On the Existence of Minimizers of Proximity Functions for Split Feasibility Problems

Many inverse problems can be formulated as split feasibility problems. To find feasible solutions, one has to minimize proximity functions. We show that the existence of minimizers to the proximity function for Censor–Elfving’s split feasibility problem is equivalent to the existence of projections on appropriate convex sets and provide conditions under which such projections exist. These projections turn out to be the unique optimal solution of their Fenchel–Rockafellar duals and can be computed by the proximal point algorithm efficiently. Applications to linear equations and linear feasibility problems are given.