PPSN.

[32]  Kit Po Wong,et al.  A Differential Evolution Based Method for Power System Planning , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[33]  Samir W. Mahfoud.  A Comparison of Parallel and Sequential Niching Methods , 1995, ICGA.

[34]  David E. Goldberg,et al.  Genetic Algorithms with Sharing for Multimodalfunction Optimization , 1987, ICGA.

[35]  Cara MacNish.  Towards unbiased benchmarking of evolutionary and hybrid algorithms for real-valued optimisation , 2007, Connect. Sci..

[36]  R. Salomon.  Re-evaluating genetic algorithm performance under coordinate rotation of benchmark functions. A survey of some theoretical and practical aspects of genetic algorithms. , 1996, Bio Systems.

[37]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1992, Artificial Intelligence.

[38]  Jing J. Liang,et al.  Novel composition test functions for numerical global optimization , 2005, Proceedings 2005 IEEE Swarm Intelligence Symposium, 2005. SIS 2005..

[39]  R. Storn,et al.  Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series) , 2005 .

[40]  Jani Rönkkönen,et al.  Comparing the Uni-Modal Scaling Performance of Global and Local Selection in a Mutation-Only Differential Evolution Algorithm , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[41]  Pierre E. Bézier,et al.  How Renault Uses Numerical Control for Car Body Design and Tooling , 1968 .

[42]  F. Wilcoxon.  Individual Comparisons by Ranking Methods , 1945 .

[43]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[44]  Ronald W. Morrison,et al.  Designing Evolutionary Algorithms for Dynamic Environments , 2004, Natural Computing Series.

[45]  Samir W. Mahfoud.  Genetic drift in sharing methods , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.