An Evolutionary Approach for the Computation of ϵ-Locally Optimal Solutions for Multiobjective Multimodal Optimization

Abstract

In this article, we address the problem of efficiently computing finite-size approximations of the set of $\epsilon $ -locally optimal solutions of a given multiobjective optimization problem (MOP). Such sets are in particular interesting in the context of multiobjective multimodal optimization (MMO). To this end, we first propose a bounded archiver, $ArchiveUpdateL_{Q,\epsilon }B$ , that is, a modification of a previously proposed unbounded archiver. These archivers can be used as external archivers to in principle any multiobjective evolutionary algorithm (MOEA). In order to reduce the computational cost compared to such archive equipped MOEAs, we propose, in a next step $L_{Q,\epsilon }$ MOEA. This evolutionary algorithm directly uses $ArchiveUpdateL_{Q,\epsilon }B$ for the selection process and hence does not need an external archive for the computation of $\epsilon $ -locally optimal solutions. We further propose a hybrid of $L_{Q,\epsilon }$ MOEA with a multiobjective continuation method, which significantly improves the accuracy of the obtained solutions in case the gradient information is at hand. Finally, we show some numerical results that demonstrate the benefit of both the bounded archiver and the new MOEAs.