Finding ϵ-Locally Optimal Solutions for Multi-Objective Multimodal Optimization

In this paper, we address the problem of computing all locally optimal solutions of a given multi-objective problem whose images are sufficiently close to the Pareto front. Such -locally optimal solutions are particularly interesting in the context of multi-objective multimodal optimization (MMO). To accomplish this task, we first define a new set of interest, LQ, that is strongly related to the recently proposed set of -acceptable solutions. Next, we propose a new unbounded archiver, ArchiveUpdateLQ, aiming to capture LQ,in the limit. This archiver can in principle be used in combination with any multi-objective evolutionary algorithm (MOEA). Further, we equip numerous MOEAs with ArchiveUpdateLQ, investigate their performances across several benchmark functions, and compare the enhanced MOEAs with their archive-free counterparts. For our experiments, we utilize the well-established metrics HV, IGDX, and p. Additionally, we propose and use a new performance indicator, IEDR, which results in comparable performances but which is applicable to problems defined in higher dimensions (in particular in decision variable space).