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

Optimization and Control