Maximal monotone operators with a unique extension to the bidual

Journal of Convex Analysis

Volumn 16, Issue 2, 2009, Pages 409-421

  Alves, M Marques ^a   Svaiter, B F ^a  

^a IMPA   (Brazil)

Author keywords

Banach spaces; Bidual; Br ndsted Rockafellar property; Extension; Maximal monotone operators; S function

Indexed keywords

EID: 70349761459     PISSN: 09446532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. H. H. Bauschke, X. Wang, L. Yao: Monotone linear relations: maximality and Fitzpatrick functions, ArXiv:0805.4256v1 (May 2008).
2. G. Birkhoff: Lattice Theory, 3rd Ed., Colloquium Publications 25, American Mathematical Society, Providence (1967).
3. J. M. Borwein: Maximal monotonicity via convex analysis, J. Convex Analysis 13 (3-4) (2006) 561-586.
4. J. M. Borwein: Maximality of sums of two maximal monotone operators in general Banach space, Proc. Amer. Math. Soc. 135 (12) (2007) 3917-3924.
5. (Pubitemid 127430574) R. S. Burachik, A. N. Iusem, B. F. Svaiter: Enlargement of monotone operators with applications to variational inequalities, Set-Valued Anal. 5 (2) (1997) 159-180.
6. R. S. Burachik, B. F. Svaiter: ε-enlargements of maximal monotone operators in Banach spaces, Set-Valued Anal. 7 (2) (1999) 117-132.
7. R. S. Burachik: Maximal monotone operators, convex functions and a special family of enlargements, in: International Workshop on Optimization and Control with Applications (Erice, July 2001), short talk.
8. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Technical Report A094, IMPA, Rio de Janeiro (August 2001), available at http://www.preprint.impa.br/Shadows/ SERIE-A/2001/94.html.
9. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (4) (2002) 297-316.
10. R. S. Burachik, B. F. Svaiter: Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc. 131 (8) (2003) 2379-2383.
11. R. S. Burachik, A. N. Iusem: On non-enlargeable and fully enlargeable monotone operators, J. Convex Analysis 13 (3-4) (2006) 603-622.
12. (Pubitemid 40669839) R. Correa, P. Gajardo: Eigenvalues of set-valued operators in Banach spaces, Set-Valued Anal. 13 (1) (2005) 1-19.
13. S. Fitzpatrick: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ. 20, Austral. Nat. Univ., Canberra (1988) 59-65.
14. J.-P. Gossez: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971) 371-395.
15. J.-P. Gossez; On the range of a coercive maximal monotone operator in a nonreflexive Banach space, Proc. Amer. Math. Soc. 35 (1972) 88-92.
16. J.-P. Gossez: On a convexity property of the range of a maximal monotone operator, Proc. Amer. Math. Soc. 55 (2) (1976) 359-360.
17. J.-P. Gossez: On the extensions to the bidual of a maximal monotone operator, Proc. Amer. Math. Soc. 62 (1) 67-71 (1977).
18. T. T. Hue, J. J. Strodiot: Convergence analysis of a relaxed extragradient-proximal point algorithm application to variational inequalities, Optimization 54 (2) (2005) 191-213.
19. A. Kaplan, R. Tichatschke: Bregman-like functions and proximal methods for variational problems with nonlinear constraints, Optimization 56 (1-2) (2007) 253-265.
20. P.-E. Maingé, A. Moudafi: A proximal method for maximal monotone operators via discretization of a first order dissipative dynamical system, J. Convex Analysis 14 (4) (2007) 869-878.
21. M. Marques Alves, B. F. Svaiter: Brøndsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces, J. Convex Analysis 15 (4) (2008) 693-706.
22. (Pubitemid 40669840) J.-E. Martínez-Legaz, B. F. Svaiter: Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13 (1) (2005) 21-46.
23. J.-P. Penot: The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., Theory Methods Appl. 58 A (7-8) (2004) 855-871.
24. J.-P. Penot: The use of nonsmooth analysis and of duality methods for the study of hamilton-jacobi equations, in: International Workshop on Optimization and Control with Applications (Erice, July 2001), short talk.
25. S. Simons: The range of a monotone operator, J. Math. Anal. Appl. 199 (1) (1996) 176-201.
26. M. V. Solodov, B. F. Svaiter: A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator, Set-Valued Anal. 7 (4) (1999) 323-345.
27. M. V. Solodov, B. F. Svaiter: A unified framework for some inexact proximal point algorithms, Numer. Funct. Anal. Optimization 22 (7-8) (2001) 1013-1035.
28. B. F. Svaiter: A family of enlargements of maximal monotone operators, Set-Valued Anal. 8 (4) (2000) 311-328.
29. B. F. Svaiter: Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc. 131 (12) (2003) 3851-3859.
30. D. Torralba: Convergence Épigraphique et Changements d'Échelle en Analyse Variationnelle et Optimisation, Ph. D. Thesis, Universite Montpellier II, Montpellier (1996).
31. M. D. Voisei: The sum and chain rules for maximal monotone operators, Set-Valued Anal., submitted (2006).