Additive Schwarz Methods for Convex Optimization as Gradient Methods

TL;DR

This paper unifies the convergence analysis of additive Schwarz methods for convex optimization, showing they function as gradient methods and applying the framework to various problem types.

Contribution

It introduces a unified convergence framework for additive Schwarz methods in convex optimization, revealing their gradient method nature and extending classical theory.

Findings

Additive Schwarz methods are equivalent to gradient methods for convex problems.

The framework applies to linear, nonlinear, nonsmooth, and nonsharp convex problems.

Classical convergence results are recovered for linear elliptic problems.

Abstract

This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling to the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove that additive Schwarz methods for general convex optimization are in fact gradient methods. Then an abstract framework for convergence analysis of additive Schwarz methods is proposed. The proposed framework applied to linear elliptic problems agrees with the classical theory. We present applications of the proposed framework to various interesting convex optimization problems such as nonlinear elliptic problems, nonsmooth problems, and nonsharp problems.