Finding ϵ -Locally Optimal Solutions for Multiobjective Multimodal Optimization
Angel E. Rodríguez-Fernandez, Lennart Schäpermeier, Carlos Hernández, Pascal Kerschke, Heike Trautmann, Oliver Schütze · IEEE Transactions on Evolutionary Computation · 2024
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).