Conical regularization for multi-objective optimization problems

Abstract

This paper focuses on multi-objective optimization problem with linear infinite-dimensional constraints. The main novelty of the contribution is that we do not assume that the ordering cone has a nonempty interior and consequently there might not exist KKT system to compute the solution numerically. We devise a regularization framework which consists of replacing the ordering cone by a family of solid cones. We provide general features of the original problem and the family of regularized problems. Using tools from the set-valued analysis, we prove that the weak outer limit of regularized solutions is the set of minimal elements of the original problem. We scalarize the original problem as well as the regularized analogs and establish that the corresponding scalarized solutions parametrize the regularized minimal elements. We demonstrate the convergence, in the Kuratowski-Mosco sense, of the regularized solutions to the efficient set of the original problem. We prove an a priori Hölder continuity estimate for the regularized efficient set-valued map when the original problem is regular, and a verifiable norm boundedness property of the multipliers holds. We present detailed numerical examples to show the efficacy of the developed framework.