Block-coordinate primal-dual method for the nonsmooth minimization over linear constraints

TL;DR

This paper introduces a block-coordinate primal-dual algorithm for nonsmooth convex minimization with linear constraints, providing convergence guarantees even in the presence of noise or misspecification.

Contribution

It extends the Chambolle-Pock primal-dual method to block-coordinate updates without requiring smoothness, strong convexity, or system consistency, broadening its applicability.

Findings

Proves convergence without smoothness or strong convexity assumptions

Guarantees convergence for noisy or misspecified systems

Applicable to a wide range of nonsmooth convex problems

Abstract

We consider the problem of minimizing a convex, separable, nonsmooth function subject to linear constraints. The numerical method we propose is a block-coordinate extension of the Chambolle-Pock primal-dual algorithm. We prove convergence of the method without resorting to assumptions like smoothness or strong convexity of the objective, full-rank condition on the matrix, strong duality or even consistency of the linear system. Freedom from imposing the latter assumption permits convergence guarantees for misspecified or noisy systems.