Minimal convex functions bounded below by the duality product
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- by J.-E. Martínez-Legaz and B. F. Svaiter PDF
- Proc. Amer. Math. Soc. 136 (2008), 873-878 Request permission
Abstract:
It is well known that the Fitzpatrick function of a maximal monotone operator is minimal in the class of convex functions bounded below by the duality product. Our main result establishes that, in the setting of reflexive Banach spaces, the converse also holds; that is, every such minimal function is the Fitzpatrick function of some maximal monotone operator. Whether this converse also holds in a nonreflexive Banach space remains an open problem.References
- R. S. Burachik and S. Fitzpatrick, On a family of convex functions associated to subdifferentials, J. Nonlinear Convex Anal. 6 (2005), no. 1, 165–171. MR 2138108
- Regina Sandra Burachik and B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002), no. 4, 297–316. MR 1934748, DOI 10.1023/A:1020639314056
- Regina Sandra Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2379–2383. MR 1974634, DOI 10.1090/S0002-9939-03-07053-9
- Simon Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65. MR 1009594
- J.-E. Martínez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13 (2005), no. 1, 21–46. MR 2128696, DOI 10.1007/s11228-004-4170-4
- Juan-Enrique Martinez-Legaz and Michel Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001), no. 2, 243–247. Special issue for Professor Ky Fan. MR 1848704
- Jean-Paul Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal. 58 (2004), no. 7-8, 855–871. MR 2086060, DOI 10.1016/j.na.2004.05.018
- Simeon Reich and Stephen Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2657–2660. MR 2146211, DOI 10.1090/S0002-9939-05-07983-9
- Julian P. Revalski and Michel Théra, Enlargements and sums of monotone operators, Nonlinear Anal. 48 (2002), no. 4, Ser. A: Theory Methods, 505–519. MR 1871464, DOI 10.1016/S0362-546X(00)00201-7
- S. Simons and C. Zălinescu, A new proof for Rockafellar’s characterization of maximal monotone operators, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2969–2972. MR 2063117, DOI 10.1090/S0002-9939-04-07462-3
- S. Simons and C. Zălinescu, Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005), no. 1, 1–22. MR 2138099
- B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3851–3859. MR 1999934, DOI 10.1090/S0002-9939-03-07083-7
- C. Zălinescu. A new proof of the maximal monotonicity of the sum using the Fitzpatrick function. In Variational analysis and applications, volume 79 of Nonconvex Optim. Appl., pages 1159–1172. Springer, New York, 2005.
Additional Information
- J.-E. Martínez-Legaz
- Affiliation: Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barce- lona, 08193 Bellaterra, Spain
- Email: JuanEnrique.Martinez@uab.es
- B. F. Svaiter
- Affiliation: Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorinha 110, Jardim Botânico, Rio de Janeiro, CEP 22460-320, Brazil
- MR Author ID: 304617
- Email: benar@impa.br
- Received by editor(s): June 20, 2006
- Published electronically: November 30, 2007
- Additional Notes: The first author was partially supported by the Ministerio de Ciencia y Tecnología, Project MTM2005-08572-C03-03. He also thanks the support of the Barcelona Economics Program of CREA
The second author was partially suported by CNPq grant n. 300755/2005-8 and Edital Universal 476842/03-2
This work was initiated during a visit of the second author to the Universitat Autònoma de Barcelona in March 2006 - Communicated by: Jonathan M. Borwein
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 873-878
- MSC (2000): Primary 47H05; Secondary 52A41, 26B25
- DOI: https://doi.org/10.1090/S0002-9939-07-09176-9
- MathSciNet review: 2361859