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Chebyshev Sets, Klee Sets, and Chebyshev Centers with Respect to Bregman Distances: Recent Results and Open Problems

In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Author

Listed:
  • Heinz H. Bauschke

    (University of British Columbia)

  • Mason S. Macklem
  • Xianfu Wang

Abstract

In Euclidean spaces, the geometric notions of nearest-points map, farthest-points map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical results from Euclidean space to Banach space settings. In all these results, the distance between points is induced by some underlying norm. Recently, these notions have been revisited from a different viewpoint in which the discrepancy between points is measured by Bregman distances induced by Legendre functions. The associated framework covers the well-known Kullback–Leibler divergence and the Itakura–Saito distance. In this survey, we review known results and we present new results on Klee sets and Chebyshev centers with respect to Bregman distances. Examples are provided and connections to recent work on Chebyshev functions are made. We also identify several intriguing open problems.

Suggested Citation

  • Heinz H. Bauschke & Mason S. Macklem & Xianfu Wang, 2011. "Chebyshev Sets, Klee Sets, and Chebyshev Centers with Respect to Bregman Distances: Recent Results and Open Problems," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 1-21, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-9569-8_1
    DOI: 10.1007/978-1-4419-9569-8_1
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    Cited by:

    1. Emanuel Laude & Peter Ochs & Daniel Cremers, 2020. "Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 724-761, March.

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