Finding ϵ-Locally Optimal Solutions for Multiobjective Multimodal Optimization

In this article, we address the problem of computing all locally optimal solutions of a given multiobjective problem whose images are sufficiently close to the Pareto front. Such $\epsilon $ -locally optimal solutions are particularly interesting in the context of multiobjective multimodal optimization (MMO). To accomplish this task, we first define a new set of interest, $L_{Q,\epsilon }$ , that is strongly related to the recently proposed set of $\epsilon $ -acceptable solutions. Next, we propose a new unbounded archiver, $ArchiveUpdateL_{Q,\epsilon }$ , aiming to capture $L_{Q,\epsilon }$ in the limit. This archiver can in principle be used in combination with any multiobjective evolutionary algorithm (MOEA). Further, we equip numerous MOEAs with $ArchiveUpdateL_{Q,\epsilon }$ , 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 $\Delta _{p}$ . Additionally, we propose and use a new performance indicator, $I_{\mathrm { EDR}}$ , which results in comparable performances but which is applicable to problems defined in higher dimensions (in particular in decision variable space).