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Approximate Douglas–Rachford algorithm for two-sets convex feasibility problems

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  • R. Díaz Millán

    (Deakin University)

  • O. P. Ferreira

    (Universidade Federal de Goiás)

  • J. Ugon

    (Deakin University)

Abstract

In this paper, we propose a new algorithm combining the Douglas–Rachford (DR) algorithm and the Frank–Wolfe algorithm, also known as the conditional gradient (CondG) method, for solving the classic convex feasibility problem. Within the algorithm, which will be named Approximate Douglas–Rachford (ApDR) algorithm, the CondG method is used as a subroutine to compute feasible inexact projections on the sets under consideration, and the ApDR iteration is defined based on the DR iteration. The ApDR algorithm generates two sequences, the main sequence, based on the DR iteration, and its corresponding shadow sequence. When the intersection of the feasible sets is nonempty, the main sequence converges to a fixed point of the usual DR operator, and the shadow sequence converges to the solution set. We provide some numerical experiments to illustrate the behaviour of the sequences produced by the proposed algorithm.

Suggested Citation

  • R. Díaz Millán & O. P. Ferreira & J. Ugon, 2023. "Approximate Douglas–Rachford algorithm for two-sets convex feasibility problems," Journal of Global Optimization, Springer, vol. 86(3), pages 621-636, July.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:3:d:10.1007_s10898-022-01264-7
    DOI: 10.1007/s10898-022-01264-7
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    References listed on IDEAS

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    1. R. Díaz Millán & O. P. Ferreira & L. F. Prudente, 2021. "Alternating conditional gradient method for convex feasibility problems," Computational Optimization and Applications, Springer, vol. 80(1), pages 245-269, September.
    2. Amir Beck & Marc Teboulle, 2004. "A conditional gradient method with linear rate of convergence for solving convex linear systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(2), pages 235-247, June.
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    4. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
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