Primal-dual algorithm for weakly convex functions under sharpness conditions

TL;DR

This paper analyzes the convergence of a primal-dual algorithm for weakly convex functions under sharpness conditions, introducing a new duality gap measure and demonstrating its effectiveness in image processing applications.

Contribution

It introduces a modified duality gap function and establishes convergence conditions for primal-dual algorithms on weakly convex problems under sharpness assumptions.

Findings

Convergence is guaranteed near saddle points under sharpness.

Numerical examples in image denoising and deblurring validate the theoretical results.

The new duality gap function improves analysis of weakly convex optimization.

Abstract

We investigate the convergence of the primal-dual algorithm for composite optimization problems when the objective functions are weakly convex. We introduce a modified duality gap function, which is a lower bound of the standard duality gap function. Under the sharpness condition of this new function, we identify the area around the set of saddle points where we obtain the convergence of the primal-dual algorithm. We give numerical examples and applications in image denoising and deblurring to demonstrate our results.