We consider a class of nonconvex nonsmooth optimization problems whoseobjective is the sum of a nonnegative smooth function and a bunch ofnonnegative proper closed possibly nonsmooth functions (whose proximal mappingsare easy to compute), some of which are further composed with linear maps. Thiskind of problems arises naturally in various applications when differentregularizers are introduced for inducing simultaneous structures in thesolutions. Solving these problems, however, can be challenging because of thecoupled nonsmooth functions: the corresponding proximal mapping can be hard tocompute so that standard first-order methods such as the proximal gradientalgorithm cannot be applied efficiently. In this paper, we propose a successivedifference-of-convex approximation method for solving this kind of problems. Inthis algorithm, we approximate the nonsmooth functions by their Moreauenvelopes in each iteration. Making use of the simple observation that Moreauenvelopes of nonnegative proper closed functions are continuousdifference-of-convex functions, we can then approximately minimize theapproximation function by first-order methods with suitable majorizationtechniques. These first-order methods can be implemented efficiently thanks tothe fact that the proximal mapping of each nonsmooth function is easy tocompute. Under suitable assumptions, we prove that the sequence generated byour method is bounded and clusters at a stationary point of the objective. Wealso discuss how our method can be applied to concrete applications such asnonconvex fused regularized optimization problems and simultaneously structuredmatrix optimization problems, and illustrate the performance numerically forthese two specific applications.
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