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Differentiability Properties of Optimal Value Functions

Published online by Cambridge University Press:  20 November 2018

Jean-Paul Penot
Jean-Paul Penot*
Affiliation:
Laboratoire de Mathématiques Appliquées, CNRS, ERS 2055, Faculté des Sciences, av. de l’Université, 64000 PAU, France e-mail: jean-paul.penot@univ-pau.fr
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Abstract

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Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.

Keywords

26B0565K1054C6090C2690C48differentiabilitygenericmarginalperformance functionsubdifferential

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 4 , 01 August 2004, pp. 825 - 842
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Asplund, E. and Rockafellar, R. T., Gradients of convex functions. Trans. Amer. Math. Soc. 139(1969), 443–467.Google Scholar
[2] Aubin, J.-P. and Ekeland, I., Applied Nonlinear Analysis, Wiley, New York, 1984.Google Scholar
[3] Bardi, M. and Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi- Bellman Equations, Birkhäuser, Basel, 1997.Google Scholar
[4] Baranger, J. and Témam, R., Nonconvex optimization problems depending on a parameter. SIAM J. Control 13(1975), 146–152.Google Scholar
[5] Bonnans, J. F. and Shapiro, A., Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.Google Scholar
[6] Bougeard, M., Penot, J.-P. and Pommellet, A., Towards minimal assumptions for the infimal convolution regularization. Approx. Theory 64(1991), 245–270.Google Scholar
[7] Bourbaki, N., Variétés différentielles et analytiques. Fascicule de résultats, Hermann, Paris, 1967.Google Scholar
[8] Boiso, M. Cepedello, Approximation of Lipschitz functions by Δ-convex functions in Banach spaces. Israel J. Math. 106(1998), 269–284.Google Scholar
[9] Boiso, M. Cepedello, On regularization in superreflexive Banach spaces by infimal convolution formulas. Studia Math. 129(1998), 265–284.Google Scholar
[10] Clarke, F. H., A new approach to Lagrange multipliers. Math. Oper. Research 1(1976), 97–102.Google Scholar
[11] Clarke, F. H., Optimization and Nonsmooth Analysis. Wiley, New York, 1983.Google Scholar
[12] Clarke, F. H., Ledyaev, Yu. S., Stern, R. J. and Wolenski, P. R., Nonsmooth Analysis and Optimal Control Theory. Springer-Verlag, New York, 1997.Google Scholar
[13] Ekeland, I., Nonconvex minimization problems. Bull. Amer. Math. Soc. 1(1979), 443–474.Google Scholar
[14] Ekeland, I. and Lebourg, G., Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces. Trans. Amer. Math. Soc. 224(1976), 193–216.Google Scholar
[15] Ekeland, I. and Témam, R., Analyse convexe et problèmes variationnels, Dunod et Gauthier-Villars, Paris, (1974), English translation North Holland, Amsterdam, 1976.Google Scholar
[16] Fitzpatrick, S.. Metric projection and the differentiability of the distance functions. Bull. Austral. Math. Soc. 22(1980), 291–312.Google Scholar
[17] Giles, J., Convex Analysis with Applications in Differentiation of Convex Functions. Pitman Research Lecture Notes in Mathematicss 58, Boston, 1982.Google Scholar
[18] Hiriart-Urruty, J.-B., Gradients généralisés de fonctions marginales. SIAM J. Control Optim. 16(1978), 301–316.Google Scholar
[19] Ioffe, A. D. and Tihomirov, V. M., Theory of extremal problems. (Russian: Nauka, Moscow, 1974), North Holland, Amsterdam, 1979.Google Scholar
[20] Ioffe, A. D. and Zaslavski, A. J., Variational principles and well-posedness in optimization and calculus of variations. SIAM J. Control Optim. 38(2000), 566–581.Google Scholar
