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An Evolutionary Approach for the Computation of <italic>ϵ</italic>-Locally Optimal Solutions for Multiobjective Multimodal Optimization
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DOI：10.1109/tevc.2025.3637276.png)
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Abstract
En 中文
In this article, we address the problem of efficiently computing finite-size approximations of the set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-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, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ArchiveUpdateL_{Q,\epsilon }B$ </tex-math></inline-formula>, 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{Q,\epsilon }$ </tex-math></inline-formula>MOEA. This evolutionary algorithm directly uses <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ArchiveUpdateL_{Q,\epsilon }B$ </tex-math></inline-formula> for the selection process and hence does not need an external archive for the computation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-locally optimal solutions. We further propose a hybrid of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{Q,\epsilon }$ </tex-math></inline-formula>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.
Keywords:
Archiving
continuation
evolutionary algorithms
multimodal optimization
multiobjective optimization
nearly optimal solutions
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