# Primal-dual proximal splitting and generalized conjugation in non-smooth non-convex optimization

## TL;DR

This paper extends primal-dual proximal splitting methods to solve challenging non-convex, non-smooth problems by leveraging generalized conjugates and saddle-point reformulations, with proven local linear convergence under certain conditions.

## Contribution

It introduces a novel framework applying primal-dual proximal splitting to non-convex, non-smooth problems using generalized conjugates and saddle-point formulations, with convergence analysis.

## Key findings

- Method successfully applied to Nash equilibrium and Potts segmentation problems.
- Proven local linear convergence under strong convexity assumptions.
- Numerical experiments confirm theoretical convergence results.

## Abstract

We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation model, can be written in terms of generalized conjugates of convex functionals. These, in turn, can be formulated as saddle-point problems involving convex non-smooth functionals and a general smooth but non-bilinear coupling term. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal--dual proximal splitting method of Chambolle and Pock is applicable to the solution of such problems. Under sufficient local strong convexity assumptions of the functionals -- but still with a non-bilinear coupling term -- we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems.