A full splitting algorithm for structured difference-of-convex programs

TL;DR

This paper introduces a novel full-splitting algorithm for structured difference-of-convex programs, proving its convergence and demonstrating its effectiveness with a practical nonmonotone line search enhancement.

Contribution

It proposes an adaptive double-proximal full-splitting algorithm with convergence guarantees for a class of nonconvex nonsmooth DC programs.

Findings

Algorithm converges to an approximate stationary point.

Global convergence established under Kurdyka-ojasiewicz property.

Practical nonmonotone line search improves convergence performance.

Abstract

In this paper, we study a class of nonconvex and nonsmooth structured difference-of-convex (DC) programs, which contain in the convex part the sum of a nonsmooth linearly composed convex function and a differentiable function, and in the concave part another nonsmooth linearly composed convex function. Among the various areas in which such problems occur, we would like to mention in particular the recovery of sparse signals. We propose an adaptive double-proximal, full-splitting algorithm with a moving center approach in the final subproblem, which addresses the challenge of evaluating compositions by decoupling the linear operator from the nonsmooth component. We establish the subsequential convergence of the generated sequence of iterates to an approximate stationary point and prove its global convergence under the Kurdyka-\L ojasiewicz property. We also discuss the tightness of the convergence results and provide insights into the rationale for seeking an approximate KKT point. This is illustrated by constructing a counterexample showing that the algorithm can diverge when seeking exact solutions. Finally, we present a practical version of the algorithm that incorporates a nonmonotone line search, which significantly improves the convergence performance.