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Alternating conditional gradient method for convex feasibility problems

Author

Listed:
  • R. Díaz Millán

    (Deakin University)

  • O. P. Ferreira

    (Universidade Federal de Goiás)

  • L. F. Prudente

    (Universidade Federal de Goiás)

Abstract

The classical convex feasibility problem in a finite dimensional Euclidean space consists of finding a point in the intersection of two convex sets. In the present paper we are interested in two particular instances of this problem. First, we assume to know how to compute an exact projection onto one of the sets involved and the other set is compact such that the conditional gradient (CondG) method can be used for computing efficiently an inexact projection on it. Second, we assume that both sets involved are compact such that the CondG method can be used for computing efficiently inexact projections on them. We combine alternating projection method with CondG method to design a new method, which can be seen as an inexact feasible version of alternate projection method. The proposed method generates two different sequences belonging to each involved set, which converge to a point in the intersection of them whenever it is not empty. If the intersection is empty, then the sequences converge to points in the respective sets whose distance between them is equal to the distance between the sets in consideration. Numerical experiments are provided to illustrate the practical behavior of the method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:1:d:10.1007_s10589-021-00293-4
    DOI: 10.1007/s10589-021-00293-4
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    References listed on IDEAS

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    1. Alfredo N. Iusem & Alejandro Jofré & Philip Thompson, 2019. "Incremental Constraint Projection Methods for Monotone Stochastic Variational Inequalities," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 236-263, February.
    2. 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.
    3. 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.
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    Cited by:

    1. 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.
    2. R. Díaz Millán & O. P. Ferreira & J. Ugon, 2024. "Extragradient method with feasible inexact projection to variational inequality problem," Computational Optimization and Applications, Springer, vol. 89(2), pages 459-484, November.

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