An MM Algorithm for Split Feasibility Problems

TL;DR

This paper introduces a majorization-minimization algorithm for solving generalized split feasibility problems involving non-linear mappings, with convergence guarantees and applications in radiation therapy optimization.

Contribution

It extends the proximity function approach to non-linear mappings and incorporates Bregman divergences, providing a flexible and convergent solution method.

Findings

Algorithm converges under mild assumptions.

Non-linear formulations outperform linear cases in certain applications.

Applicable to intensity-modulated radiation therapy optimization.

Abstract

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.