Abstract
We give estimates of the modulus of continuity of the Legendre-Fenchel Transform with respect to the Attouch-Wets uniformity. We point out the local Lipschitz behaviour of the conjugacy operation when endowing the set of closed proper convex functions defined on a normed vector space with suitable families of semimetrics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Attouch, H.:Variational Convergence for Functions and Operators, Appl. Math. Ser., 129, Pitman, Boston, London, 1984.
Attouch, H., Azé, D., and Beer, G.: On some inverse problems for the epigraphical sum,Nonlinear Anal., Theory Meth. Appl. 16 (1991), 241–254.
Attouch, H., Lucchetti R., and Wets, R. J.-B.: The topology of the ρ-Hausdorff distance,Ann. Mat. Pura Appl., IV. Ser. CLX (1991), 303–320.
Attouch, H. and Wets, R. J.-B.: Isometries for the Legendre-Fenchel transform,Trans. Amer. Math. Soc. 296 (1986), 33–60.
Attouch, H. and Wets, R. J.-B.: Lipschitzian stability of the ε-approximate solutions in convex optimization, IIASA working paper WP 87-25, Laxenburg, Austria, 1987.
Attouch, H. and Wets, R. J.-B.: in H. Attouch, J.-P. Aubin, F. Clarke, and I. Ekeland (eds.),Epigraphical analysis, analyse non linéaire, Gauthier-Villars, Paris, 1989, pp. 73–100.
Attouch, H. and Wets, R. J.-B.: Quantitative stability of variational systems: I. The epi graphical distance,Trans. Am. Math. Soc. (to appear).
Attouch, H. and Wets, R. J.-B.: Quantitative stability of variational systems: II. A framework for nonlinear conditioning,SIAM J. Control Optim. (to appear).
Aubin, J.-P.: Lipschitz behaviour of solutions to convex optimization problems,Math. Oper. Res. 9 (1984), 87–111.
Azé, D.: On some metric aspects of set convergence, Preprint, Université de Perpignan 1988.
Azé, D.: Convergences variationnelles et dualité. Applications en calcul des variations et en programmation mathématique, Thèse d'État, Université de Perpignan, 1986.
Azé, D.: Caractérisation de la convergence au sens de Mosco en termes d'approximations infconvolutives,Ann. Fac. Sci. Toulouse 8 (1986–1987), 293–314.
Azé, D. and Penot, J.-P.: Operations on convergent families of sets and functions,Optimization 21 (1990), 521–534.
Azé, D. and Penot, J.-P.: Recent quantitative results about the convergence of convex sets and functions, in P.-L. Papini (ed.),Functional Analysis and Approximation Pitagora, Bologna, 1989, pp. 90–110.
Azé, D. and Penot, J.-P.: Uniformly convex functions and uniformly smooth convex functions, preprint, University of Perpignan, 1987.
Azé, D. and Rahmouni, A.: Intrinsic bounds for Kuhn-Tucker points of perturbed convex programs, preprint, University of Perpignan, 1992.
Back, K.: Convergence of Lagrange multipliers and dual variables for convex optimization problems,Math. Oper. Res. 13 (1990), 74–79.
Beer, G.: Conjugate convex functions and the epi-distance topology,Proc. Amer. Math. Soc. 108 (1990), 117–126.
Beer, G. and Lucchetti, R.: The epi-distance topology: continuity and stability results with applications to convex optimization problems,Math. Oper. Res. 17 (1992), 715–726.
Beer, G. and Lucchetti, R.: Convex optimization and the epi-distance topology,Trans. Amer. Math. Soc. 327 (1991), 795–813.
Beer, G. and Théra, M.: Attouch-Wets convergence and a differential operator for convex functions, to appear,Proc. Amer. Math. Soc.
Borwein, J. M.: A note on ε-subgradients and maximal monotonicity,Pacific J. Math. 103 (1982), 307–314.
Ekeland, I. and Temam, R.:Analyse convexe et problèmes variationnels, Gauthier-Villars, Paris, 1974.
Fougères, A.: Coercivité des intégrandes convexes normales. Applications à la minimisation des fonctionnelles intégrales et du calcul des variations, Semin. Anal. Convexe, Univ. Sci. Tech. Languedoc, exp. nO 19, 1976.
Hiriart-Urruty, J.-B.: Lipschitz r-continuity of the approximate subdifferential of a convex function,Math. Scand. 47 (1980), 123–134.
Hiriart-Urruty, J.-B.: Extension of Lipschitz functions,J. Math. Anal. Appl. 77 (1980), 539–554.
Hiriart-Urruty, J.-B.: A general formula on the conjugate of the difference of functions,Canad. Math. Bull. 29 (1986), 482–485.
Hörmander, L.: Sur la fonction d'appui des ensembles convexes dans un espace localement convexe,Ark. Mat. 3 (1954), 181–186.
Joly, J.-L.: Une famille de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarité est bicontinue,J. Math. Pures Appl., IX. Sér. 52 (1973), 421–441.
Mc Shane, E. J.: Extension of range of functions,Bull. Amer. Math. Soc. 40 (1934), 837–842.
Moreau, J.-J.: Fonctionnelles convexes, Séminaire sur les équations aux dérivées partielles, Lecture Notes, Collège de France, Paris, 1966.
Penot, J.-P.: Preservation of persistence and stability under intersections and operations,J. Optim. Theory Appl. (to appear).
Penot, J.-P.: Topologies and convergences on the set of convex functions,Nonlinear Anal. Theory Meth. Appl. 18 (10) (1992), 905–916.
Penot, J.-P.: The cosmic Hausdorff topology; the bounded Hausdorff topology and continuity of polarities,Proc. Amer. Math. Soc. 113 (1991), 275–285.
Radström, H.: An imbedding theorem for spaces of convex sets,Proc. Amer. Math. Soc. 3 (1952), 165–169.
Rockafellar, R. T.:Conjugate Duality and Optimization, SIAM Regional Conference Series in Applied Mathematics 16, 1974.
Volle, M.: private communication.
Walkup, D. W. and Wets, R. J.-B.: Continuity of some convex-cone-valued mappings,Proc. Amer. Math. Soc. 18 (1967), 229–235.
Wets, R. J.-B.: Convergence of convex functions, variational inequalities and convex optimization problems, in P. Cottle, F. Giannessie and J.-L. Lions (eds.),Variational Inequalities and Complementarity Problems, Wiley, New York, 1980, pp. 373–403.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Azé, D., Rahmouni, A. Lipschitz behaviour of the Legendre-Fenchel Transform. Set-Valued Anal 2, 35–48 (1994). https://doi.org/10.1007/BF01027091
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01027091