Monotone and nonmonotone linearized block coordinate descent methods for nonsmooth composite optimization problems

TL;DR

This paper introduces monotone and nonmonotone linearized block coordinate descent methods for nonsmooth composite optimization, providing convergence guarantees and demonstrating promising numerical performance.

Contribution

It develops novel monotone and nonmonotone LiBCoD algorithms with convergence analysis for nonsmooth composite problems.

Findings

Global sublinear convergence rates established for both variants.

Nonmonotone variant does not require Lipschitz continuity or bounded iterates.

Numerical experiments show promising performance.

Abstract

In this paper, we introduce both monotone and nonmonotone variants of LiBCoD, a \textbf{Li}nearized \textbf{B}lock \textbf{Co}ordinate \textbf{D}escent method for solving composite optimization problems. At each iteration, a random block is selected, and the smooth components of the objective are linearized along the chosen block in a Gauss-Newton approach. For the monotone variant, we establish a global sublinear convergence rate to a stationary point under the assumption of bounded iterates. For the nonmonotone variant, we derive a global sublinear convergence rate without requiring global Lipschitz continuity or bounded iterates. Preliminary numerical experiments highlight the promising performance of the proposed approach.