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On the circumcentered-reflection method for the convex feasibility problem

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Abstract

The ancient concept of circumcenter has recently given birth to the circumcentered-reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection-based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.

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Acknowledgments

We thank the anonymous referees for their valuable suggestions which significantly improved this manuscript.

Funding

RB was partially supported by the Brazilian Agency Conselho Nacional de Desenvolvimento Científico e Tecnoógico (CNPq), Grants 304392/2018-9 and 429915/2018-7; YBC was partially supported by the National Science Foundation (NSF), Grant DMS – 1816449.

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Authors and Affiliations

  1. School of Applied Mathematics, Fundação Getúlio Vargas, Rio de Janeiro, RJ, 22250-900, Brazil

    Roger Behling

  2. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115-2828, USA

    Yunier Bello-Cruz

  3. Department of Mathematics, Federal University of Santa Catarina, Blumenau, SC, 8040-900, Brazil

    Luiz-Rafael Santos

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  1. Roger Behling
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  2. Yunier Bello-Cruz
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  3. Luiz-Rafael Santos
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Correspondence to Roger Behling.

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Behling, R., Bello-Cruz, Y. & Santos, LR. On the circumcentered-reflection method for the convex feasibility problem. Numer Algor 86, 1475–1494 (2021). https://doi.org/10.1007/s11075-020-00941-6

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