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Autoconjugate functions and representations of monotone operators

Published online by Cambridge University Press:  17 April 2009

Jean-Paul Penot
Jean-Paul Penot
Affiliation:
Laboratoire de Mathématiques Appliquées, CNRS ERS 2055, Faculté des Sciences, Université de Pau, BP 1155, 64013 PAU Cedex, France
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Abstract

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We show the existence of a convex representation of a maximal monotone operator by a convex function which is invariant with respect to the Fenchel conjugacy (up to an interchange of variables). We use the framework of generalized convexity.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 2 , April 2003 , pp. 277 - 284
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Balder, E.J., ‘An extension of duality-stability relations to nonconvex problems’, SIAM J. Control Optim. 15 (1977), 329–343.CrossRefGoogle Scholar
[2]Burachik, R.S. and Svaiter, B.F., ‘Maximal monotonicity, conjugation and the duality product’, (preprint, IMPA Rio, 10 2001).Google Scholar
[3]Burachik, R.S. and Svaiter, B.F., ‘Maximal monotone operators, convex functions and a special family of enlargements’, Set-Valued Anal. (2002) (to appear).Google Scholar
[4]Fitzpatrick, S., ‘Representing monotone operators by convex functions’,in Functional Analysis and Optimization, (workshop and miniconference,Canberra, Australia1988), Proc. Center Math. Anal. Austral. Nat. Univ. 20 (Australian National University, Canberra ACT, 1988), pp. 59–65.Google Scholar
[5]Ioffe, A.D., ‘Abstract convexity and non-smooth analysis’, in Advances in Mathematical Economics 3 (Springer-Verlag, Tokyo, 2001), pp. 45–61.CrossRefGoogle Scholar
[6]Janin, R., ‘Sur la dualité en programmation dynamique’, C.R. Acad. Sci. Paris A 277 (1973), 1195–1197.Google Scholar
[7]Krauss, E., ‘On the maximality of the sum of monotone operators’, Math. Nachr. 101 (1981), 199–206.CrossRefGoogle Scholar
[8]Krauss, E., ‘A representation of maximal monotone operators by saddle functions’, Rev. Roumaine Math. Pures Appl. 30 (1985), 823–836.Google Scholar
[9]Krauss, E., ‘A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions’, Nonlinear Anal. 9 (1985), 1381–1399.CrossRefGoogle Scholar
[10]Krauss, E., ‘Maximal monotone operators and saddle functions. I’, Z. Anal. Anwendungen 5 (1986), 333–346.CrossRefGoogle Scholar
[11]Martínez-Legaz, J.-E., ‘On lower subdifferentiable functions’, in Trends in Mathematical Optimization, Internat. Series Numer. Math. 84 (Birkhauser, Basel, 1988), pp. 197–232.CrossRefGoogle Scholar
[12]Martínez-Legaz, J.-E. and Singer, I., ‘Subdifferentials with respect to dualities’, ZOR-Math. Methods Oper. Res. 42 (1995), 109–125.CrossRefGoogle Scholar
[13]Martínez-Legaz, J.E. and Théra, M., ‘A convex representation of maximal monotone operators’, J. Nonlinear Convex Anal. 2 (2001), 243–247.Google Scholar
[14]Moreau, J.-J., ‘Inf-convolution, sous-additivité, convexité des functions numériques’, J. Math. Pures Appl. 49 (1970), 109–154.Google Scholar
[15]Pallaschke, D. and Rolewicz, S., Foundations of mathematical optimization, Mathematics and its Applications 83 (Kluwer, Dordrecht, 1998).Google Scholar
[16]Penot, J.-P., ‘What is quasiconvex analysis?’, Optimization 47 (2000), 35–110.CrossRefGoogle Scholar
[17]Penot, J.-P., ‘A Lagrangian approach to quasiconvex analysis’, (preprint, Univ. of Pau 1998), J. Optim. Theory Appl. 117 (2003) (to appear).CrossRefGoogle Scholar
[18]Penot, J.-P., ‘Augmented Lagrangians, duality and growth conditions’, J. Nonlinear Convex Anal. (to appear).Google Scholar
[19]Penot, J.-P., ‘Rotundity, smoothness and duality’, (preprint, Univ. of Pau, 10 2001).Google Scholar
[20]Penot, J.-P., ‘The relevance of convex analysis for the study of monotonicity’, (preprint, Univ. of Pau, 04 2002).Google Scholar
[21]Penot, J.-P., and Volle, M., ‘Another duality scheme for quasiconvex problems’, in Trends in Mathematical Optimization, Internat. Series Number. Math. 84 (Birkhauser, Basel, 1988), pp. 259–275.CrossRefGoogle Scholar
[22]Penot, J.-P., and Volle, M., ‘On quasi-convex duality’, Math. Oper. Res. 15 (1990), 597–625.CrossRefGoogle Scholar
[23]Penot, J.-P., and Volle, M., On strongly convex and paraconvex dualities, in Generalized convexity and fractional programming with economic applications, (Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P., Schaible, S., Editors), Lecture notes in Econom. and Math. Systems 345, (Proceedings, Pisa, Italy 1988) (Springer-Verlag, Berlin, 198–218).CrossRefGoogle Scholar
[24]Penot, J.-P., and Volle, M., ‘Surrogate programming and multipliers in quasiconvex programming’, (preprint, Univ. of Avignon, 1997).Google Scholar
[25]Rockafellar, R.T., ‘Augmented Lagrange multiplier functions and duality in nonconvex programming’, SIAM J. Control Optim. 12 (1974), 268–285.CrossRefGoogle Scholar
[26]Rockafellar, R.T. and Wets, R.J.-B., Variational analysis (Springer-Verlag, Berlin, 1998).CrossRefGoogle Scholar
[27]Rubinov, A.M., Abstract convexity and global optimization (Kluwer, Dordrecht, 2000).CrossRefGoogle Scholar
[28]Simons, S., Minimax and monotonicity, Lecture Notes in Mathematics 693 (Springer-Verlag, Berlin, 1998).CrossRefGoogle Scholar
[29]Singer, I., Abstract convex analysis (J. Wiley and Sons, New York, 1997).Google Scholar