Abstract

We study and compare the notions of uniform convexity of functions at a point and on bounded sets with the notions of total convexity at a point and sequential consistency of functions, respectively. We establish connections between these concepts of strict convexity in infinite dimensional settings and use the connections in order to obtain improved convergence results concerning the outer Bregman projection algorithm for solving convex feasibility problems and the generalized proximal point algorithm for optimization in Banach spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Alfredo N. Iusem

Institute of Pure and Applied Mathematics (IMPA), Estrada Dona Castorina 110,
Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil

iusp@impa.br

Constantin Zălinescu

Faculty of Mathematics, "Al. I. Cuza" University, 6600 Iasi, Romania

zalinesc@uaic.ro

Keywords

  • Uniform convexity at a point
  • total convexity at a point
  • uniform convexity on bounded sets
  • sequential consistency
  • generalized proximal point algorithm for optimization
  • outer Bregman projection algorithm for feasibility

Mathematics Subject Classification

90C25, 90C48, 26B25, 49J52, 46N10, 46N20, 90C30

References

  1. Y. Alber, A. N. Iusem: Extensions of subgradient techniques for nonsmooth optimization in Banach spaces, Set-Valued Analysis 9 (2001) 315–335.https://doi.org/10.1023/A:1012665832688Google Scholar
  2. Y. Alber, A. N. Iusem, M. V. Solodov: Minimization of nonsmooth convex functionals in Banach spaces, Journal of Convex Analysis 4 (1997) 235–255.https://doi.org/10.68381/jca04012Google Scholar
  3. E. Asplund: Averaged norms, Israel Journal of Mathematics 5 (1967) 227–233.https://doi.org/10.1007/BF02771611Google Scholar
  4. E. Asplund, R. T. Rockafellar: Gradients of convex functions, Transactions of the American Mathematical Society 139 (1969) 443–467.https://doi.org/10.1090/S0002-9947-1969-0240621-XGoogle Scholar
  5. J. B. Baillon: Un example concernant le comportement asymptotique de la solution du problème 0 ∈ du/dt + ∂ϕ(u), Journal of Functional Analysis 28 (1978) 369–376.https://doi.org/10.1016/0022-1236(78)90093-9Google Scholar
  6. H. H. Bauschke, J. M. Borwein, P. L. Combettes: Essential smoothness, essential strict convexity and Legendre functions in Banach spaces, Communications in Contemporary Mathematics 3 (2001) 615–647.https://doi.org/10.1142/S0219199701000524Google Scholar
  7. H. H. Bauschke, A. S. Lewis: Dykstra’s algorithm with Bregman projections: A convergence proof, Optimization 48 (2000) 409–428.https://doi.org/10.1080/02331930008844513Google Scholar
  8. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer-Verlag, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  9. R. E. Bruck, S. Reich: Nonexpansive projections and resolvents of acretive operators in Banach spaces, Houston Journal of Mathematics 3 (1977) 459–470.Google Scholar
  10. R. Burachik: Generalized Proximal Point Methods for the Variational Inequality Problem, Ph.D. Thesis, The Institute of Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil (1995).Google Scholar
  11. R. Burachik, S. Scheimberg: A proximal point algorithm for the variational inequality problem, SIAM Journal of Control and Optimization 39 (2000) 1633–1649.https://doi.org/10.1137/S0363012998339745Google Scholar
  12. D. Butnariu, Y. Censor, S. Reich: Iterative averaging of entropic projections for solving stochastic convex feasibility problems, Computational Optimization and Applications 8 (1997) 21–39.https://doi.org/10.1023/A:1008654413997Google Scholar
  13. D. Butnariu, A. N. Iusem: Totally Convex Functions for Fixed Point Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers (2000).https://doi.org/10.1007/978-94-011-4066-9Google Scholar
  14. D. Butnariu, A. N. Iusem: On a proximal point method for convex optimization in Banach spaces, Numerical Functional Analysis and Optimization 18 (1997) 723–744.https://doi.org/10.1080/01630569708816788Google Scholar
  15. D. Butnariu, A. N. Iusem, R. Burachik: Iterative methods for solving stochastic convex feasibility problems and applications, Computational Optimization and Applications 15 (2000) 269–307.https://doi.org/10.1023/A:1008795702124Google Scholar
