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Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions

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Abstract

In this paper we compare the performance of the Differential Evolution (DE) and the Repulsive Particle Swarm (RPS) methods of global optimization. To this end, seventy test functions have been chosen. Among these test functions, some are new while others are well known in the literature; some are unimodal, the others multi-modal; some are small in dimension (no. of variables, x in f(x)), while the others are large in dimension; some are algebraic polynomial equations, while the other are transcendental, etc. FORTRAN programs of DE and RPS have been appended. Among 70 functions, a few have been run for small as well as large dimensions. In total, 73 optimization exercises have been done. DE has succeeded in 63 cases while RPS has succeeded in 55 cases. In almost all cases, DE has converged faster and given much more accurate results. The convergence of RPS is much slower even for lesser stringency on accuracy. Some test functions have been hard for both the methods. These are: Zero-Sum (30D), Perm#1, Perm#2, Power and Bukin functions, Weierstrass, and Michalewicz functions. From what we find, one cannot reach at the definite conclusion that the DE performs better or worse than the RPS. None could assure a supremacy over the other. Each one faltered in some cases; each one succeeded in some others. However, DE is unquestionably faster, more accurate and more frequently successful than the RPS. It may be argued, nevertheless, that alternative choice of adjustable parameters could have yielded better results in either method’s case. The protagonists of either method could suggest that. Our purpose is not to join with the one or the other. We simply want to highlight that in certain cases they both succeed, in certain other case they both fail and each one has some selective preference over some particular type of surfaces. What is needed is to identify such structures and surfaces that suit a particular method most. It is needed that we find out some criteria to classify the problems that suit (or does not suit) a particular method. This classification will highlight the comparative advantages of using a particular method for dealing with a particular class of problems.

Suggested Citation

  • Mishra, SK, 2006. "Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions," MPRA Paper 1005, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:1005
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    File URL: https://mpra.ub.uni-muenchen.de/1005/1/MPRA_paper_1005.pdf
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    Cited by:

    1. Sudhanshu K MISHRA, 2009. "Representation-Constrained Canonical Correlation-Analysis: A Hybridization Of Canonical Correlation And Principal Component Analysis," Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 4(1(7)_ Spr).
    2. Piotrowski, Adam P. & Napiorkowski, Jaroslaw J. & Kiczko, Adam, 2012. "Differential Evolution algorithm with Separated Groups for multi-dimensional optimization problems," European Journal of Operational Research, Elsevier, vol. 216(1), pages 33-46.
    3. S. K. Mishra, 2010. "(Computer Algorithms) The Most Representative Composite Rank Ordering of Multi-Attribute Objects by the Particle Swarm Optimization Method," Journal of Quantitative Economics, The Indian Econometric Society, vol. 8(2), pages 165-200.
    4. SK Mishra, 2007. "Estimation of Zellner-Revankar Production Function Revisited," Economics Bulletin, AccessEcon, vol. 3(14), pages 1-7.
    5. Sudhanshu K Mishra, 2013. "Global Optimization of Some Difficult Benchmark Functions by Host-Parasite Coevolutionary Algorithm," Economics Bulletin, AccessEcon, vol. 33(1), pages 1-18.
    6. Mishra, SK, 2012. "Construction of Pena’s DP2-based ordinal synthetic indicator when partial indicators are rank scores," MPRA Paper 39088, University Library of Munich, Germany.
    7. S K Mishra, 2007. "Globalization and Structural Changes in the Indian Industrial Sector: An Analysis of Production Functions," The IUP Journal of Managerial Economics, IUP Publications, vol. 0(4), pages 56-81, November.
    8. Mishra, SK, 2007. "Completing correlation matrices of arbitrary order by differential evolution method of global optimization: A Fortran program," MPRA Paper 2000, University Library of Munich, Germany.
    9. Mishra, SK, 2008. "A note on the sub-optimality of rank ordering of objects on the basis of the leading principal component factor scores," MPRA Paper 12419, University Library of Munich, Germany.
    10. repec:ebl:ecbull:v:3:y:2007:i:14:p:1-7 is not listed on IDEAS

    More about this item

    Keywords

    : Global optimization; Stochastic search; Repulsive particle swarm; Differential Evolution; Clustering algorithm; Simulated annealing; Genetic algorithm; Tabu search; Ant Colony algorithm; Monte Carlo method; Box algorithm; Nelder-Mead; Nonlinear programming; FORTRAN computer program; local optima; Benchmark; test functions;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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