Convergence Rate Comparison of Proximal Algorithms for Non-Smooth Convex Optimization With an Application to Texture Segmentation

Abstract

In this paper we provide a theoretical and numerical comparison of convergence rates of forward-backward, Douglas-Rachford, and Peaceman-Rachford algorithms for minimizing the sum of a convex proper lower semicontinuous function and a strongly convex differentiable function with Lipschitz continuous gradient. Our results extend the comparison made in [1], when both functions are smooth, to the context where only one is assumed differentiable. Optimal step-sizes and rates of the three algorithms are compared theoretically and numerically in the context of texture segmentation problem, obtaining very sharp estimations and illustrating the high efficiency of Peaceman-Rachford splitting.