  16. D. Butnariu, A. N. Iusem, E. Resmerita: Total convexity for the powers of the norm in uniformly convex Banach spaces, Journal of Convex Analysis 7 (2000) 319–334.https://doi.org/10.68381/jca07016Google Scholar
  17. D. Butnariu, I. Markowicz: An evaluation of the moduli of total convexity for the powers of the norms in locally uniformly convex Banach spaces, Technical Report, Department of Mathematics, University of Haifa (March 1997).Google Scholar
  18. D. Butnariu, 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, D. Butnariu, Y. Censor, S. Reich (eds.), Elsevier, Amsterdam (2001) 69–87.https://doi.org/10.1016/S1570-579X(01)80007-0Google Scholar
  19. D. Butnariu, E. Resmerita: Averaged subgradient methods for constrained convex optimization and Nash equilibria computation, Optimization, to appear.Google Scholar
  20. Y. Censor, A. Lent: An iterative row action method for interval convex programming, Journal of Optimization Theory and Applications 34 (1981) 321–353.https://doi.org/10.1007/BF00934676Google Scholar
  21. Y. Censor, S. A. Zenios: Proximal minimization algorithm with D-functions, Journal of Optimization Theory and Applications 73 (1992) 455–468.https://doi.org/10.1007/BF00940051Google Scholar
  22. I. Ciorănescu: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht (1990).https://doi.org/10.1007/978-94-009-2121-4Google Scholar
  23. J. Diestel: Geometry of Banach Spaces – Selected Topics, Springer-Verlag, Berlin (1975).https://doi.org/10.1007/BFb0082079Google Scholar
  24. R. Gárciga Otero, A. N. Iusem: Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numerical Functional Analysis and Optimization 22 (2001) 609–640; erratum 23 (2002) 227–228.https://doi.org/10.1081/NFA-100105310Google Scholar
  25. O. Güler: On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization 29 (1991) 403–419.https://doi.org/10.1137/0329022Google Scholar
  26. A. N. Iusem, C. Isnard: On mixed Hölder-Minkowski inequalities and total convexity of certain functions in Lp (Ω), Mathematical Inequalities and Applications 3 (2000) 519–537.https://doi.org/10.7153/mia-03-50Google Scholar
  27. E. S. Levitin, B. T. Polyak: Convergence of minimizing sequences in conditional extremum problems, Soviet Mathematics-Doklady 7 (1966) 764–767.Google Scholar
  28. O. Nevalinna, S. Reich: Strong convergence of contraction semigroups and of iterative methods for acretive operators in Banach spaces, Israel Journal of Mathematics 32 (1979), 44–58.https://doi.org/10.1007/BF02761184Google Scholar
  29. J.-P. Penot, M. Volle: Inversion of real valued functions and applications, Zeitschrift für Operations Research 34 (1990) 117–141.https://doi.org/10.1007/BF01415975Google Scholar
  30. B. T. Polyak: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Mathematics-Doklady 7 (1966) 73–75.Google Scholar
  31. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton, New Jersey (1970).Google Scholar
  32. R. T. Rockafellar: Local boundedness of nonlinear monotone operators, Michigan Mathematical Journal, 16 (1969) 397–407.https://doi.org/10.1307/mmj/1029000324Google Scholar
  33. R. T. Rockafellar: Monotone operators and the proximal point algorithm, SIAM Journal of Control and Optimization 14 (1976) 877–898.https://doi.org/10.1137/0314056Google Scholar
  34. M. V. Solodov, B. F. Svaiter: Forcing strong convergence of proximal point iterations in a Hilbert space, Mathematical Programming 87 (2000) 189–202.https://doi.org/10.1007/s101079900113Google Scholar
  35. A. A. Vladimirov, Y. E. Nesterov, Y. N. Chekanov: On uniformly convex functionals (Russian), Vestnik Moskovskaya Universiteta, Seria Matematika y Kibernetika 3 (1978) 12–23.Google Scholar
  36. C. Zălinescu: On uniformly convex functions, Journal of Mathematical Analysis and Applications 95 (1983) 344–374.https://doi.org/10.1016/0022-247X(83)90112-9Google Scholar
  37. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, New Jersey (2002).https://doi.org/10.1142/5021Google Scholar
  38. T. Zolezzi: On equiwellset minimum problems, Applied Mathematics and Optimization 4 (1978) 209–223.https://doi.org/10.1007/BF01442140Google Scholar

Suggested citation

D. Butnariu, A. N. Iusem, 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 10 (2003), No. 1, 35–61. https://doi.org/10.68381/jca1002

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.