A majorization–minimization algorithm for split feasibility problems

Jason Xu (), Eric C. Chi, Meng Yang and Kenneth Lange
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Jason Xu: Duke University
Eric C. Chi: North Carolina State University
Meng Yang: North Carolina State University
Kenneth Lange: University of California

Computational Optimization and Applications, 2018, vol. 71, issue 3, No 8, 795-828

Abstract: 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.

Keywords: Majorize–minimize; Nonlinear split feasibility; Intensity modulated radiation therapy; Proximity function minimization; Constrained regression (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10589-018-0025-z

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