Optimal Leveraging of Smoothness and Strong Convexity for Peaceman--Rachford Splitting

TL;DR

This paper presents a novel methodology to enhance the convergence rate of Peaceman--Rachford splitting by optimally leveraging smoothness and strong convexity, leading to faster solutions in convex optimization.

Contribution

The paper introduces a new approach to modify the optimization problem, enabling the Peaceman--Rachford splitting method to achieve an optimal linear convergence rate.

Findings

Achieves the best known linear convergence rate for PRS in strongly convex settings.

Demonstrates improved practical performance in image processing applications.

Provides a theoretical framework for parameter optimization in splitting methods.

Abstract

In this paper, we introduce a simple methodology to leverage strong convexity and smoothness in order to obtain an optimal linear convergence rate for the Peaceman--Rachford splitting (PRS) scheme applied to optimization problems involving two smooth strongly convex functions. The approach consists of adding and subtracting suitable quadratic terms from one function to the other so as to redistribute strong convexity in the primal formulation and smoothness in the dual formulation. This yields an equivalent modified optimization problem in which each term has adjustable levels of strong convexity and smoothness. In this setting, the Peaceman--Rachford splitting method converges linearly to the solution of the modified problem with a convergence rate that can be optimized with respect to the introduced parameters. Upon returning to the original formulation, this procedure gives rise to a modified variant of PRS. The optimal linear rate established in this work is strictly better than the best rates previously available in the general setting. The practical performance of the method is illustrated through an academic example and applications in image processing.