  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.
