Multimodal Optimization Using a Bi-Objective Evolutionary Algorithm

In a multimodal optimization task, the main purpose is to find multiple optimal solutions (global and local), so that the user can have better knowledge about different optimal solutions in the search space and as and when needed, the current solution may be switched to another suitable optimum solution. To this end, evolutionary optimization algorithms (EA) stand as viable methodologies mainly due to their ability to find and capture multiple solutions within a population in a single simulation run. With the preselection method suggested in 1970, there has been a steady suggestion of new algorithms. Most of these methodologies employed a niching scheme in an existing single-objective evolutionary algorithm framework so that similar solutions in a population are deemphasized in order to focus and maintain multiple distant yet near-optimal solutions. In this paper, we use a completely different strategy in which the single-objective multimodal optimization problem is converted into a suitable bi-objective optimization problem so that all optimal solutions become members of the resulting weak Pareto-optimal set. With the modified definitions of domination and different formulations of an artificially created additional objective function, we present successful results on problems with as large as 500 optima. Most past multimodal EA studies considered problems having only a few variables. In this paper, we have solved up to 16-variable test problems having as many as 48 optimal solutions and for the first time suggested multimodal constrained test problems which are scalable in terms of number of optima, constraints, and variables. The concept of using bi-objective optimization for solving single-objective multimodal optimization problems seems novel and interesting, and more importantly opens up further avenues for research and application.

[1]  Lawrence Davis,et al.  Using a genetic algorithm to optimize problems with feasibility constraints , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[2]  J. Kennedy.  Stereotyping: improving particle swarm performance with cluster analysis , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[3]  David E. Goldberg,et al.  Probabilistic Crowding: Deterministic Crowding with Probabilistic Replacement , 1999 .

[4]  Kalyanmoy Deb,et al.  Omni-optimizer: A generic evolutionary algorithm for single and multi-objective optimization , 2008, Eur. J. Oper. Res..

[5]  Lothar Thiele,et al.  Multiobjective genetic programming: reducing bloat using SPEA2 , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[6]  Kang Li,et al.  A Sequential Niching Technique for Particle Swarm Optimization , 2005, ICIC.

[7]  L. Darrell Whitley,et al.  Lamarckian Evolution, The Baldwin Effect and Function Optimization , 1994, PPSN.

[8]  Aravind Srinivasan,et al.  Innovization: innovating design principles through optimization , 2006, GECCO.

[9]  Xiaodong Li,et al.  This article has been accepted for inclusion in a future issue. IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION 1 Locating and Tracking Multiple Dynamic Optima by a Particle Swarm Model Using Speciation , 2022 .

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

[11]  Patrick D. Surry,et al.  A Multi-objective Approach to Constrained Optimisation of Gas Supply Networks: the COMOGA Method , 1995, Evolutionary Computing, AISB Workshop.

[12]  Ofer M. Shir,et al.  Niching in evolution strategies , 2005, GECCO '05.

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

[14]  Zachary V. Hendershot.  A Differential Evolution Algorithm for Automatically Discovering Multiple Global Optima in Multidimensional, Discontinuous Spaces , 2004, MAICS.

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

[16]  Kalyanmoy Deb,et al.  RePAMO: Recursive Perturbation Approach for Multimodal Optimization , 2013 .

[17]  Xiaodong Li,et al.  A Generator for Multimodal Test Functions with Multiple Global Optima , 2008, SEAL.

[18]  Kalyanmoy Deb,et al.  Multiobjective Problem Solving from Nature: From Concepts to Applications (Natural Computing Series) , 2008 .

[19]  Carlos A. Coello Coello,et al.  Constraint-handling in genetic algorithms through the use of dominance-based tournament selection , 2002, Adv. Eng. Informatics.

[20]  Kalyanmoy Deb,et al.  A Bi-criterion Approach to Multimodal Optimization: Self-adaptive Approach , 2010, SEAL.

