Nonconvex differential calculus for infinitedimensional multifunctions
 Boris S. Mordukhovich,
 Yongheng Shao
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The paper is concerned with generalized differentiation of setvalued mappings between Banach spaces. Our basic object is the socalled coderivative of multifunctions that was introduced earlier by the first author and has had a number of useful applications to nonlinear analysis, optimization, and control. This coderivative is a nonconvexvalued mapping which is related to sequential limits of Fréchetlike graphical normals but is not dual to any tangentially generated derivative of multifunctions. Using a variational approach, we develop a full calculus for the coderivative in the framework of Asplund spaces. The latter class is sufficiently broad and convenient for many important applications. Some useful calculus results are also obtained in general Banach spaces.
 Aubin, J.P. Contingent derivatives of setvalued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. eds. (1981) Mathematical Analysis and Applications. Academic Press, New York, pp. 159229
 Aubin, J.P. (1984) Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9: pp. 87111
 Aubin, J.P., Frankowska, H. (1990) SetValued Analysis. Birkhäuser, Boston
 Borwein, J. M. (1987) EpiLipschitzlike sets in Banach spaces: theorems and examples. Nonlinear Anal. 11: pp. 12071217
 Borwein, J. M., Fitzpatrick, S. P. (1995) Weakstar sequential compactness and bornological limit derivatives. Convex Anal. 2: pp. 5968
 Borwein, J. M., Strojwas, H. M. (1985) Tangential approximations. Nonlinear Anal. 9: pp. 13471366
 Borwein, J. M., Zhuang, D. M. (1988) Vefiable necessary and sufficient conditions for regularity of setvalued and singlevalued maps. J. Math. Anal. Appl. 134: pp. 441459
 Clarke, F. H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York
 Dontchev, A. L. and Rockafellar, R. T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim., to appear.
 Ekeland, I., Lebourg, G. (1976) Generic Fréchet differentiability and perturbed optimization problems in Banach spaces. Trans. Amer. Math. Soc. 224: pp. 193216
 Fabian, M. (1989) Subdifferentiallity and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carolinae 30: pp. 5156
 Ginsburg, B., Ioffe, A. D. The maximum principle in optimal control of systems governed by semilinear equations. In: Mordukhovich, B. S., Sussmann, H. J. eds. (1996) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. SpringerVerlag, New York, pp. 81110
 Ioffe, A. D. (1984) Approximate subdifferentials and applications. I: The finite dimensional theory. Trans. Amer. Math. Soc. 281: pp. 389416
 Ioffe, A. D. (1989) Approximate subdifferential and applications III: The metric theory. Mathematika 36: pp. 138
 Ioffe, A. D. (1990) Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41: pp. 175192
 Jourani, A., Thibault, L. (1994) A note of Fréchet and approximate subdifferentials of composite functions. Bull. Austral. Math. Soc. 49: pp. 111116
 Jourani, A., Thibault, L. (1995) Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces. Trans. Amer. Math. Soc. 347: pp. 12551268
 Kruger, A. Y. (1985) Properties of generalized differentials. Siberian Math J. 26: pp. 822832
 Kruger, A. Y., Mordukhovich, B. S. (1980) Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR 24: pp. 684687
 Kruger, A. Y. and Mordukhovich, B. S.: Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming, Part I: Depon. VINITI No. 40880; part II: Depon. VINITI No. 49480, Moscow, 1980.
 Lang, S. (1993) Real and Functional Analysis. SpringerVerlag, New York
 Leach, E. B. (1961) A note on inverse function theorem. Proc. Amer. Math. Soc. 12: pp. 694697
 Loewen, P. D. Limits of Fréchet normals in nonsmooth analysis. In: Ioffe, A. D. eds. (1992) Optimization and Nonlinear Analysis. Longman, Harlow, Essex, pp. 178188
 Loewen, P. D. and Rockafellar, R. T.: New necesary conditions for the generalized problem of Bolza, SIAM J. Control Optim., to appear.
 Mordukhovich, B. S. (1976) Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40: pp. 960969
 Mordukhovich, B. S. (1980) Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22: pp. 526530
 Mordukhovich, B. S. (1988) Approximation Methods in Problems of Optimization and Control. Nauka, Moscow
 Mordukhovich, B. S. (1993) Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340: pp. 135
 Mordukhovich, B. S. (1994) Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Trans. Amer. Math. Soc. 343: pp. 609658
 Mordukhovich, B. S. (1994) Generalized differential calculus for nonsmooth and setvalued mappings. J. Math. Anal. Appl. 183: pp. 250288
 Mordukhovich, B. S. (1995) Discrete approximations and refined EulerLagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33: pp. 882915
 Mordukhovich, B. S., Shao, Y. (1995) Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces. Nonlinear Anal. 24: pp. 14011424
 Mordukhovich, B. S., Shao, Y. (1995) On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal. 2: pp. 211227
 Mordukhovich, B. S., Shao, Y. (1996) Extremal characterizations of Asplund spaces. Proc. Amer. Math. Soc. 124: pp. 197205
 Mordukhovich, B. S., Shao, Y. (1996) Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348: pp. 12351280
 Mordukhovich, B. S. and Shao, Y.: Stability of setvalued mappings in infinite dimensions: point criteria and applications, SIAM J. Control Optim., to appear (Preprint, November 1994).
 Mordukhovich, B. S. and Shao, Y.: Fuzzy calculus for coderivatives of multifunctions, Nonlinear Anal., to appear (Preprint, August 1995).
 Penot, J.P. (1989) Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13: pp. 629643
 Phelps, R. R. (1993) Convex Functions, Monotone Operators and differentiability. SpringerVerlag, Berlin
 Rockafellar, R. T. (1980) Generalzied directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32: pp. 257280
 Rockafellar, R. T. (1985) Lipschitzian properties of multifunctions. Nonlinear Anal. 9: pp. 867885
 Rockafellar, R. T. (1985) Maximal monotone relations and the second derivatives of nonsmooth functions. Ann. Inst. H. Poincaré Anal. Non Linéaire 2: pp. 167184
 Rockafellar, R. T. Protodifferentiability of setvalued mappings and its applications in optimization. In: Attouch, H. eds. (1989) Analyse non linéaire. GauthierVillars, Paris, pp. 449482
 Rockafellar, R. T. and Wets, R. J.B.: Variational Analysis, SpringerVerlag, New York, to appear.
 Thibault, L. (1980) Subdifferentials of compactly Lipschitzian vectorvalued functions. Ann. Mat. Pura Appl. 125: pp. 157192
 Thibault, L. (1991) On subdifferentials of optimal value functions. SIAM J. Control Optim. 29: pp. 10191036
 Title
 Nonconvex differential calculus for infinitedimensional multifunctions
 Journal

SetValued Analysis
Volume 4, Issue 3 , pp 205236
 Cover Date
 19960901
 DOI
 10.1007/BF00419366
 Print ISSN
 09276947
 Online ISSN
 1572932X
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 49J52
 58C06
 58C20
 coderivaties of multifunctions
 Frechet normals
 sequential limits
 Asplund spaces
 Authors

 Boris S. Mordukhovich ^{(1)}
 Yongheng Shao ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA