On duality for nonconvex minimization problems within the framework of abstract convexity

TL;DR

This paper develops a duality framework for nonconvex minimization problems using abstract convexity, introducing new dual concepts and conditions for zero duality gap and optimality.

Contribution

It extends duality theory to nonconvex problems via $\

Findings

Provides conditions for zero duality gap

Introduces $\

Discusses relationships with existing conjugate duals

Abstract

By applying the perturbation function approach, we propose the Lagrangian and the conjugate duals for minimization problems of the sum of two, generally nonconvex, functions. The main tools are the $\Phi$-convexity theory and minimax theorems for $\Phi$-convex functions. We provide conditions ensuring zero duality gap and introduce $\Phi$-Karush-Kuhn-Tucker conditions that characterize solutions to primal and dual problems. We also discuss the relationship between the dual problems introduced in the present investigation and some conjugate-type duals existing in the literature.