A Perturbed DCA for Computing d-Stationary Points of Nonsmooth DC Programs

TL;DR

This paper presents a perturbed DCA algorithm that efficiently computes d-stationary points in nonsmooth DC programs, requiring fewer subproblem solutions and demonstrating strong convergence and practical effectiveness.

Contribution

The paper introduces pDCA, a novel perturbed difference-of-convex algorithm that reduces computational complexity and guarantees convergence to d-stationary points in nonsmooth DC problems.

Findings

pDCA requires solving only one subproblem per iteration.

The algorithm converges to d-stationary points almost surely.

Numerical experiments show pDCA's efficiency in practical problems.

Abstract

This paper introduces an efficient perturbed difference-of-convex algorithm (pDCA) for computing d-stationary points of an important class of structured nonsmooth difference-of-convex problems. Compared to the principal algorithms introduced in [J.-S. Pang, M. Razaviyayn, and A. Alvarado, Math. Oper. Res. 42(1):95--118 (2017)], which may require solving several subproblems for a one-step update, pDCA only requires solving a single subproblem. Therefore, the computational cost of pDCA for one-step update is comparable to the widely used difference-of-convex algorithm (DCA) introduced in [D. T. Pham and H. A. Le Thi, Acta Math. Vietnam. 22(1):289--355 (1997)] for computing a critical point. Importantly, under practical assumptions, we prove that every accumulation point of the sequence generated by pDCA is a d-stationary point almost surely. Numerical experiment results on several important examples of nonsmooth DC programs demonstrate the efficiency of pDCA for computing d-stationary points.