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

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 65 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-sum, and Bukin-6 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, DES 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.