Abstract
Epi/hypo-convergence is introduced from a variational view-point. The known topological properties are reviewed and extended. Finally, it is shown that the (partial) Legendre-Fenchel transform is biocontinuous with respect to the topology induced by epi/hypoconvergence on the space of convex-concave bivariate functions.
Partially supported by a Guggenheim Fellowship.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Glowinski, R., Lions, J.L., and Trémolières, R., Analyse Numérique des Inéquations Variationnelles, Tome 1, Dunod.
De Giorgi, E., and Spagnolo, S., Sulla convergenza degli integrali dell’energia per operatori ellittici del 2e ordine, Boll. Un. Mat. Ital., (4) 8 (1973, 391–411.
Tartar, L., Cours Peccot, Mars 1979, Collège de France; and Murat, F., H-convergence, Seminaire Alger, 1977–78.
Fiacco, A., and Hutzler, W., Optimal value differential stability results for general inequality constrained differentiable mathematical programs, in Mathematical Programming with Data Perturbations, I, ed. A. Fiacco, Marcel Dekker, New-York, 1982, 29–44.
Zolezzi, T., On the stability analysis in mathematical programming, Manuscript, IMA-Genova, 1981.
De Giorgi, E., Convergence problems for functionals and operators, in Recent Methods in Nonlinear Analysis, Pitagora Editrice, Bologna, 1979, pp. 131–138.
Attouch, H., and Wets, R., A convergence theory for saddle functions, Trans. Amer. Math. Soc. (1983).
Cavazzuti, E., γ-convergence multiple, convergenza di punti di sella e di max-min, Boll. Un. Mat. Ital., to appear.
Cavazzuti, E., Alcune caraterizzazioni della γ-convergenza multipla, Manuscript, Università di Modena, 1981.
Sonntag, Y., Convergence au sens de U. Mosco: théorie et applications à l’approximation des solutions d’inéquations, Thèse, Univ. Provence, 1982.
De Giorgi, E., and Franzoni, T., Su un tipo di convergenza variazionale, Atti Acc. Naz. Lincei (8), 58 (1975), 842–850. See also: Fascicolo delle demostrazioni, Scuola Normale Superiore di Pisa, May 1976.
Rockafellar, R.T., A general correspondence between dual minimax problems and convex programs, Pacific J. Math., 25 (1968), 597–611.
Rockafellar, R.T., Monotone operators associated with saddle functions and minimax problems, in Nonlinear Functional Analysis, ed. F. Browder, Amer. Math Soc. Providence, 1970.
Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton, 1970.
McLinden, L., A minimax theorem, Math. Operations Res., (to appear).
McLinden, L., Dual operations on saddle functions, Trans. Amer. Math. Soc., 179 (1973), 363–381.
Ekeland, I., and Temam, R., Analyse Convexe et Problèmes Variationnels, Dunod, Paris, 1974.
Aubin, J.P., Mathematical methods of game and economic theory, North-Holland, Amsterdam, 1979.
Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Advances Math., 3 (1969), 273–299.
Attouch, H., Familles d’opérateurs maximaux monotones et mesurabilité, Annali di Matematica pura ed applicata (IV), CXX (1979), 35–111.
Wijsman, R., Convergence of sequences of convex sets, cones and functions, II, Trans. Amer. Math. Soc., 123 (1966), 32–45.
Joly, J.L., Une famille de topologies et de convergences sur l’ensemble des fonctionelles convexes, Thèse Grenoble, France, 1970.
Attouch, H., and Wets, R., Convergence of convex-concave saddle functions, to appear.
Attouch, H., Variational Convergence for Functions and Operators, Research Notes in Mathematics, Pittman, London, to appear.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Attouch, H., Wets, R.JB. (1983). A convergence for bivariate functions aimed at the convergence of saddle values. In: Cecconi, J.P., Zolezzi, T. (eds) Mathematical Theories of Optimization. Lecture Notes in Mathematics, vol 979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066247
Download citation
DOI: https://doi.org/10.1007/BFb0066247
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11999-9
Online ISBN: 978-3-540-39473-0
eBook Packages: Springer Book Archive