On the improved conditions for some primal-dual algorithms

TL;DR

This paper improves the convergence conditions for primal-dual algorithms solving convex problems, expanding the allowable stepsizes and unifying several algorithms under a common framework.

Contribution

It introduces a base algorithm connecting multiple primal-dual methods and relaxes assumptions to enlarge stepsize bounds, enhancing algorithm efficiency.

Findings

Chambolle-Pock stepsize limit increased to 4/3 of previous bound

Unified framework for AFBA, PD3O, and Chambolle-Pock algorithms

Convergence proven under relaxed linear operator assumptions

Abstract

The convex minimization of $f(\mathbf{x})+g(\mathbf{x})+h(\mathbf{A}\mathbf{x})$ over $\mathbb{R}^n$ with differentiable $f$ and linear operator $\mathbf{A}: \mathbb{R}^n\rightarrow \mathbb{R}^m$, has been well-studied in the literature. By considering the primal-dual optimality of the problem, many algorithms are proposed from different perspectives such as monotone operator scheme and fixed point theory. In this paper, we start with a base algorithm to reveal the connection between several algorithms such as AFBA, PD3O and Chambolle-Pock. Then, we prove its convergence under a relaxed assumption associated with the linear operator and characterize the general constraint on primal and dual stepsizes. The result improves the upper bound of stepsizes of AFBA and indicates that Chambolle-Pock, as the special case of the base algorithm when $f=0$, can take the stepsize of the dual iteration up to $4/3$ of the previously proven one.