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A majorization–minimization algorithm for split feasibility problems

Author

Listed:
  • Jason Xu

    (Duke University)

  • Eric C. Chi

    (North Carolina State University)

  • Meng Yang

    (North Carolina State University)

  • Kenneth Lange

    (University of California)

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.

Suggested Citation

  • Jason Xu & Eric C. Chi & Meng Yang & Kenneth Lange, 2018. "A majorization–minimization algorithm for split feasibility problems," Computational Optimization and Applications, Springer, vol. 71(3), pages 795-828, December.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:3:d:10.1007_s10589-018-0025-z
    DOI: 10.1007/s10589-018-0025-z
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    References listed on IDEAS

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    Cited by:

    1. Li-Jun Zhu & Yonghong Yao, 2019. "An Iterative Approach to the Solutions of Proximal Split Feasibility Problems," Mathematics, MDPI, vol. 7(2), pages 1-9, February.
    2. Chen Chen & Ting Kei Pong & Lulin Tan & Liaoyuan Zeng, 2020. "A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection," Journal of Global Optimization, Springer, vol. 78(1), pages 107-136, September.
    3. Wu, Runxiong & Chen, Xin, 2021. "MM algorithms for distance covariance based sufficient dimension reduction and sufficient variable selection," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).

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