Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-modal Benchmark Functions

Our objective in this paper is to compare the performance of the Differential Evolution (DE) and the Repulsive Particle Swarm (RPS) methods of global optimization. To this end, some relatively difficult test functions have been chosen. These functions are: Perm, Power-Sum, Bukin, Zero-Sum, Hougen, Giunta, DCS, Kowalik, Fletcher-Powell and some now functions. Our results show that DE (with the exponential crossover scheme) mostly fails to find the optimum of most of these functions. Of course, it succeeds in case of some functions (perm#2, zero-sum) for very small dimension (m), but begins to falter as soon as the dimension is increased. In case of DCS function, it works well up to m (dimension) = 5. When we use no crossover (only probabilistic replacement) we obtain better results in case of several of the functions under study. Thus, overall, table #2 presents better results than what table #1 does. In case of Perm#1, Perm#2, Zero-sum, Kowalik, Hougen and Power-sum functions the advantage is clear. Whether crossover or no crossover, DE falters when the optimand function has some element of randomness. This is indicated by the functions: Yao-Liu#7, Fletcher-Powell, and “New function#2”. DE has no problems in optimizing the “New function#1”. But the “New function #2” proves to be a hard nut. However, RPS performs much better for such stochastic functions. When the Fletcher-Powell function is optimized with non-stochastic c vector, DE works fine. But as soon as c is stochastic, it becomes unstable. Thus, it may be observed that an introduction of stochasticity into the decision variables (or simply added to the function as in Yao-Liu#7) interferes with the fundamentals of DE, which works through attainment of better and better (in the sense of Pareto improvement) population at each successive iteration.