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The split feasibility problem is to find an element in the intersection of aclosed set $C$ and the linear preimage of another closed set $D$, assuming theprojections onto $C$ and $D$ are easy to compute. This class of problems arisesnaturally in many contemporary applications such as compressed sensing. Whilethe sets $C$ and $D$ are typically assumed to be convex in the literature, inthis paper, we allow both sets to be possibly nonconvex. We observe that, inthis setting, the split feasibility problem can be formulated as anoptimization problem with a difference-of-convex objective so that standardmajorization-minimization type algorithms can be applied. Here we focus on thenonmonotone proximal gradient algorithm with majorization studied in [15,Appendix A]. We show that, when this algorithm is applied to a splitfeasibility problem, the sequence generated clusters at a stationary point ofthe problem under mild assumptions. We also study local convergence property ofthe sequence under suitable assumptions on the closed sets involved. Finally,we perform numerical experiments to illustrate the efficiency of our approachon solving split feasibility problems that arise in completely positive matrixfactorization, (uniformly) sparse matrix factorization, and outlier detection.

A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection