  The classical multi-set split feasibility problem seeks a point in the
intersection of finitely many closed convex domain constraints, whose image
under a linear mapping also lies in the intersection of finitely many closed
convex range constraints. Split feasibility generalizes important inverse
problems including convex feasibility, linear complementarity, and regression
with constraint sets. When a feasible point does not exist, solution methods
that proceed by minimizing a proximity function can be used to obtain optimal
approximate solutions to the problem. We present an extension of the proximity
function approach that generalizes the linear split feasibility problem to
allow for non-linear mappings. Our algorithm is based on the principle of
majorization-minimization, is amenable to quasi-Newton acceleration, and comes
complete with convergence guarantees under mild assumptions. Furthermore, we
show that the Euclidean norm appearing in the proximity function of the
non-linear split feasibility problem can be replaced by arbitrary Bregman
divergences. We explore several examples illustrating the merits of non-linear
formulations over the linear case, with a focus on optimization for
intensity-modulated radiation therapy.
