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

Abstract:     
This paper aims at comparing 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. Among these test functions, some are new while others are well known in the literature.

We use DE method with the exponential crossover scheme as well as with no crossover (only probabilistic replacement). Our findings suggest that DE (with the exponential crossover scheme) mostly fails to find the optimum in case of the functions under study. Of course, it succeeds in case of some functions (Perm #2, Zero-sum) for very small dimension, but begins to falter as soon as the dimension is increased. In case of DCS function, it works well up to dimension=5. When we use no crossover (only probabilistic replacement) we obtain better results in case of several of the functions under study. In case of Perm #1, Perm #2, Zero-sum, Kowalik, Hougen and Power-sum functions, a remarkable advantage is there.

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 vector, DE works fine. But as soon as 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.

The paper concludes: (1) for different types of problems, different schemes of crossover (including none) may be suitable or unsuitable, (2) Stochasticity entering into the optimand function may make DE unstable, but RPS may function well.

Keywords: Differential Evolution, Repulsive Particle Swarm, Global optimization, non-convex functions, Fortran, computer program, benchmark, test, Stochastic functions, Fletcher-Powell, Kowalik, Hougen, Power-sum, Perm, Zero-sum, New functions, Bukin function

JEL Classifications: C15, C63

Working Paper Series