Computing B-Stationary Points of Nonsmooth DC Programs

Abstract: Motivated by a class of applied problems arising from physical layer based security in a digital communication system, in particular, by a secrecy sum-rate maximization problem, this paper studies a nonsmooth, difference-of-convex (dc) minimization problem. The contributions of this paper are: (i) clarify several kinds of stationary solutions and their relations; (ii) develop and establish the convergence of a novel algorithm for computing a d-stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions; (iii) extend the algorithm in several directions including: a randomized choice of the subproblems that could help the practical convergence of the algorithm, a distributed penalty approach for problems whose objective functions are sums of dc functions, and problems with a specially structured (nonconvex) dc constraint. For the latter class of problems, a pointwise Slater constraint qualification is introduced that facilitates the verification and computation of a B(ouligand)-stationary point.