Abstract

Let KK be a convex subset of a normed linear space and let R1R^1 denote the real line. We show that there are many (in the sense of Baire category) strictly convex and totally convex functions f ⁣:K→R1f \colon K \to R^1. It is known that the existence of such functions is crucial in numerous optimization algorithms.

Author information

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

Alexander J. Zaslavski

Dept. of Mathematics, Technion - Israel Inst. Technology, 32000 Haifa, Israel

ajzasl@tx.technion.ac.il

Keywords

  • Complete metric space
  • essentially strictly convex function
  • generic property
  • strictly convex function
  • totally convex function

Mathematics Subject Classification

46N10, 52A41, 54E50, 54E52

References

  1. Y. Alber, R. Burachik, A. N. Iusem: A proximal point method for nonsmooth optimization in Banach spaces, Abstr. Appl. Anal. 2 (1997) 97–120.https://doi.org/10.1155/S1085337597000298Google Scholar
  2. Y. Alber, D. Butnariu, I. Ryazantseva: Regularization and resolution of monotone variational inequalities with operators given by hypomonotone approximations, J. Nonlinear Convex Anal. 6 (2005) 23–53.Google Scholar
  3. H. Bauschke, J. M. Borwein, P. L. Combettes: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001) 615–647.https://doi.org/10.1142/S0219199701000524Google Scholar
  4. H. Bauschke, J. M. Borwein, P. L. Combettes: Bregman monotone optimization algorithms, SIAM J. Control Optim. 42 (2003) 596–636.https://doi.org/10.1137/S0363012902407120Google Scholar
  5. H. Bauschke, P. L. Combettes: Construction of best Bregman approximations in reflexive Banach spaces, Proc. Amer. Math. Soc. 131 (2003) 3757–3766.https://doi.org/10.1090/S0002-9939-03-07050-3Google Scholar
  6. D. Butnariu, Y. Censor, S. Reich: Iterative averaging of entropic projections for solving stochastic convex feasibility problems, Comput. Optim. Appl. 8 (1997) 21–39.https://doi.org/10.1023/A:1008654413997Google Scholar
  7. D. Butnariu, 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, Y. Censor, S. Reich (eds.), Contemporary Mathematics 204, American Mathematical Society, Providence (1997) 61–91.https://doi.org/10.1090/conm/204/02623Google Scholar
  8. D. Butnariu, A. N. Iusem: On a proximal point method for convex optimization in Banach spaces, Numer. Funct. Anal. Optim. 18 (1998) 723–744.https://doi.org/10.1080/01630569708816788Google Scholar
  9. D. Butnariu, A. N. Iusem: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-94-011-4066-9Google Scholar
  10. D. Butnariu, A. N. Iusem, R. S. Burachik: Iterative methods of solving stochastic convex feasibility problems and applications, Comput. Optim. Appl. 15 (2000) 269–307.https://doi.org/10.1023/A:1008795702124Google Scholar
  11. D. Butnariu, A. N. Iusem, E. Resmerita: Total convexity of the powers of the norm in uniformly convex Banach spaces, J. Convex Analysis 7 (2000) 319–334.https://doi.org/10.68381/jca07016Google Scholar
  12. D. Butnariu, A. N. Iusem, C. Zalinescu: Uniform convexity, total convexity and the convergence of the outer Bregman projection and proximal point algorithms in Banach spaces, J. Convex Analysis 10 (2003) 35–61.https://doi.org/10.68381/jca1002Google Scholar
  13. D. Butnariu, S. Reich, A. J. Zaslavski: Generic power convergence of operators in Banach spaces, Numer. Funct. Anal. Optim. 20 (1999) 629–650.https://doi.org/10.1080/01630569908816915Google Scholar
  14. D. Butnariu, S. Reich, A. J. Zaslavski: Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001) 151–174.https://doi.org/10.1515/JAA.2001.151Google Scholar
  15. D. Butnariu, S. Reich, A. J. Zaslavski: Asymptotic behavior of quasi-nonexpansive mappings, in: Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, D. Butnariu, Y. Censor, S. Reich (eds.), Elsevier, Amsterdam (2001) 49–68.https://doi.org/10.1016/S1570-579X(01)80006-9Google Scholar
  16. D. Butnariu, E. Resmerita: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006.https://doi.org/10.1155/AAA/2006/84919Google Scholar
  17. Y. Censor, A. Lent: An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981) 321–353.https://doi.org/10.1007/BF00934676Google Scholar
  18. Y. Censor, S. Reich: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996) 323–339.https://doi.org/10.1080/02331939608844225Google Scholar
  19. Y. Censor, S. A. Zenios: Parallel Optimization, Oxford, New York (1997).Google Scholar
  20. F. S. De Blasi, J. Myjak: Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach, C. R. Acad. Sci., Paris, Sér. A 283 (1976) 185–187.Google Scholar
  21. A. N. Iusem, R. G. Otero: Augmented Lagrangian methods for cone constrained convex optimization in Banach spaces, J. Nonlinear Convex Anal. 3 (2002) 155–176.Google Scholar
  22. R. G. Otero, B. F. Svaiter: A strongly convergent hybrid proximal method in Banach spaces, J. Math. Anal. Appl. 289 (2004) 700–711.https://doi.org/10.1016/j.jmaa.2003.09.030Google Scholar
  23. 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, A. G. Kartsatos (ed.), Marcel Dekker, New York (1996) 313–318.Google Scholar
  24. E. Resmerita: On total convexity, Bregman projections and stability in Banach spaces, J. Convex Analysis 11 (2004) 1–16.https://doi.org/10.68381/jca11001Google Scholar
  25. R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  26. R. T. Rockafellar: Augmented Lagrangeans and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1 (1976) 97–116.https://doi.org/10.1287/moor.1.2.97Google Scholar
  27. A. N. Tikhonov: Regularization of incorrectly posed problems, Soviet Math. Dokl. 4 (1963) 1035–1038.Google Scholar
  28. G. Vidossich: Existence, uniqueness and approximation of fixed points as a generic property, Bol. Soc. Bras. Mat. 5 (1974) 17–29.https://doi.org/10.1007/BF02584769Google Scholar
  29. A. J. Zaslavski: Well-posedness and porosity in optimal control without convexity assumptions, Calc. Var. Partial Differ. Equ. 13 (2001) 265–293.https://doi.org/10.1007/s005260000073Google Scholar

Suggested citation

D. Butnariu, S. Reich, A. J. Zaslavski. “There are Many Totally Convex Functions.” Journal of Convex Analysis 13 (2006), No. 3/4, 623–632. https://doi.org/10.68381/jca13048

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.