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Alternating conditional gradient method for convex feasibility problems

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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.

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Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments, which have helped to substantially improve the presentation of the paper.

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Authors and Affiliations

  1. School of Information Technology, Deakin University, Melbourne, Australia

    R. Díaz Millán

  2. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    O. P. Ferreira & L. F. Prudente

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  1. R. Díaz Millán
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  2. O. P. Ferreira
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  3. L. F. Prudente
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Correspondence to O. P. Ferreira.

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The authors was supported in part by CNPq Grants 305158/2014-7 and 302473/2017-3, FAPEG/PRONEM- 201710267000532, CNPq Grants 424860/2018-0, 309628/2020-2, FAPEG PPP03/15-201810267001725.

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Díaz Millán, R., Ferreira, O.P. & Prudente, L.F. Alternating conditional gradient method for convex feasibility problems. Comput Optim Appl 80, 245–269 (2021). https://doi.org/10.1007/s10589-021-00293-4

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