Regularized Multiobjective Optimization with Directionally Lipschitzian Data

TL;DR

This paper explores regularized multiobjective optimization problems involving directionally Lipschitzian functions, providing theoretical insights and necessary conditions for Pareto optimality relevant to machine learning and physics applications.

Contribution

It introduces new properties of directionally Lipschitzian functions and derives necessary optimality conditions using advanced variational analysis tools.

Findings

Directionally Lipschitzian functions differ from locally Lipschitzian ones.

Necessary conditions for Pareto optimality are established.

Applications include proximal algorithms and models in machine learning and 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.