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Nonconvex differential calculus for infinite-dimensional multifunctions. (English) Zbl 0866.49024
The authors start from a notion of normal cone built by means of Fréchet $\epsilon$-normals and a procedure of sequential upper limit. Then they build a related notion of coderivative for a multifunction $F$ from a Banach space $X$ to another Banach space $Y$. When $X$ and $Y$ are both Asplund spaces, such a notion turns out to have a rich calculus. Their approach should be compared with that of A. D. Ioffe [Mathematika 36, No. 1, 1-38 (1989; Zbl 0713.49022)], which is otherwise based on topological upper limits. In the particular case of an extended real-valued function $\varphi :X\to \overline{ℝ}$, corresponding notions have been introduced by the same authors in J. Convex Anal. 2, No. 1-2, 211-227 (1995; Zbl 0838.49013).
MSC:
 49J52 Nonsmooth analysis (other weak concepts of optimality) 58C06 Set-valued and function-space valued mappings on manifolds 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
References:
 [1] AubinJ.-P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, in: L.Nachbin (ed.), Mathematical Analysis and Applications, Academic Press, New York, 1981, pp. 159-229. [2] AubinJ.-P.: Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87-111. · Zbl 0539.90085 · doi:10.1287/moor.9.1.87 [3] AubinJ.-P. and FrankowskaH.: Set-Valued Analysis, Birkhäuser, Boston, 1990. [4] BorweinJ. M.: Epi-Lipschitz-like sets in Banach spaces: theorems and examples, Nonlinear Anal. 11 (1987), 1207-1217. · Zbl 0639.49014 · doi:10.1016/0362-546X(87)90008-3 [5] BorweinJ. M., and FitzpatrickS. P.: Weak-star sequential compactness and bornological limit derivatives, Convex Anal. 2 (1995), 59-68. [6] BorweinJ. M. and StrojwasH. M.: Tangential approximations, Nonlinear Anal. 9 (1985), 1347-1366. · Zbl 0613.49016 · doi:10.1016/0362-546X(85)90095-1 [7] BorweinJ. M. and ZhuangD. M.: Vefiable necessary and sufficient conditions for regularity of set-valued and single-valued maps. J. Math. Anal. Appl. 134 (1988), 441-459. · Zbl 0654.49004 · doi:10.1016/0022-247X(88)90034-0 [8] ClarkeF. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [9] Dontchev, A. L. and Rockafellar, R. T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim., to appear. [10] EkelandI. and LebourgG.: Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193-216. [11] FabianM.: Subdifferentiallity and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolinae 30 (1989), 51-56. [12] GinsburgB. and IoffeA. D.: The maximum principle in optimal control of systems governed by semilinear equations, in: B. S.Mordukhovich and H. J.Sussmann (eds), Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, IMA Volumes in Mathematics and its Applications 78, Springer-Verlag, New York, 1996, pp. 81-110. [13] IoffeA. D.: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984) 389-416. [14] IoffeA. D.: Approximate subdifferential and applications III: The metric theory, Mathematika 36 (1989), 1-38. · Zbl 0713.49022 · doi:10.1112/S0025579300013541 [15] IoffeA. D.: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175-192. · Zbl 0725.46045 · doi:10.1112/jlms/s2-41.1.175 [16] JouraniA. and ThibaultL.: A note of Fréchet and approximate subdifferentials of composite functions, Bull. Austral. Math. Soc. 49 (1994), 111-116. · Zbl 0806.49014 · doi:10.1017/S0004972700016142 [17] JouraniA. and ThibaultL.: Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces, Trans. Amer. Math. Soc. 347, (1995), 1255-1268. · Zbl 0827.54013 · doi:10.2307/2154809 [18] KrugerA. Y.: Properties of generalized differentials, Siberian Math J. 26 (1985), 822-832. · Zbl 0596.46038 · doi:10.1007/BF00969103 [19] KrugerA. Y. and MordukhovichB. S.: Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684-687. [20] Kruger, A. Y. and