A proximal splitting algorithm for generalized DC programming with applications in signal recovery

TL;DR

This paper introduces a novel proximal splitting algorithm for a broad class of nonconvex DC programs, with proven convergence properties and effective application to signal recovery tasks.

Contribution

It develops a new algorithm combining proximal evaluation and Douglas--Rachford splitting for generalized DC problems, with convergence guarantees and practical effectiveness.

Findings

Algorithm converges to critical points under Kurdyka--Łojasiewicz property.

Proven global convergence of the entire sequence of iterates.

Competitive performance demonstrated on signal recovery problems.

Abstract

The difference-of-convex (DC) program is an important model in nonconvex optimization due to its structure, which encompasses a wide range of practical applications. In this paper, we aim to tackle a generalized class of DC programs, where the objective function is formed by summing a possibly nonsmooth nonconvex function and a differentiable nonconvex function with Lipschitz continuous gradient, and then subtracting a nonsmooth continuous convex function. We develop a proximal splitting algorithm that utilizes proximal evaluation for the concave part and Douglas--Rachford splitting for the remaining components. The algorithm guarantees subsequential convergence to a {\color{black}critical} point of the problem model. Under the widely used Kurdyka--{\L}ojasiewicz property, we establish global convergence of the full sequence of iterates and derive convergence rates for both the iterates and the objective function values, without assuming the concave part is differentiable. The performance of the proposed algorithm is tested on signal recovery problems with a nonconvex regularization term and exhibits competitive results compared to notable algorithms in the literature on both synthetic data and real-world data.