On proximal subgradient splitting method for minimizing the sum of two nonsmooth convex functions

TL;DR

This paper introduces the Proximal Subgradient Splitting Method, a novel iterative approach for minimizing sums of two nonsmooth convex functions, extending classical subgradient methods with convergence and complexity analysis.

Contribution

It proposes a new proximal subgradient splitting algorithm for nonsmooth convex optimization, with proven weak convergence and complexity analysis, extending existing methods.

Findings

Weak convergence of the method was established.

The method extends classical subgradient iteration.

Complexity analysis of the iterates was provided.

Abstract

In this paper we present a variant of the proximal forward-backward splitting iteration for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be called Proximal Subgradient Splitting Method, extends the classical subgradient iteration for important classes of problems, exploiting the additive structure of the objective function. The weak convergence of the generated sequence was established using different stepsizes and under suitable assumptions. Moreover, we analyze the complexity of the iterates.