Acceleration and global convergence of a first-order primal--dual method for nonconvex problems

TL;DR

This paper extends the primal--dual hybrid gradient method to nonconvex problems with nonlinear operators, establishing convergence conditions, acceleration strategies, and demonstrating effectiveness in PDE-constrained optimization.

Contribution

It introduces new step length conditions and acceleration rules for nonconvex primal--dual methods, with proven convergence and applicability to PDE problems.

Findings

Proved linear convergence rates for nonconvex problems.

Established global convergence under certain conditions.

Validated step length rules on PDE-constrained optimization.

Abstract

The primal--dual hybrid gradient method (PDHGM, also known as the Chambolle--Pock method) has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step length rules for PDE-constrained optimization problems.