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

  • Heinz H. Bauschke
  • Mason S. Macklem
  • Xianfu Wang
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)

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.

Keywords

Bregman distance Chebyshev center Chebyshev function Chebyshev point of a function Chebyshev set Convex function Farthest point Fenchel conjugate Itakura–Saito distance Klee set Klee function Kullback–Leibler divergence Legendre function Nearest point Projection 

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Notes

Acknowledgements

The authors thank two referees for their careful reading and pertinent comments. Heinz Bauschke was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program. Xianfu Wang was partially supported by the Natural Sciences and Engineering Research Council of Canada.

References

  1. 1.
    Asplund, E.: Sets with unique farthest points. Israel J. Math. 5, 201–209 (1967)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M.: Legendre functions and the method of random Bregman projections. J. Convex Anal. 4, 27–67 (1997)MathSciNetMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Borwein, J.M.: Joint and separate convexity of the Bregman distance. In: D. Butnariu, Y. Censor, S. Reich (ed.) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (Haifa 2000), pp. 23–36. Elsevier (2001)Google Scholar
  4. 4.
    Bauschke, H.H., Noll, D.: The method of forward projections. J. Nonlin. Convex Anal. 3, 191–205 (2002)MathSciNetMATHGoogle Scholar
  5. 5.
    Bauschke, H.H., Borwein, J.M, Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3, 615–647 (2001)Google Scholar
  6. 6.
    Bauschke, H.H., Wang, X., Ye, J., Yuan, X.: Bregman distances and Chebyshev sets. J. Approx. Theory 159, 3–25 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bauschke, H.H., Wang, X., Ye, J., Yuan, X.: Bregman distances and Klee sets. J. Approx. Theory 158, 170–183 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bauschke, H.H., Macklem, M.S., Sewell, J.B., Wang, X.: Klee sets and Chebyshev centers for the right Bregman distance. J. Approx. Theory 162, 1225–1244 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Berens, H., Westphal, U.: Kodissipative metrische Projektionen in normierten linearen Räumen. In: P. L. Butzer and B. Sz.-Nagy (eds.) Linear Spaces and Approximation, vol. 40, pp. 119–130, Birkhäuser (1980)Google Scholar
  10. 10.
    Borwein, J.M.: Proximity and Chebyshev sets. Optim. Lett. 1, 21–32 (2007)MATHGoogle Scholar
  11. 11.
    Borwein, J.M, Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd edn. Springer (2006)Google Scholar
  12. 12.
    Borwein, J.M., Vanderwerff, J.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press (2010)MATHGoogle Scholar
  13. 13.
    Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. U.S.S.R. Comp. Math. Math 7, 200–217 (1967)Google Scholar
  14. 14.
    Bunt, L.N.H.: Bijdrage tot de theorie de convexe puntverzamelingen. Thesis, Univ. of Groningen, Amsterdam, 1934Google Scholar
  15. 15.
    Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation in Infinite Dimensional Optimization. Kluwer, Dordrecht (2000)Google Scholar
  16. 16.
    Censor, Y., Zenios, S.A.: Parallel Optimization. Oxford University Press (1997)Google Scholar
  17. 17.
    Deutsch, F.: Best Approximation in Inner Product Spaces. Springer (2001)Google Scholar
  18. 18.
    Fremlin, D.H.: Measure Theory, vol. 2. Broad Foundations, 2nd edn. Torres Fremlin, Colchester (2010)Google Scholar
  19. 19.
    Garkavi, A.L.: On the Čebyšev center and convex hull of a set. Usp. Mat. Nauk 19, 139–145 (1964)MathSciNetMATHGoogle Scholar
  20. 20.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press (1990)Google Scholar
  21. 21.
    De Guzmán, M.: A change-of-variables formula without continuity. Am. Math. Mon. 87, 736–739 (1980)MATHCrossRefGoogle Scholar
  22. 22.
    Hiriart-Urruty, J.-B.: Ensembles de Tchebychev vs. ensembles convexes: l’etat de la situation vu via l’analyse convexe non lisse. Ann. Sci. Math. Québec 22, 47–62 (1998)Google Scholar
  23. 23.
    Hiriart-Urruty, J.-B.: La conjecture des points les plus éloignés revisitée. Ann. Sci. Math. Québec 29, 197–214 (2005)MathSciNetMATHGoogle Scholar
  24. 24.
    Hiriart-Urruty, J.-B.: Potpourri of conjectures and open questions in nonlinear analysis and optimization. SIAM Rev. 49, 255–273 (2007)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer (1996)Google Scholar
  26. 26.
    Klee, V.: Circumspheres and inner products. Math. Scand. 8, 363–370 (1960)MathSciNetGoogle Scholar
  27. 27.
    Klee, V.: Convexity of Chebyshev sets. Math. Ann. 142, 292–304 (1960/1961)Google Scholar
  28. 28.
    Motzkin, T.: Sur quelques propriétés caractéristiques des ensembles convexes. Atti. Accad. Naz. Lincei, Rend., VI. Ser. 21, 562–567 (1935)Google Scholar
  29. 29.
    Motzkin, T.S., Straus, E.G., Valentine, F.A.: The number of farthest points. Pac. J. Math. 3, 221–232 (1953)MathSciNetMATHGoogle Scholar
  30. 30.
    Nielsen, F., Nock, R.: On the smallest enclosing information disk. Inform. Process. Lett. 105, 93–97 (2008)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Nock, R., Nielsen, F.: Fitting the smallest enclosing Bregman ball. In: J. Gama, R. Camacho, P. Brazdil, A. Jorge and L. Torgo (eds.) Machine Learning: 16th European Conference on Machine Learning (Porto 2005), pp. 649–656, Springer Lecture Notes in Computer Science vol. 3720 (2005)Google Scholar
  32. 32.
    Reich, S., Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 73, 122–135 (2010)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  34. 34.
    Rockafellar, R.T., Wets, R. J.-B.: Variational Analysis. Springer, New York (1998)MATHCrossRefGoogle Scholar
  35. 35.
    Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer (1970)Google Scholar
  36. 36.
    Singer, I.: The Theory of Best Approximation and Functional Analysis. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 13. Society for Industrial and Applied Mathematics (1974)Google Scholar
  37. 37.
    Vlasov, L.P.: Approximate properties of sets in normed linear spaces. Russian Math. Surv. 28, 1–66 (1973)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Wang, X.: On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368, 293–310 (2010)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Westphal, U., Schwartz, T.: Farthest points and monotone operators. B. Aust. Math. Soc. 58, 75–92 (1998)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing (2002)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Mason S. Macklem
  • Xianfu Wang
  1. 1.Department of Mathematics, Irving K. Barber SchoolUniversity of British ColumbiaKelownaCanada

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