Essential smoothness, essential strict convexity, and legendre functions in banach spaces

Communications in Contemporary Mathematics

Volumn 3, Issue 4, 2001, Pages 615-647

  Bauschke, Heinz H ^a   Borwein, Jonathan M ^b   Combettes, Patrick L ^c  

^a UNIVERSITY OF GUELPH   (Canada)

^b SIMON FRASER UNIVERSITY   (Canada)

^c SORBONNE UNIVERSITÉ   (France)

Author keywords

Bregman distance, Bregman projection; Coercive; Cofinite function; Convex function of Legendre type; Essentially smooth; Essentially strictly convex; Legendre function; Schur property; Schur space; Subdifferential; Supercoercive; Weak Asplund space

Indexed keywords

EID: 23044531274     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219199701000524     Document Type: Article

References (49)

1. J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons Inc., New York (1984).
2. S. Banach, Théorie des Opérations Linéaires, second edition, Chelsea, New York (1978).
3. English translation: Theory of Linear Operations, North-Holland Publishing Co., Amsterdam (1987).
4. V. Barbu and Th. Precupanu, Convexity and Optimization in Banach Spaces, D. Reidel Publishing Co., second edition, Dordrecht (1986).
5. H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38(3) (1996) 367-426.
6. H. H. Bauschke and J. M. Borwein, Legendre functions and the method of random Bregman projections, J. Convex Anal. 4(1) (1997) 27-67.
7. H. H. Bauschke, J. M. Borwein and P. L. Combettes, Bregman monotone optimization algorithms, CECM preprint 01:171. Available at www.cecm.sfu.ca/preprints.
8. J. Borwein, M. Fabian and J. Vanderwerff, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam. 22(1) (1997) 53-69.
9. J. M. Borwein, Asplund spaces are "sequentially reflexive", 1991. Combinatorics and Optimization Research Report CORR 91-14, University of Waterloo, Canada.
10. J. M. Borwein, Minimal CUSCOS and subgradients of Lipschitz functions, in Fixed Point Theory and Applications (Marseille, 1989), Longman Sci. Tech., Harlow (1991), pp. 57-81.
11. J. M. Borwein and A. S. Lewis, Strong rotundity and optimization, SIAM J. Optim. 4(1) (1994) 146-158.
12. J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, Springer-Verlag, New York (2000).
13. J. M. Borwein, A. S. Lewis and Q. J. Zhu, Convex spectral functions of compact operators, Part II: Lower semicontinuity and rearrangement invariance, in Optimization and Related Topics, eds. A. Rubinov and B. Glover, Kluwer (2001), pp. 179-196.
14. J. M. Borwein, J. Read, A. S. Lewis and Q. J. Zhu, Convex spectral functions of compact operators, J. Nonlinear Convex Anal. 1(1) (2000) 17-35.
15. J. M. Borwein and J. D. Vanderwerff, Convex functions of Legendre type in general Banach spaces, 2000. Preprint 00:151, Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada. Available at www.cecm.sfu.ca/preprints, to appear in J. Convex Anal.
16. N. Bourbaki, Topologie Générale, Chapitres 5 à 10, Hermann, Paris (1974).
17. English Translation: General Topology, Chapters 5-10, Springer-Verlag, New York (1989).
18. L. M. Bregman, A relaxation method of finding a common point of convex sets and its application to the solution of problems in convex programming, Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967) 620-631.
19. English Translation in U.S.S.R. Comput. Math. Math. Phys. 7(3) (1967) 200-217.
20. D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer (2000).
21. D. Butnariu, A. N. Iusem and E. Resmerita, Total convexity for powers of the norm in uniformly convex Banach spaces, J. Convex Anal. 7(2) (2000) 319-334.
22. Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34(3) (1981) 321-353.
23. Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, New York (1997).
24. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht (1990).
25. F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition (1990).
26. F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag (1998).
27. R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Longman Scientific & Technical, Harlow (1993).
28. J. Diestel, Geometry of Banach Spaces - Selected Topics, Springer-Verlag, Berlin (1975). Lecture Notes in Mathematics, Vol. 485.
29. J. Diestel, Sequences and Series in Banach spaces, Springer-Verlag, New York (1984).
30. J. R. Giles, Convex Analysis with Application in the Differentiation of Convex Functions, Pitman, Boston, MA (1982).
31. R. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22(3) (1990) 261-268.
32. R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, New York (1975). Graduate Texts in Mathematics, No. 24.
33. P. S. Kenderov, The set-valued monotone mappings are almost everywhere single-valued, C. R. Acad. Bulgare Sci. 27 (1974) 1173-1175.
34. H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York (1974). Die Grundlehren der mathematischen Wissenschaften, Band 208.
35. D. G. Larman and R. R. Phelps, Gateaux differentiability of convex functions on Banach spaces, J. London Math. Soc. (2) 20(1) (1979) 115-127.
36. E. S. Levitin and B. T. Poljak, Convergence of minimizing sequences in conditional extremum problems, Dokl. Akad. Nauk SSSR 168 (1966) 997-1000.
37. English translation in Soviet Math. Dokl. 7 (1966) 764-767.
38. A. S. Lewis, Convex analysis on the Hermitian matrices, SIAM J. Optim. 6(1) (1996) 164-177.
39. J.-J. Moreau, Sur la fonction polaire d'une fonction semi-continue supérieurement, C. R. Acad. Sci. Paris 258 (1964) 1128-1130.
40. J.-J. Moreau, Fonctionnelles convexes, Collège de France, Paris, 1966-1967. Séminaire sur les Equations aux Dérivées Partielles II.
41. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Springer-Verlag, Berlin, second edition (1993).
42. 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, Dekker, New York (1996), pp. 313-318.
43. R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966) 46-63.
44. R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969) 397-407.
45. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ (1970). Princeton Mathematical Series, No. 28.
46. R. T. Rockafellar, Conjugate Duality and Optimization, SIAM, Philadelphia, PA (1974).
47. Stephen Simons, Minimax and Monotonicity, Springer-Verlag, Berlin (1998).
48. A. A. Vladimirov, Ju. E. Nesterov and Ju. N. Čekanov, Uniformly convex functionals, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 3 (1978) 12-23.
49. C. Zǎlinescu, On uniformly convex functions, J. Math. Anal. Appl. 95(2) (1983) 344-374.