Computational Optimization and Applications

, Volume 71, Issue 3, pp 795–828 | Cite as

A majorization–minimization algorithm for split feasibility problems

  • Jason Xu
  • Eric C. Chi
  • Meng Yang
  • Kenneth Lange


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.


Majorize–minimize Nonlinear split feasibility Intensity modulated radiation therapy Proximity function minimization Constrained regression 



We thank Steve Wright and Dávid Papp for their help with the IMRT data examples, and thank Patrick Combettes for helpful comments. We also thank two anonymous referees for their constructive comments and thoughtful feedback.


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Authors and Affiliations

  1. 1.Department of Statistical ScienceDuke UniversityDurhamUSA
  2. 2.Department of StatisticsNorth Carolina State UniversityRaleighUSA
  3. 3.Departments of Biomathematics, Human Genetics, and StatisticsUniversity of CaliforniaLos AngelesUSA

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