New global optimality conditions for nonsmooth DC optimization problems

TL;DR

This paper introduces new global optimality conditions for nonsmooth DC optimization problems using affine support sets, providing both theoretical insights and practical tools for solving constrained and unconstrained problems.

Contribution

It develops a novel approach based on affine support sets for deriving necessary and sufficient global optimality conditions in DC optimization, improving upon existing methods.

Findings

New global optimality conditions for DC problems.

Simple criteria for the exactness of the $\,l_1$ penalty function.

Constructive examples demonstrating the application of the conditions.

Abstract

In this article we propose a new approach to an analysis of DC optimization problems. This approach was largely inspired by codifferential calculus and the method of codifferential descent and is based on the use of a so-called affine support set of a convex function instead of the Frenchel conjugate function. With the use of affine support sets we define a global codifferential mapping of a DC function and derive new necessary and sufficient global optimality conditions for DC optimization problems. We also provide new simple necessary and sufficient conditions for the global exactness of the $\ell_1$ penalty function for DC optimization problems with equality and inequality constraints and present a series of simple examples demonstrating a constructive nature of the new global optimality conditions. These examples show that when the optimality conditions are not satisfied, they can be easily utilised in order to find "global descent" directions of both constrained and unconstrained problems. As an interesting theoretical example, we apply our approach to the analysis of a nonsmooth problem of Bolza.