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Finding best approximation pairs for two intersections of closed convex sets

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Abstract

The problem of finding a best approximation pair of two sets, which in turn generalizes the well known convex feasibility problem, has a long history that dates back to work by Cheney and Goldstein in 1959. In 2018, Aharoni, Censor, and Jiang revisited this problem and proposed an algorithm that can be used when the two sets are finite intersections of halfspaces. Motivated by their work, we present alternative algorithms that utilize projection and proximity operators. Our modeling framework is able to accommodate even convex sets. Numerical experiments indicate that these methods are competitive and sometimes superior to the one proposed by Aharoni et al.

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Acknowledgements

The authors thank the editor and the reviewers for helpful suggestions and constructive feedback which helped us to improve the presentation of the results, and Patrick Combettes for pointing out the relevant reference [13]. HHB and XW are supported by the Natural Sciences and Engineering Research Council of Canada.

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  1. Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Heinz H. Bauschke, Shambhavi Singh & Xianfu Wang

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  1. Heinz H. Bauschke
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  2. Shambhavi Singh
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  3. Xianfu Wang
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Correspondence to Heinz H. Bauschke.

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Bauschke, H.H., Singh, S. & Wang, X. Finding best approximation pairs for two intersections of closed convex sets. Comput Optim Appl 81, 289–308 (2022). https://doi.org/10.1007/s10589-021-00324-0

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