New Bregman proximal type algorithms for solving DC optimization problems

Abstract: Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have $L$-smoothness. In this article, we propose the Bregman Proximal DC Algorithm (BPDCA) for solving large-scale DC optimization problems that do not possess $L$-smoothness. Instead, it requires that the convex part of the objective function has the $L$-smooth adaptable property that is exploited in Bregman proximal gradient algorithms. In addition, we propose an accelerated version, the Bregman Proximal DC Algorithm with extrapolation (BPDCAe), with a new restart scheme. We show the global convergence of the iterates generated by BPDCA(e) to a limiting critical point under the assumption of the Kurdyka-{\L}ojasiewicz property or subanalyticity of the objective function and other weaker conditions than those of the existing methods. We applied our algorithms to phase retrieval, which can be described both as a nonconvex optimization problem and as a DC optimization problem. Numerical experiments showed that BPDCAe outperformed existing Bregman proximal-type algorithms because the DC formulation allows for larger admissible step sizes.