A primal dual variational formulation suitable for a large class of non-convex problems in optimization

TL;DR

This paper introduces a new primal-dual variational framework for non-convex optimization problems, leveraging convex analysis and duality theory to establish dual formulations with no duality gap near critical points.

Contribution

It develops a dual formulation applicable to a broad class of non-convex problems, including dual variables with large concave domains, and proves the absence of duality gap locally.

Findings

Dual formulation extends to primal variables with large concave domains.

No duality gap exists at local extremal points.

Framework applicable to various non-convex variational problems.

Abstract

In this article we develop a new primal dual variational formulation suitable for a large class of non-convex problems in the calculus of variations. The results are obtained through basic tools of convex analysis, duality theory, the Legendre transform concept and the respective relations between the primal and dual variables. The novelty here is that the dual formulation is established also for the primal variables, however with a large domain region of concavity about a critical point. Finally, we formally prove there is no duality gap between the primal and dual formulations in a local extremal context.