An Evolutionary Approach for the Computation of ∈-Locally Optimal Solutions for Multi-Objective Multimodal Optimization

In this paper, we address the problem of efficiently computing finite-size approximations of the set of -locally optimal solutions of a given multi-objective optimization problem (MOP). Such sets are in particular interesting in the context of multi-objective multimodal optimization (MMO). To this end, we first propose a bounded archiver, ArchiveUpdateLQ,∈B, that is, a modification of a previously proposed unbounded archiver. These archivers can be used as external archivers to in principle any multi-objective evolutionary algorithm (MOEA). In order to reduce the computational cost compared to such archive equipped MOEAs, we propose, in a next step LQ,∈MOEA. This evolutionary algorithm directly uses ArchiveUpdateLQ,∈B for the selection process and hence does not need an external archive for the computation of ∈-locally optimal solutions. We further propose a hybrid of LQ,∈MOEA with a multi-objective 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.