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Alternating Minimization as Sequential Unconstrained Minimization: A Survey

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  • Charles L. Byrne

    (University of Massachusetts Lowell)

Abstract

Sequential unconstrained minimization is a general iterative method for minimizing a function over a given set. At each step of the iteration we minimize the sum of the objective function and an auxiliary function. The aim is to select the auxiliary functions so that, at least, we get convergence in function value to the constrained minimum. The SUMMA is a broad class of these methods for which such convergence holds. Included in the SUMMA class are the barrier-function methods, entropic and other proximal minimization algorithms, the simultaneous multiplicative algebraic reconstruction technique, and, after some reformulation, penalty-function methods. The alternating minimization method of Csiszár and Tusnády also falls within the SUMMA class, whenever their five-point property holds. Therefore, the expectation maximization maximum likelihood algorithm for the Poisson case is also in the SUMMA class.

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  • Charles L. Byrne, 2013. "Alternating Minimization as Sequential Unconstrained Minimization: A Survey," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 554-566, March.
    • Handle: RePEc:spr:joptap:v:156:y:2013:i:3:d:10.1007_s10957-012-0134-2
    DOI: 10.1007/s10957-012-0134-2
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    References listed on IDEAS

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    1. Charles Byrne & Yair Censor, 2001. "Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization," Annals of Operations Research, Springer, vol. 105(1), pages 77-98, July.
    2. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
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    Cited by:

    1. 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.

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