Abstract
A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birkhoff, G.: Lattice Theory, Corrected reprint of the 1967 third edition, Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc., Providence, RI, 1979.
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, Notas de Matemática (50), North-Holland, Amsterdam, American Elsevier, New York, 1973.
Brézis, H.: Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983.
Burachik, R. S., Iusem, A. N. and Svaiter, B. F.: Enlargement of monotone operators with applications to variational inequalities, Set-Valued Anal. 5 (1997), 159–180.
Burachik, R. S. and Svaiter, B. F.: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002), 297–316.
Fitzpatrick, S.: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization, Canberra, Proc. Centre Math. Anal. Austral. Nat. Univ. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65.
Mahey, P. and Tao, PhD: Partial regularization of the sum of two maximal monotone operators, RAIRO Modél. Math. Anal. Numér. 27 (1993), 375–392.
Martínez-Legaz, J.-E. and Théra, M.: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001), 243–247.
Moudafi, A.: On the regularization of the sum of two maximal monotone operators, Nonlinear Anal., Ser. A: Theory Methods 42 (2000), 1203–1208.
Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, Berlin, 1989.
Revalski, J. P. and Théra, M.: Variational and extended sums of monotone operators, in: Ill-posed Variational Problems and Regularization Techniques (Trier, 1998), Springer, Berlin, 1998, pp. 229–246.
Revalski, J. P. and Théra, M.: Generalized sums of monotone operators, C.R. Acad. Sci. Paris Sér. I Math. 329 (1999), 979–984.
Rockafellar, R. T.: Convex Analysis, Princeton Math. Ser. 28, Princeton University Press, Princeton, NJ, 1970.
Rockafellar, R. T.: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216.
Simons, S.: Minimax and Monotonicity, Lecture Notes in Mathematics 1693, Springer-Verlag, Berlin, 1998.
Svaiter, B. F.: A family of enlargements of maximal monotone operators, Set-Valued Anal. 8(4) (2000), 311–328.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classifications (2000)
47H05, 46B99, 47H17.
Rights and permissions
About this article
Cite this article
MartÍnez-Legaz, J.E., Svaiter, B.F. Monotone Operators Representable by l.s.c. Convex Functions. Set-Valued Anal 13, 21–46 (2005). https://doi.org/10.1007/s11228-004-4170-4
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11228-004-4170-4