Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces

Butnariu, Dan; Resmerita, Elena

doi:https://doi.org/10.1155/AAA/2006/84919

Abstract and Applied Analysis

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Volume 2006 | Article ID 084919 | https://doi.org/10.1155/AAA/2006/84919

Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces

Dan Butnariu¹and Elena Resmerita²

Received25 Jul 2004

Accepted06 Apr 2005

Published27 Feb 2006

Abstract

The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations.

References

Y. I. Alber, “Some iterative methods for nonregular minimization problems and estimates of the speed of convergence,” Technicheskaya Kybernetika, vol. 3, pp. 22–30, 1988 (Russian).
View at: Google Scholar
Y. I. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Functional-Differential Equations, M. E. Drakhlin and E. Litsyn, Eds., vol. 1 of Funct. Differential Equations Israel Sem., pp. 1–21, Coll. Judea Samaria, Ariel, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 15–50, Dekker, New York, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. I. Alber, “Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces,” Applied Mathematics Letters, vol. 11, no. 6, pp. 115–121, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. I. Alber, “Some new characteristics of Banach spaces,” preprint, 2004.
View at: Google Scholar
Y. I. Alber and D. Butnariu, “Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces,” Journal of Optimization Theory and Applications, vol. 92, no. 1, pp. 33–61, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. I. Alber, D. Butnariu, and I. Ryazantseva, “Regularization and resolution of monotone variational inequalities with operators given by hypomonotone approximations,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 23–53, 2005.
View at: Google Scholar
Y. I. Alber, A. N. Iusem, and M. V. Solodov, “Minimization of nonsmooth convex functionals in Banach spaces,” Journal of Convex Analysis, vol. 4, no. 2, pp. 235–255, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Asplund, “Averaged norms,” Israel Journal of Mathematics, vol. 5, pp. 227–233, 1967.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J.-B. Baillon and G. Haddad, “Quelques propriétés des opérateurs angle-bornés et $n$ -cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977 (French).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. H. Bauschke and J. M. Borwein, “Legendre functions and the method of random Bregman projections,” Journal of Convex Analysis, vol. 4, no. 1, pp. 27–67, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces,” Communications in Contemporary Mathematics, vol. 3, no. 4, pp. 615–647, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Bregman monotone optimization algorithms,” SIAM Journal on Control and Optimization, vol. 42, no. 2, p. 596–636, 2003.
View at: Google Scholar | MathSciNet
H. H. Bauschke and P. L. Combettes, “Construction of best Bregman approximations in reflexive Banach spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 12, p. 3757–3766, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. H. Bauschke and A. S. Lewis, “Dykstra's algorithm with Bregman projections: a convergence proof,” Optimization, vol. 48, no. 4, pp. 409–427, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Bertsekas, Nonlinear Programming, Athena Scientific, Massachusetts, 2nd edition, 1999.
View at: Google Scholar | Zentralblatt MATH
J. M. Borwein and A. S. Lewis, “Convergence of best entropy estimates,” SIAM Journal on Optimization, vol. 1, no. 2, pp. 191–205, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. M. Borwein and M. A. Limber, “On entropy maximization via convex programming,” preprint, 1996, available at http://citeseer.ist.psu.edu/71881.html.
View at: Google Scholar
L. M. Bregman, “The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, pp. 200–217, 1967.
View at: Google Scholar
R. E. Bruck, Jr., “A strongly convergent iterative solution of $0 \in U(x)$ for a maximal monotone operator $U$ in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 48, pp. 114–126, 1974.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. E. Bruck, Jr., “An iterative solution of a variational inequality for certain monotone operators in Hilbert space,” American Mathematical Society Bulletin, vol. 81, no. 5, pp. 890–892, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. S. Burachik, Generalized proximal point methods for the variational inequality problem, M.S. thesis, Instituto de Mathematica Pura e Aplicada (IMPA), Rio de Janeiro, 1995.
View at: Google Scholar
R. S. Burachik and S. Scheimberg, “A proximal point method for the variational inequality problem in Banach spaces,” SIAM Journal on Control and Optimization, vol. 39, no. 5, pp. 1633–1649, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu, Y. Censor, and S. Reich, “Iterative averaging of entropic projections for solving stochastic convex feasibility problems,” Computational Optimization and Applications, vol. 8, no. 1, pp. 21–39, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu and A. N. Iusem, “Local moduli of convexity and their application to finding almost common fixed points of measurable families of operators,” in Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995), Y. Censor and S. Reich, Eds., vol. 204 of Contemp. Math., pp. 61–91, American Mathematical Society, Rhode Island, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu and A. N. Iusem, “On a proximal point method for convex optimization in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 18, no. 7-8, pp. 723–744, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, vol. 40 of Applied Optimization, Kluwer Academic, Dordrecht, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu, A. N. Iusem, and R. S. Burachik, “Iterative methods of solving stochastic convex feasibility problems and applications,” Computational Optimization and Applications, vol. 15, no. 3, pp. 269–307, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu, A. N. Iusem, and E. Resmerita, “Total convexity for powers of the norm in uniformly convex Banach spaces,” Journal of Convex Analysis, vol. 7, no. 2, pp. 319–334, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu, A. N. Iusem, and C. Zălinescu, “On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces,” Journal of Convex Analysis, vol. 10, no. 1, pp. 35–61, 2003.
