Monotone linear relations: Maximality and fitzpatrick functions

Journal of Convex Analysis

Volumn 16, Issue 3-4, 2009, Pages 673-686

  Bauschke, Heinz H ^a   Wang, Xianfu ^a   Yao, Liangjin ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Adjoint process; Fenchel conjugate; Fitzpatrick family; Fitzpatrick function; Linear relation; Maximal monotone operator; Monotone operator; Skew linear relation

Indexed keywords

EID: 73349135630     PISSN: 09446532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. R. Arens: Operation calculus on linear relations, Pac. J. Math. 19 (1961) 9-23.
2. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).
3. S. Bartz, H. H. Bauschke, J. M. Borwein, S. Reich, X. Wang: Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative, Nonlinear Anal., Theory Methods Appl. 66A (2007) 1198-1223.
4. H. H. Bauschke, J. M. Borwein, X. Wang: Fitzpatrick functions and continuous linear monotone operators, SIAM J. Optim. 18 (2007) 789-809.
5. H. H. Bauschke, D. A. McLaren, H. S. Sendov: Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick, J. Convex Analysis 13 (2006) 499-523.
6. J. M. Borwein: Adjoint process duality, Math. Oper. Res. 8 (1983) 403-434.
7. H. Brézis, F. E. Browder: Linear maximal monotone operators and singular nonlinear integral equations of Hammerstein type, in: Nonlinear Analysis, L. Cesari et al. (ed.), Academic Press, New York (1978) 31-42.
8. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002) 297-316.
9. R. S. Burachik, B. F. Svaiter: Operating enlargements of monotone operators, Pac. J. Optim. 2 (2006) 425-445.
10. D. Butnariu, G. Kassay: A proximal-projection method for finding zeros of set-valued operators, SIAM J. Control Optimization 4 (2008) 2096-2136.
11. R. Cross: Multivalued Linear Operators, Marcel Dekker, New York (1998).
12. S. Fitzpatrick: Representing monotone operators by convex functions, in: Functional Analysis and Optimization, Workshop / Miniconference (Canberra, 1988), Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, Australian National University, Canberra (1988) 59-65.
13. M. Marques Alves, B. F. Svaiter: Maximal monotone operators with a unique extension to the bidual, J. Convex Analysis 16 (2008) 409-421.
14. J.-E. Martínez-Legaz, M. Théra: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001) 243-247.
15. J.-P. Penot: The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., Theory Methods Appl. 58A (2004) 855-871.
16. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Springer, Berlin (1993).
17. R. R. Phelps, S. Simons: Unbounded linear monotone operators on nonreflexive Banach spaces, J. Convex Analysis 5 (1998) 303-328.
18. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).
19. R. T. Rockafellar, R. J.-B, Wets: Variational Analysis, Springer, Berlin (1998).
20. S. Simons: From Hahn-Banach to Monotonicity, Springer, Berlin (2008).
21. S. Simons, C. Zǎlinescu: Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005) 1-22.
22. M. D. Voisei: The sum theorem for linear maximal monotone operators, Math. Sci. Res. J. 10 (2006) 83-85.
23. M. D. Voisei, C. Zǎlinescu: Linear monotone subspaces of locally convex spaces, preprint, available at: http://arxiv.org/abs/0809.5287v1 (September 2008).
24. L. Yao: Decompositions and Representations of Monotone Operators with Linear Graphs, M. Sc. Thesis, University of British Columbia Okanagan, Kelowna (December 2007).
25. C. Zǎlinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).