Regularized Multiobjective Optimization with Directionally Lipschitzian Data

TL;DR

This work addresses regularized multiobjective optimization with directionally Lipschitzian data, focusing on proximal-type regularization and the derivation of necessary Pareto optimality conditions. It develops a variational-analysis framework based on the Mordukhovich limiting subdifferential and distance-function normals, and uses Ekeland’s variational principle to obtain Lagrange-type necessary conditions for proximal-regularized problems. The main contributions include exact penalization results and structured subdifferential inclusions that capture the interaction between multiple objectives, the proximal term, and geometric constraints, under local Lipschitz conditions around Pareto minimizers. The framework extends non-Clarke approaches to non-Lipschitz settings and informs the design and analysis of proximal-type algorithms in engineering, statistics, and medical physics.

Abstract

The paper is devoted to the study of regularized versions of multiobjective optimization problems described by directionally Lipschitzian functions. Such regularizations appear in proximal-type algorithms of multiobjective optimization, various models of machine learning, medical physics, etc. We investigate and illustrate several useful properties of directionally Lipschitzian functions, which distinguish them from locally Lipschitzian ones. By using advanced tools of variational analysis and generalized differentiation revolving around the limiting/Mordukhovich subdifferential, we derive necessary conditions for Pareto optimality in regularized multiobjective problems.

Figures (2)

• Figure 1: Graph of the functions $f_1$ and $f_2$.
• Figure 2: Graph of the functions $f_1$ and $f_2$.

Theorems & Definitions (5)

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• Remark 1