Mordukhovich, B. S.: Generalized normals and derivatives, and necessary optimality conditions in nondifferentiable programming, Part I: Depon. VINITI No. 408-80; part II: Depon. VINITI No. 494-80, Moscow, 1980. [21] LangS.: Real and Functional Analysis, 3rd edn, Springer-Verlag, New York, 1993. [22] LeachE. B.. A note on inverse function theorem, Proc. Amer. Math. Soc. 12 (1961), 694-697. · doi:10.1090/S0002-9939-1961-0126146-9 [23] LoewenP. D.: Limits of Fréchet normals in nonsmooth analysis, in: A. D.Ioffe et al. (eds), Optimization and Nonlinear Analysis, Pitman Research Notes in Math. Series No. 244, Longman, Harlow, Essex, 1992, pp. 178-188. [24] Loewen, P. D. and Rockafellar, R. T.: New necesary conditions for the generalized problem of Bolza, SIAM J. Control Optim., to appear. [25] MordukhovichB. S.: Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40, (1976), 960-969. · Zbl 0362.49017 · doi:10.1016/0021-8928(76)90136-2 [26] MordukhovichB. S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980) 526-530. [27] MordukhovichB. S.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988. [28] MordukhovichB. S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340 (1993), 1-35. · Zbl 0791.49018 · doi:10.2307/2154544 [29] MordukhovichB. S.: Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc. 343 (1994), 609-658. · Zbl 0826.49008 · doi:10.2307/2154734 [30] MordukhovichB. S.: Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250-288. · Zbl 0807.49016 · doi:10.1006/jmaa.1994.1144 [31] MordukhovichB. S.: Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM J. Control Optim. 33 (1995), 882-915. · Zbl 0844.49017 · doi:10.1137/S0363012993245665 [32] MordukhovichB. S. and ShaoY.: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Anal. 24 (1995), 1401-1424. · Zbl 0863.47030 · doi:10.1016/0362-546X(94)00256-H [33] MordukhovichB. S. and ShaoY.: On nonconvex subdifferential calculus in Banach spaces, J. Convex Anal. 2 (1995), 211-227. [34] MordukhovichB. S. and ShaoY.: Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1996), 197-205. · Zbl 0849.46010 · doi:10.1090/S0002-9939-96-03049-3 [35] MordukhovichB. S. and ShaoY.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235-1280. · Zbl 0881.49009 · doi:10.1090/S0002-9947-96-01543-7 [36] Mordukhovich, B. S. and Shao, Y.: Stability of set-valued mappings in infinite dimensions: point criteria and applications, SIAM J. Control Optim., to appear (Preprint, November 1994). [37] Mordukhovich, B. S. and Shao, Y.: Fuzzy calculus for coderivatives of multifunctions, Nonlinear Anal., to appear (Preprint, August 1995). [38] PenotJ.-P.: Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), 629-643. · Zbl 0687.54015 · doi:10.1016/0362-546X(89)90083-7 [39] PhelpsR. R.: Convex Functions, Monotone Operators and differentiability, 2nd edn, Springer-Verlag, Berlin, 1993. [40] RockafellarR. T.: Generalzied directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32 (1980), 257-280. · Zbl 0447.49009 · doi:10.4153/CJM-1980-020-7 [41] RockafellarR. T.: Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985), 867-885. · Zbl 0573.54011 · doi:10.1016/0362-546X(85)90024-0 [42] RockafellarR. T.: Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 167-184. [43] RockafellarR. T. Proto-differentiability of set-valued mappings and its applications in optimization, in: H.Attouch et al. (eds.), Analyse non linéaire, Gauthier-Villars, Paris, 1989, pp. 449-482. [44] Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer-Verlag, New York, to appear. [45] ThibaultL.: Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl. 125 (1980), 157-192. · Zbl 0486.46037 · doi:10.1007/BF01789411 [46] ThibaultL.: On subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), 1019-1036. · Zbl 0734.90093 · doi:10.1137/0329056