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

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.