View at: Google Scholar | MathSciNet
D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508, 2003.
View at: Google Scholar | MathSciNet
D. Butnariu and E. Resmerita, “The outer Bregman projection method for stochastic feasibility problems in Banach spaces,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Stud. Comput. Math., pp. 69–86, North-Holland, Amsterdam, 2001.
View at: Google Scholar | MathSciNet
C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,” Journal of Optimization Theory and Applications, vol. 34, no. 3, pp. 321–353, 1981.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. Censor and S. A. Zenios, Parallel Optimization. Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. A. Clarkson, “Uniformly convex spaces,” Transactions of the American Mathematical Society, vol. 40, no. 3, pp. 396–414, 1936.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, vol. 1543 of Lecture Notes in Mathematics, Springer, Berlin, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Eckstein, “Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming,” Mathematics of Operations Research, vol. 18, no. 1, pp. 202–226, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. P. B. Eggermont, “Maximum entropy regularization for Fredholm integral equations of the first kind,” SIAM Journal on Mathematical Analysis, vol. 24, no. 6, pp. 1557–1576, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. W. Engl and G. Landl, “Convergence rates for maximum entropy regularization,” SIAM Journal on Numerical Analysis, vol. 30, no. 5, pp. 1509–1536, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
K. Fan and I. Glicksberg, “Some geometric properties of the spheres in a normed linear space,” Duke Mathematical Journal, vol. 25, pp. 553–568, 1958.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, 1981.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. B. Holmes, A Course on Optimization and Best Approximation, Lecture Notes in Mathematics, Vol. 257, Springer, Berlin, 1972.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
T. Ibaraki, Y. Kimura, and W. Takahashi, “Convergence theorems for generalized projections and maximal monotone operators in Banach spaces,” Abstract and Applied Analysis, vol. 10, pp. 621–629, 2003.
View at: Google Scholar | MathSciNet
A. N. Iusem and R. Gárciga Otero, “Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 609–640, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. N. Iusem and R. Gárciga Otero, “Augmented Lagrangian methods for cone-constrained convex optimization in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 3, no. 2, pp. 155–176, 2002.
View at: Google Scholar | MathSciNet
M. Ĭ. Kadec', “Spaces isomorphic to a locally uniformly convex space,” Izvestija Vysš Učebnyh ZavedeniĭMatematika, vol. 6, no. 13, pp. 51–57, 1959 (Russian).
View at: Google Scholar | MathSciNet
K. C. Kiwiel, “Proximal minimization methods with generalized Bregman functions,” SIAM Journal on Control and Optimization, vol. 35, no. 4, pp. 1142–1168, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” American Journal of Mathematics, vol. 73, pp. 615–624, 1951.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
F. Liu and M. Z. Nashed, “Regularization of nonlinear ill-posed variational inequalities and convergence rates,” Set-Valued Analysis, vol. 6, no. 4, pp. 313–344, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
U. Mosco, “Convergence of convex sets and of solutions of variational inequalities,” Advances in Mathematics, vol. 3, pp. 510–585, 1969.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” American Mathematical Society Bulletin, vol. 73, pp. 591–597, 1967.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff, The Hague; Sijthoff & Noordhoff International, Alphen aan den Rijn, 1978.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer, Berlin, 2nd edition, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
B. Polyak, “A general method for solving extremum problems,” Doklady Akedamii Nauk SSSR, vol. 174, pp. 593–597, 1967.
View at: Google Scholar
S. Reich, “Extension problems for accretive sets in Banach spaces,” Journal of Functional Analysis, vol. 26, no. 4, pp. 378–395, 1977.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 313–318, Dekker, New York, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Resmerita, Fixed point and optimization methods in infinite dimensional Banach spaces, M.S. thesis, University of Haifa, Israel, 2003.
View at: Google Scholar
E. Resmerita, “On total convexity, Bregman projections and stability in Banach spaces,” Journal of Convex Analysis, vol. 11, pp. 1–16, 2004.
View at: Google Scholar
R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, no. 28, Princeton University Press, New Jersey, 1970.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. V. Solodov and B. F. Svaiter, “An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions,” Mathematics of Operations Research, vol. 25, no. 2, pp. 214–230, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. L. Troyanski, “On locally uniformly convex and differentiable norms in certain non-separable Banach spaces,” Studia Mathematica, vol. 37, pp. 173–180, 1970.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. A. Vladimirov, Ju. E. Nesterov, and Ju. N. Čekanov, “Uniformly convex functionals,” Vestnik Moskovskogo Universiteta. Seriya XV. Vychislitel'naya Matematika i Kibernetika, vol. 3, pp. 12–23, 1978 (Russian).
View at: Google Scholar | MathSciNet
C. Zălinescu, “On uniformly convex functions,” Journal of Mathematical Analysis and Applications, vol. 95, no. 2, pp. 344–374, 1983.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, New Jersey, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

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Copyright © 2006 Dan Butnariu and Elena Resmerita. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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