A primal dual formulation through a proximal approach for non-convex variational optimization

TL;DR

This paper introduces a primal dual formulation using a proximal approach tailored for non-convex variational models, with theoretical foundations and applications to Ginzburg-Landau type models.

Contribution

It develops a novel primal dual framework for non-convex variational problems using proximal methods, expanding duality theory in this context.

Findings

Established a primal dual formulation for non-convex models

Applied the framework to Ginzburg-Landau type models

Presented optimality conditions and duality principles

Abstract

This article develops a primal dual formulation for a primal proximal approach suitable for a large class of non-convex models in the calculus of variations. The results are established through standard tools of functional analysis, convex analysis and duality theory and are applied to a Ginzburg-Landau type model. Finally, in the last two sections, we present concerning optimal conditions and another related duality principle for the model in question.