A globally convergent difference-of-convex algorithmic framework and application to log-determinant optimization problems

Abstract

The difference-of-convex algorithm (DCA) is a conceptually simple method for the minimization of (possibly) nonconvex functions that are expressed as the difference of two convex functions. At each iteration, DCA constructs a global overestimator of the objective and solves the resulting convex subproblem. Despite its conceptual simplicity, the theoretical understanding and algorithmic framework of DCA needs further investigation. In this paper, global convergence of DCA at a linear rate is established under an extended Polyak--{\L}ojasiewicz condition. The proposed condition holds for a class of DC programs with a bounded, closed, and convex constraint set, for which global convergence of DCA cannot be covered by existing analyses. Moreover, the DCProx computational framework is proposed, in which the DCA subproblems are solved by a primal--dual proximal algorithm with Bregman distances. With a suitable choice of Bregman distances, DCProx has simple update rules with cheap per-iteration complexity. As an application, DCA is applied to several fundamental problems in network information theory, for which no existing numerical methods are able to compute the global optimum. For these problems, our analysis proves the global convergence of DCA, and more importantly, DCProx solves the DCA subproblems efficiently. Numerical experiments are conducted to verify the efficiency of DCProx.