${\varepsilon}$-optimality in reverse convex optimization

TL;DR

This paper introduces a novel characterization of approximate solutions in reverse convex optimization problems, extending existing results to broader cases including additional convex constraints and nonlinear equalities.

Contribution

It develops a new approach converting reverse problems into unconstrained bicriteria DC programs using Fenchel's ${\

Findings

Provides a characterization of ${\

Extends results to problems with convex constraints and nonlinear equalities

Applicable to functions with extended values and under certain constraint qualifications

Abstract

We characterize approximate global optimal solutions (${\varepsilon}$-optima) to reverse optimization problems, namely, problems whose non-convex constraint is of the form $h(x) \geq 0$. This issue has not been addressed previously in the literature. Our idea consists of converting the reverse program into an unconstrained bicriteria DC program. The main condition presented is obtained in terms of Fenchel's ${\varepsilon}$-subdifferentials thanks to an earlier result in difference vector optimization by El Maghri. This extends and improves similar results from the literature dealing with exact (${\varepsilon} = 0$) solutions. Moreover, as we consider functions with extended values, our approach also applies to reverse problems subject to additional convex constraints, provided that Moreau-Rockafellar or Attouch-Br\'ezis constraint qualification conditions are satisfied. Similarly, new results for the special case of a nonlinear equality constraint $h(x) = 0$ are also obtained.