Construction of best Bregman approximations in reflexive Banach spaces

Bauschke, Heinz; Combettes, Patrick

doi:10.1090/S0002-9939-03-07050-3

Construction of best Bregman approximations in reflexive Banach spaces
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by Heinz H. Bauschke and Patrick L. Combettes PDF

Proc. Amer. Math. Soc. 131 (2003), 3757-3766 Request permission

Abstract:

An iterative method is proposed to construct the Bregman projection of a point onto a countable intersection of closed convex sets in a reflexive Banach space.

References

Heinz H. Bauschke, Jonathan M. Borwein, and Patrick L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), no. 4, 615–647. MR 1869107, DOI 10.1142/S0219199701000524
H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Bregman monotone optimization algorithms, SIAM J. Control Optim., to appear.
H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), 248–264.
Heinz H. Bauschke and Adrian S. Lewis, Dykstra’s algorithm with Bregman projections: a convergence proof, Optimization 48 (2000), no. 4, 409–427. MR 1811866, DOI 10.1080/02331930008844513
J. M. Borwein and M. Fabián, On convex functions having points of Gateaux differentiability which are not points of Fréchet differentiability, Canad. J. Math. 45 (1993), no. 6, 1121–1134. MR 1247537, DOI 10.4153/CJM-1993-062-8
James P. Boyle and Richard L. Dykstra, A method for finding projections onto the intersection of convex sets in Hilbert spaces, Advances in order restricted statistical inference (Iowa City, Iowa, 1985) Lect. Notes Stat., vol. 37, Springer, Berlin, 1986, pp. 28–47. MR 875647, DOI 10.1007/978-1-4613-9940-7_{3}
Lev M. Bregman, Yair Censor, and Simeon Reich, Dykstra’s algorithm as the nonlinear extension of Bregman’s optimization method, J. Convex Anal. 6 (1999), no. 2, 319–333. MR 1736245
L. M. Bregman, Y. Censor, S. Reich, and Y. Zepkowitz-Malachi, Finding the projection of a point onto the intersection of convex sets via projections onto halfspaces, preprint, 2002.
Dan Butnariu and Alfredo N. Iusem, Totally convex functions for fixed points computation and infinite dimensional optimization, Applied Optimization, vol. 40, Kluwer Academic Publishers, Dordrecht, 2000. MR 1774818, DOI 10.1007/978-94-011-4066-9
D. Butnariu, A. Iusem, and C. Zălinescu, On uniform convexity, total convexity and the convergence of the proximal point and outer Bregman projection algorithms in Banach spaces, J. Convex Anal., to appear.
Yair Censor and Simeon Reich, The Dykstra algorithm with Bregman projections, Commun. Appl. Anal. 2 (1998), no. 3, 407–419. MR 1626725
Yair Censor and Stavros A. Zenios, Parallel optimization, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1997. Theory, algorithms, and applications; With a foreword by George B. Dantzig. MR 1486040
Patrick L. Combettes, Strong convergence of block-iterative outer approximation methods for convex optimization, SIAM J. Control Optim. 38 (2000), no. 2, 538–565. MR 1741152, DOI 10.1137/S036301299732626X
Frank Deutsch, Best approximation in inner product spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 7, Springer-Verlag, New York, 2001. MR 1823556, DOI 10.1007/978-1-4684-9298-9
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
T. Dumont, Décomposition par Projection de Certains Problèmes aux Limites Elliptiques non Linéaires, Thèse, Université Claude Bernard, Lyon, France, 1978.
R. Gárciga-Otero, A strongly convergent hybrid proximal point method in Banach spaces, conference talk (presented at the IV Brazilian Workshop on Continuous Optimization, Rio de Janeiro, July 15, 2002) reporting on a forthcoming paper with B. F. Svaiter.
Y. Haugazeau, Sur les Inéquations Variationnelles et la Minimisation de Fonctionnelles Convexes, Thèse, Université de Paris, Paris, France, 1968.
G. Pierra, Eclatement de contraintes en parallèle pour la minimisation d’une forme quadratique, in Lecture Notes in Computer Science, Vol. 41, Springer-Verlag, New York, 1976, 200–218.
E. Resmerita, On total convexity, Bregman projections and stability in Banach spaces, preprint, 2002.
Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Translated from the Romanian by Radu Georgescu. MR 0270044, DOI 10.1007/978-3-662-41583-2
M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. 87 (2000), no. 1, Ser. A, 189–202. MR 1734665, DOI 10.1007/s101079900113
J. D. Vanderwerff, personal communication, 2002.
C. Zălinescu, On uniformly convex functions, J. Math. Anal. Appl. 95 (1983), no. 2, 344–374. MR 716088, DOI 10.1016/0022-247X(83)90112-9
Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. MR 768749, DOI 10.1007/978-1-4612-5020-3