Tangent cones and quasi-interiorly tangent cones to multifunctions
HTML articles powered by AMS MathViewer
- by Lionel Thibault PDF
- Trans. Amer. Math. Soc. 277 (1983), 601-621 Request permission
Abstract:
R. T. Rockafellar has proved a number of rules of subdifferential calculus for nonlocally lipschitzian real-valued functions by investigating the Clarke tangent cones to the epigraphs of such functions. Following these lines we study in this paper the tangent cones to the sum and the composition of two multifunctions. This will be made possible thanks to the notion of quasi-interiorly tangent cone which has been introduced by the author for vector-valued functions in [29] and whose properties in the context of multifunctions are studied. The results are strong enough to cover the cases of real-valued or vector-valued functions.References
- Jean-Pierre Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 159–229. MR 634239
- Claude Berge, Espaces topologiques: Fonctions multivoques, Collection Universitaire de Mathématiques, Vol. III, Dunod, Paris, 1959 (French). MR 0105663
- J. M. Borwein, A Lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand. 48 (1981), no. 2, 189–204. MR 631335, DOI 10.7146/math.scand.a-11911 —, Multivalued convexity; a unified approach to equality and inequality constraints, Math. Programming 13 (1977), 163-180.
- Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. MR 367131, DOI 10.1090/S0002-9947-1975-0367131-6
- Frank H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), no. 2, 165–174. MR 414104, DOI 10.1287/moor.1.2.165
- Szymon Dolecki, Tangency and differentiation: some applications of convergence theory, Ann. Mat. Pura Appl. (4) 130 (1982), 223–255. MR 663973, DOI 10.1007/BF01761497
- Phạm Canh Du’o’ng and Hoàng Tụy, Stability, surjectivity and local invertibility of nondifferentiable mappings, Acta Math. Vietnam. 3 (1978), no. 1, 89–105. MR 527479 S. Gautier, Différentiabilité des multiapplications, Publ. Math. de Pau 5 (1978), 1-17.
- J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res. 4 (1979), no. 1, 79–97. MR 543611, DOI 10.1287/moor.4.1.79
- Jean-Baptiste Hiriart-Urruty and Lionel Thibault, Existence et caractérisation de différentielles généralisées d’applications localement lipschitziennes d’un espace de Banach séparable dans un espace de Banach réflexif séparable, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 23, 1091–1094 (French, with English summary). MR 587863
- A. D. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), no. 1, 1–56. MR 613784, DOI 10.1090/S0002-9947-1981-0613784-7
- John L. Kelley, General topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. MR 0370454
- Ştefan Mirică, The contingent and the paratingent as generalized derivatives for vector-valued and set-valued mappings, Nonlinear Anal. 6 (1982), no. 12, 1335–1368. MR 684969, DOI 10.1016/0362-546X(82)90108-0
- Jean-Paul Penot, Differentiability of relations and differential stability of perturbed optimization problems, SIAM J. Control Optim. 22 (1984), no. 4, 529–551. MR 747968, DOI 10.1137/0322033
- Jean-Paul Penot, On regularity conditions in mathematical programming, Math. Programming Stud. 19 (1982), 167–199. Optimality and stability in mathematical programming. MR 669731, DOI 10.1007/bfb0120988
- Jean-Paul Penot and Michel Théra, Polarité des applications convexes à valeurs vectorielles, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 7, A419–A422 (French, with English summary). MR 552067
- Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
- Stephen M. Robinson, Normed convex processes, Trans. Amer. Math. Soc. 174 (1972), 127–140. MR 313769, DOI 10.1090/S0002-9947-1972-0313769-9
- Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130–143. MR 430181, DOI 10.1287/moor.1.2.130 R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N.J., 1970.
- R. T. Rockafellar, Clarke’s tangent cones and the boundaries of closed sets in $\textbf {R}^{n}$, Nonlinear Anal. 3 (1979), no. 1, 145–154 (1978). MR 520481, DOI 10.1016/0362-546X(79)90044-0
- R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canadian J. Math. 32 (1980), no. 2, 257–280. MR 571922, DOI 10.4153/CJM-1980-020-7
- R. T. Rockafellar, Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3) 39 (1979), no. 2, 331–355. MR 548983, DOI 10.1112/plms/s3-39.2.331
- Lionel Thibault, Problème de Bolza dans un espace de Banach séparable, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 22, Aii, A1303–A1306. MR 435980
- Lionel Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl. (4) 125 (1980), 157–192. MR 605208, DOI 10.1007/BF01789411
- L. Thibault, Mathematical programming and optimal control problems defined by compactly Lipschitzian mappings, Travaux Sém. Anal. Convexe 8 (1978), no. 2, exp. no. 10, 58. MR 521153
- Lionel Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal. 6 (1982), no. 10, 1037–1053. MR 678055, DOI 10.1016/0362-546X(82)90074-8
- Lionel Thibault, Subdifferentials of nonconvex vector-valued functions, J. Math. Anal. Appl. 86 (1982), no. 2, 319–344. MR 652180, DOI 10.1016/0022-247X(82)90226-8
- Lionel Thibault, Épidifférentiels de fonctions vectorielles, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 2, A87–A90 (French, with English summary). MR 563945 —, Lagrange multipliers for nonconvex multifunctions (to appear).
- Corneliu Ursescu, Multifunctions with convex closed graph, Czechoslovak Math. J. 25(100) (1975), no. 3, 438–441. MR 388032
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 601-621
- MSC: Primary 58C20; Secondary 26E25, 90C30
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694379-8
- MathSciNet review: 694379