[21] Jofre, A. and Penot, J.-P., A note on the directional derivative of a marginal function. Rev.Mat. Apl. 14(1993), 37–54.Google Scholar
[22] Laurent, P.-J., Approximation et optimisation. Hermann, Paris, 1972.Google Scholar
[23] Lebourg, G., Perturbed optimization problems in Banach spaces. Bull. Soc. Math. France Mém. 60(1979), 95–111.Google Scholar
[24] Lions, P.-L. and Souganidis, P. E., Differential games, optimal control and directional derivativess of viscosity solutions of Bellman's and Isaac's equations. SIAM J. Control Optim. 23(1985), 566–583.Google Scholar
[25] Loridan, P. and Morgan, J., New results on approximate solutions in two-level optimization. Optimization 20(1989), 819–836.Google Scholar
[26] Nijenhuis, A., Strong derivatives and inverse mappings. Amer. Math. Monthly 81(1974), 969–980.Google Scholar
[27] Penot, J.-P., A characterization of tangential regularity. Nonlinear Anal. 5(1981), 625–643.Google Scholar
[28] Penot, J.-P., On regularity conditions in mathematical programming. Math. Programming Stud. 19(1982), 167–199.Google Scholar
[29] Penot, J.-P., Central and peripheral results in the study of marginal and performance functions. In Mathematical programming with data perturbations, Fiacco, A. V., ed. Lectures notes in pure and applied math, 195. Dekker, New York (1998), 305–337.Google Scholar
[30] Penot, J.-P., Proximal mappings. J. Approx. Theory 94(1998), 203–221.Google Scholar
[31] Penot, J.-P., Genericity of well-posedness, perturbations and smooth variational principles. Set-Valued Anal. 9(2001), 131–157.Google Scholar
[32] Penot, J.-P., Calmness and stability properties of marginal and performance functions. Preprint, Univ. of Pau, 2001.Google Scholar
[33] Preiss, D., Differentiability of Lipschitz functions on Banach spaces. J. Funct. Anal. 91(1990), 312–345.Google Scholar
[34] Robinson, S. M., An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16(1991), 292–309.Google Scholar
[35] Rockafellar, R. T., Conjugate duality and optimization. CBMS-NSF Regional Conf. Series in Applied Math, SIAM, Philadelphia, 1974.Google Scholar
[36] Rockafellar, R. T., Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Programming Stud. 17(1982), 28–66.Google Scholar
[37] Rockafellar, R. T., Directional differentiability of the optimal value function in a nonlinear programming problem. Math. Programming Stud. 21(1984), 213–226.Google Scholar
[38] Strömberg, T., On regularization in Banach spaces. Ark. Mat. 34 (1996), 383–406.Google Scholar
[39] Thibault, L.. On subdifferentials of optimal value functions. SIAM J. Control Optim. 29(1991), 1019–1036.Google Scholar
[40] Duc Van, Tran, Hoang, Nguyen and Tsuji, Mikio, On Hopf 's formula for Lipschitz solutions of the Cauchy problem for Hamilton-Jacobi equations. Nonlinear Anal. 29(1997), 1145–1159.Google Scholar
[41] Valadier, M., Contribution à l’analyse convexe. Thèse d’Etat, Univ. Paris, 1970.Google Scholar
[42] Vinter, R., Optimal Control, Birkhäuser, Basel, 2000.Google Scholar
[43] Ye, J., Optimal strategies for bilevel dynamic problems. SIAM J. Control Optim. 35(1997), 512–531.Google Scholar
[44] Zalinescu, C., Convex Analysis in General Vector Spaces. World Scientific, Singapore, 2002.Google Scholar
[45] Zolezzi, T., Well-posedness criteria in optimization with application to calculus of variations. Nonlinear Anal. 25(1995), 437–453.Google Scholar
[46] Zolezzi, T., Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91(1996), 257–268.Google Scholar
[47] Zolezzi, T., Tikhonov regularization under epi-convergent perturbations. Ricerche Mat. 49(2000), 155–168.Google Scholar