[21]  Kalyanmoy Deb,et al.  Real-coded Genetic Algorithms with Simulated Binary Crossover: Studies on Multimodal and Multiobjective Problems , 1995, Complex Syst..

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

[23]  Richard A. Watson,et al.  Reducing Local Optima in Single-Objective Problems by Multi-objectivization , 2001, EMO.

[24]  Jani Rönkkönen.  ContinuousMultimodal Global Optimization with Differential Evolution-Based Methods , 2009 .

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

[26]  Edwin D. de Jong,et al.  Reducing bloat and promoting diversity using multi-objective methods , 2001 .

[27]  R. K. Ursem.  Multi-objective Optimization using Evolutionary Algorithms , 2009 .

[28]  M. Jensen.  Helper-Objectives: Using Multi-Objective Evolutionary Algorithms for Single-Objective Optimisation , 2004 .

[29]  Andreas Zell,et al.  A Clustering Based Niching EA for Multimodal Search Spaces , 2003, Artificial Evolution.

[30]  Mikkel T. Jensen,et al.  Guiding Single-Objective Optimization Using Multi-objective Methods , 2003, EvoWorkshops.

[31]  Joshua D. Knowles,et al.  Multiobjectivization by Decomposition of Scalar Cost Functions , 2008, PPSN.

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

[33]  Kalyanmoy Deb,et al.  Multimodal Deceptive Functions , 1993, Complex Syst..

[34]  Kalyanmoy Deb,et al.  Optimization for Engineering Design: Algorithms and Examples , 2004 .

[35]  Xiaodong Yin,et al.  A Fast Genetic Algorithm with Sharing Scheme Using Cluster Analysis Methods in Multimodal Function Optimization , 1993 .

[36]  Kenneth Alan De Jong,et al.  An analysis of the behavior of a class of genetic adaptive systems. , 1975 .

[37]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[38]  Xiaodong Li,et al.  Evolutionary algorithms and multi-objectivization for the travelling salesman problem , 2009, GECCO.

[39]  Chang-Hwan Im,et al.  A novel algorithm for multimodal function optimization based on evolution strategy , 2004 .

[40]  Xin Yao,et al.  A dilemma for fitness sharing with a scaling function , 1995, Proceedings of 1995 IEEE International Conference on Evolutionary Computation.

[41]  Roman Hamar,et al.  Higher-order finite element modeling and optimization of actuator with non-linear materials , 2013, Computing.

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

[43]  Kalyanmoy Deb,et al.  Multiobjective Problem Solving from Nature: From Concepts to Applications , 2008, Natural Computing Series.

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

[45]  Mikkel T. Jensen,et al.  Helper-objectives: Using multi-objective evolutionary algorithms for single-objective optimisation , 2004, J. Math. Model. Algorithms.

[46]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[47]  Patrick Siarry,et al.  Island Model Cooperating with Speciation for Multimodal Optimization , 2000, PPSN.

[48]  Carlos A. Coello Coello,et al.  A Review of Particle Swarm Optimization Methods Used for Multimodal Optimization , 2009, Innovations in Swarm Intelligence.

[49]  D. J. Cavicchio,et al.  Adaptive search using simulated evolution , 1970 .

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

[51]  Michael N. Vrahatis,et al.  On the computation of all global minimizers through particle swarm optimization , 2004, IEEE Transactions on Evolutionary Computation.

[52]  Hyun-Kyo Jung,et al.  Niching genetic algorithm with restricted competition selection for multimodal function optimization , 1999 .

[53]  Kathrin Klamroth,et al.  Constrained optimization using multiple objective programming , 2007, J. Glob. Optim..

[54]  A. Ravindran,et al.  Engineering Optimization: Methods and Applications , 2006 .

[55]  Kalyanmoy Deb,et al.  An Investigation of Niche and Species Formation in Genetic Function Optimization , 1989, ICGA.