Evolutionary Multimodal Optimization Revisited

We revisit a class of multimodal function optimizations using evolutionary algorithms reformulated into a multiobjective framework where previous implementations have needed niching/sharing to ensure diversity. In this paper, we use a steady-state multiobjective algorithm which preserves diversity without niching to produce diverse sampling of the Pareto-front with significantly lower computational effort. Multimodal optimization (MMO) and multiobjective optimization (MOO) are two classes of optimizations requiring multiple (near-)optimal solutions: having found a solution set, a user makes a selection from the (hopefully) diverse options. In this context, niching/sharing techniques have been commonly employed to ensure a diverse solution set although such techniques work the best when one has a priori knowledge of the problem. In most real-problems, the analytical form is unknown and so picking good niche parameters is problematic. Consequently, most of the work related to MMO using EAs has been done to test the efficacy of the EAs in solving known problems rather than solving the problem per se. Watson [1] concluded that sharing-based GAs often perform worse than random search and questioned whether niching is really useful for identifying multiple fitness peaks in MMOs. We have revisited solving MMO using EAs without any problem-dependent parameters using the same reformulation of MMO into a MOO framework as [2], to obtain good diversity in objective space without any explicit diversity-preserving operator. Deb [2] has recast of a number of single-objective MMO problems into dualobjective MOO problems and empirically investigated the effects of sharing. Many have studied the promotion of diversity using sharing for MOO problems – see [3] for a review. We have used a MOO algorithm [3] which, to the best of our knowledge, is the only implementation which does not need any explicit sharing mechanism; we demonstrate its efficacy in achieving diversity for two sample MMO problems, F1 (Sect. 4.1 of [2]) and F2 (Sect. 5.3 of [2]) which were considered by earlier researchers using multiobjective methods. We have used the same formulation, as far as is ∗ Partially supported by the Ministry of Human Resource Development, Government of India Evolutionary Multimodal Optimization Revisited 1593 known, for fair comparison. We repeated the experiments many hundreds of times, each with a different initial population to check the consistency of the results. Typical results selected on the basis of their average performance are presented below. The multi-modal F2 function can be recast as a multiobjective problem requiring the simultaneous minimization of f21(x1) and f22; see Sect. 5.3 of reference [2] for more details. Figure 1a shows the initial population (size = 100) where we obtain individuals near the optima by random chance. Figure 1b shows the population after 100 epochs which can be compared with the results for 500 generations in [2] and are superior to those in [2] both in terms of proximity to the Pareto-optimal front and diversity. Significantly this has been achieved at reduced computational cost. We have also studied the F1 function (Sect. 4.1 of [2]) and find a result entirely consistent with that we have observed with function F2 – see [3]. Explicit diversity preserving methods need prior knowledge and their efficacy depends on parameter fine-tuning; without proper values they cannot be beneficial. Claims of the superiority of variablevs. objective space sharing are unfounded, problem dependent and nothing general can be said on the selection of proper values for niching/sharing. In conclusion, we have shown that we can solve multimodal problems by recasting them as multiobjective ones without an explicit niching/sharing. Comparing our results with previous work [2], the algorithm employed here provided superior diversity and proximity to the true Pareto-front at reduced computational cost.