Fitzpatrick functions and continuous linear monotone operators

SIAM Journal on Optimization

Volumn 18, Issue 3, 2007, Pages 789-809

  Bauschke, Heinz H ^a   Borwein, Jonathan M ^b   Wang, Xianfu ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b DALHOUSIE UNIVERSITY   (Canada)

Author keywords

Cyclic monotonicity; Fitzpatrick family; Fitzpatrick function; Linear operator; Maximal monotone operator; Moore Penrose inverse; Paramonotone operator; Rotator

Indexed keywords

BANACH SPACES; CONSTRUCTION; HILBERT SPACES;

CYCLIC MONOTONICITY; EUCLIDEAN PLANES; FITZPATRICK; FITZPATRICK FAMILY; FITZPATRICK FUNCTION; HILBERT; LINEAR OPERATOR; LOWER SEMICONTINUOUS; MAXIMAL MONOTONE OPERATOR; MAXIMAL MONOTONE OPERATORS; MONOTONICITY PROPERTIES; MOORE-PENROSE INVERSE; PARAMONOTONE OPERATOR; POSITIVE OPERATORS; ROTATOR; SUBDIFFERENTIAL OPERATORS; UPPER BOUNDS;

MATHEMATICAL OPERATORS;

EID: 49449086907     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/060655468     Document Type: Conference Paper

References (43)

1. E. ASPLUND, A monotone convergence theorem for sequences of nonlinear mappings, in Non-linear Functional Analysis, Proceedings of Symposia in Pure Mathematics, Vol. 18, Part 1, AMS, Providence, RI, 1970, pp. 1-9.
2. S. BARTZ, H. H. BAUSCHKE, J. M. BORWEIN, S. REICH, AND X. WANG, Fitzpatrick functions, cyclic monotonicity, and Rockafellar's antiderivative, Nonlinear Anal., 66 (2007), pp. 1198-1223.
3. H. H. BAUSCHKE, Projection Algorithms and Monotone Operators, Ph.D. thesis, Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, 1996; available online at http://www.cecm.sfu.ca/ preprints/1996pp.html.
4. H. H. BAUSCHKE, The composition of finitely many projections onto closed convex sets in Hilbert space is asymptotically regular, Proc Amer. Math. Soc., 131 (2003), pp. 141-146.
5. H. H. BAUSCHKE AND J. M. BORWEIN, Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators, Pacific J. Math., 189 (1999), pp. 1-20.
6. H. H. BAUSCHKE, D. A. MCLAREN, AND H. S. SENDOV, Fitzpatrick functions: Inequalities, examples and remarks on a problem by S. Fitzpatrick, J. Convex Anal., 13 (2006), pp. 499-523.
7. H. H. BAUSCHKE AND S. SIMONS, Stronger maximal monotonicity properties of linear operators, Bull. Austral. Math. Soc., 60 (1999), pp. 163-174.
8. J. M. BORWEIN, Maximal monotonicity via convex analysis, J. Convex Anal., 13 (2006), pp. 561-586.
9. J. M. BORWEIN AND Q. J. ZHU, Techniques of Variational Analysis, Springer-Verlag, New York, 2005.
10. H. BRÉZIS, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.
11. H. BRÉZIS AND A. HARAUX, Image d'une somme d'opérateurs monotones et applications, Israel J. Math., 23 (1976), pp. 165-186.
12. R. S. BURACHIK AND S. FITZPATRICK, On a family of convex functions associated to subdifferentials, J. Nonlinear Convex Anal., 6 (2005), pp. 165-171.
13. R. S. BURACHIK AND S. FITZPATRICK, Corrigendum to: "On a family of convex functions associated to subdifferentials," J. Nonlinear Convex Anal., 6 (2005), p. 535.
14. R. S. BURACHIK AND B. F. SVAITER, Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal., 10 (2002), pp. 297-316.
15. R. S. BURACHIK AND B. F. SVAITER, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), pp. 2379-2383.
16. S. CHENG AND Y. TIAN, Two sets of new characterizations for normal and EP matrices, Linear Algebra Appl., 375 (2003), pp. 181-195.
17. S. FITZPATRICK, Representing monotone operators by convex functions, in Workshop/Mini-conference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ. 20, Australian National University, Canberra, Australia, 1988, pp. 59-65.
18. Y. GARCÍA, M. LASSONDE, AND J. P. REVALSKI, Extended sums and extended compositions of monotone operators, J. Convex Anal., 13 (2006), pp. 721-738.
19. K. GOEBEL AND S. REICH, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
20. C. W. GROETSCH, Generalized Inverses of Linear Operators, Marcel Dekker, New York, 1977.
21. P. R. HALMOS, A Hilbert Space Problem Book, Van Nostrand, Princeton, NJ, 1967.
22. A. N. IUSEM, On some properties of paramonotone operators, J. Convex Anal., 5 (1998), pp. 269-278.
23. E. KREYSZIG, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1989.
24. J.-E. MARTÍNEZ-LEGAZ AND B. F. SVAITER, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), pp. 21-46.
25. J.-E. MARTÍNEZ-LEGAZ AND M. THÉ RA, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal., 2 (2001), pp. 243-247.
26. C. D. MEYER, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000.
27. B. S. MORDUKHOVICH, Variational Analysis and Generalized Differentiation I, Springer-Verlag, Berlin, 2006.
28. B. S. MORDUKHOVICH AND Y. SHAO, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 348 (1996), pp. 1235-1280.
29. J. P. PENOT, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), pp. 855-871.
30. R. R. PHELPS AND S. SIMONS, Unbounded linear monotone operators on nonreflexive Banach spaces, J. Convex Anal., 5 (1998), pp. 303-328.
31. S. REICH AND S. SIMONS, Fenchel duality, Fitzpatrick functions and the Kirszbraun- Valentine extension theorem, Proc. Amer. Math. Soc., 133 (2005), pp. 2657-2660.
32. R. T. ROCKAFELLAR, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), pp. 209-216.
33. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
34. R. T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
35. R. T. ROCKAFELLAR AND R. J. B. WETS, Variational Analysis, Springer-Verlag, Berlin, 1998.
36. S. SIMONS, Minimax and Monotonicity, Lecture Notes in Math. 1693, Springer-Verlag, Berlin, 1998.
37. S. SIMONS, Dualized and scaled Fitzpatrick functions, Proc. Amer. Math. Soc., 134 (2006), pp. 2983-2987.
38. S. SIMONS, Positive sets and monotone sets, J. Convex Anal., 14 (2007), pp. 297-318.
39. S. SIMONS, LC-functions and maximal monotonicity, J. Nonlinear Convex Anal., 7 (2006), pp. 123-137.
40. S. SIMONS, The Fitzpatrick function and nonreflexive spaces, J. Convex Anal., 13 (2006), pp. 861-881.
41. S. SIMONS AND C. ZǍLINESCU, A new proof for Rockafellar's characterization of maximal monotone operators, Proc. Amer. Math. Soc., 132 (2004), pp. 2969-2972.
42. S. SIMONS AND C. ZǍLINESCU, Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal., 6 (2005), pp. 1-22.
43. C. ZǍLINESCU, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, 2002.