Parametrization and convergence of a primal-dual block-coordinate approach to linearly-constrained nonsmooth optimization

TL;DR

This paper analyzes a randomized primal-dual block-coordinate algorithm for linearly-constrained nonsmooth convex optimization, demonstrating its robustness and deriving convergence rates under various conditions, with applications to optimal transport problems.

Contribution

It extends and specializes existing primal-dual algorithms by providing convergence analysis for a robust, inexact proximal gradient method applicable even without strong duality.

Findings

The algorithm converges under broad conditions, including inconsistent linear constraints.

Tight sublinear convergence rates are established for convex and strongly convex cases.

Numerical experiments validate the theoretical results on an optimal transport problem.

Abstract

This note is concerned with the problem of minimizing a separable, convex, composite (smooth and nonsmooth) function subject to linear constraints. We study a randomized block-coordinate interpretation of the Chambolle-Pock primal-dual algorithm, based on inexact proximal gradient steps. A specificity of the considered algorithm is its robustness, as it converges even in the absence of strong duality or when the linear program is inconsistent. Using matrix preconditiong, we derive tight sublinear convergence rates with and without duality assumptions and for both the convex and the strongly convex settings. Our developments are extensions and particularizations of original algorithms proposed by Malitsky (2019) and Luke and Malitsky (2018). Numerical experiments are provided for an optimal transport problem of service pricing.