Lagrangian duality for nonconvex optimization problems with abstract convex functions

TL;DR

This paper explores Lagrangian duality in nonconvex optimization using $\

Contribution

It introduces duality results for nonconvex problems via $\\Phi$-convexity, extending duality theory to various classes of nonconvex functions.

Findings

Conditions for zero duality gap established

Conditions for strong duality established

Applicable to prox-bounded, DC, weakly convex, and paraconvex functions

Abstract

We investigate Lagrangian duality for nonconvex optimization problems. To this aim we use the $\Phi$-convexity theory and minimax theorem for $\Phi$-convex functions. We provide conditions for zero duality gap and strong duality. Among the classes of functions, to which our duality results can be applied, are prox-bounded functions, DC functions, weakly convex functions and paraconvex functions.