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In this paper, we consider a class of nonconvex (not necessarilydifferentiable) optimization problems called generalized DC(Difference-of-Convex functions) programming, which is minimizing the sum oftwo separable DC parts and one two-block-variable coupling function. Tocircumvent the nonconvexity and nonseparability of the problem underconsideration, we accordingly introduce a Unified Bregman AlternatingMinimization Algorithm (UBAMA) by maximally exploiting the favorable DCstructure of the objective. Specifically, we first follow the spirit ofalternating minimization to update each block variable in a sequential order,which can efficiently tackle the nonseparablitity caused by the couplingfunction. Then, we employ the Fenchel-Young inequality to approximate thesecond DC components (i.e., concave parts) so that each subproblem reduces to aconvex optimization problem, thereby alleviating the computational burden ofthe nonconvex DC parts. Moreover, each subproblem absorbs a Bregman proximalregularization term, which is usually beneficial for inducing closed-formsolutions of subproblems for many cases via choosing appropriate Bregman kernelfunctions. It is remarkable that our algorithm not only provides an algorithmicframework to understand the iterative schemes of some novel existingalgorithms, but also enjoys implementable schemes with easier subproblems thansome state-of-the-art first-order algorithms developed for generic nonconvexand nonsmooth optimization problems. Theoretically, we prove that the sequencegenerated by our algorithm globally converges to a critical point under theKurdyka-{\L}ojasiewicz (K{\L}) condition. Besides, we estimate the localconvergence rates of our algorithm when we further know the prior informationof the K{\L} exponent.

A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programming with Application to Imaging Data