Chebyshev Sets, Klee Sets, and Chebyshev Centers with Respect to Bregman Distances: Recent Results and Open Problems
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.
KeywordsBregman 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
Unable to display preview. Download preview PDF.
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.
- 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
- 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
- 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
- 11.Borwein, J.M, Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd edn. Springer (2006)Google Scholar
- 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.Bunt, L.N.H.: Bijdrage tot de theorie de convexe puntverzamelingen. Thesis, Univ. of Groningen, Amsterdam, 1934Google Scholar
- 15.Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation in Infinite Dimensional Optimization. Kluwer, Dordrecht (2000)Google Scholar
- 16.Censor, Y., Zenios, S.A.: Parallel Optimization. Oxford University Press (1997)Google Scholar
- 17.Deutsch, F.: Best Approximation in Inner Product Spaces. Springer (2001)Google Scholar
- 18.Fremlin, D.H.: Measure Theory, vol. 2. Broad Foundations, 2nd edn. Torres Fremlin, Colchester (2010)Google Scholar
- 20.Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press (1990)Google Scholar
- 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
- 25.Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer (1996)Google Scholar
- 27.Klee, V.: Convexity of Chebyshev sets. Math. Ann. 142, 292–304 (1960/1961)Google Scholar
- 28.Motzkin, T.: Sur quelques propriétés caractéristiques des ensembles convexes. Atti. Accad. Naz. Lincei, Rend., VI. Ser. 21, 562–567 (1935)Google Scholar
- 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
- 35.Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer (1970)Google Scholar
- 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
- 40.Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing (2002)Google Scholar