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Extensions of the CQ Algorithm for the Split Feasibility and Split Equality Problems

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  • Charles L. Byrne and Abdellatif Moudafi

    (Department of Mathematical Sciences, University of Massachusetts
    Ceregmia-Département Scientifique Interfacultaire)

Abstract

The convex feasibility problem (CFP) is to find a member of the intersection of finitely many closed convex sets in Euclidean space. When the intersection is empty, one can minimize a proximity function to obtain an approximate solution to the problem. The split feasibility problem (SFP) and the split equality problem (SEP) are generalizations of the CFP. The approximate SFP (ASFP) and approximate SEP (ASEP) involve finding only approximate solutions to the SFP and SEP, respectively. We present here the SSEA, a simultaneous iterative algorithm for solving the ASEP. When this algorithm is applied to the ASFP it resembles closely, but is not equivalent to, the CQ algorithm. The SSEA involves orthogonal projection onto the given closed convex sets. The relaxed SSEA (RSSEA) is an easily implementable variant of the SSEA that uses orthogonal projection onto half-spaces at each step to solve the SEP. The perturbed version of the SSEA (PSSEA) is similar to the RSSEA, but uses orthogonal projection onto a sequence of epi-convergent closed convex sets.

Suggested Citation

  • Charles L. Byrne and Abdellatif Moudafi, 2013. "Extensions of the CQ Algorithm for the Split Feasibility and Split Equality Problems," Documents de Travail 2013-01, CEREGMIA, Université des Antilles et de la Guyane.
    • Handle: RePEc:crg:wpaper:dt2013-01
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    File URL: http://www2.univ-ag.fr/RePEc/DT/DT2013-01_Byrne_Moudafi.pdf
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    Cited by:

    1. Qu, Biao & Liu, Binghua & Zheng, Na, 2015. "On the computation of the step-size for the CQ-like algorithms for the split feasibility problem," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 218-223.
    2. Suthep Suantai & Suparat Kesornprom & Watcharaporn Cholamjiak & Prasit Cholamjiak, 2022. "Modified Projection Method with Inertial Technique and Hybrid Stepsize for the Split Feasibility Problem," Mathematics, MDPI, vol. 10(6), pages 1-12, March.
    3. Ismat Beg & Mujahid Abbas & Muhammad Waseem Asghar, 2023. "Approximation of the Solution of Split Equality Fixed Point Problem for Family of Multivalued Demicontractive Operators with Application," Mathematics, MDPI, vol. 11(4), pages 1-16, February.

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