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Convex subcones of the contingent cone in nonsmooth calculus and optimization. (English) Zbl 0629.58007

This paper is on convex subcones of the contingent cone in Banach spaces. It is shown that although no such subcone has the isotonicity property, there are convex subcones which are better approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is similarly strong. However, the hypotheses under which calculus rules and optimality conditions for nonsmooth mathematical programming are proven involve the Clarke tangent cone in an essential way.
Reviewer: A.Kriegl

MSC:

58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
90C30 Nonlinear programming
46G05 Derivatives of functions in infinite-dimensional spaces
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[1] Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. · Zbl 0641.47066
[2] J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9 – 52. · Zbl 0557.49020 · doi:10.1007/BF00938588
[3] Jonathan Michael Borwein, Epi-Lipschitz-like sets in Banach space: theorems and examples, Nonlinear Anal. 11 (1987), no. 10, 1207 – 1217. · Zbl 0639.49014 · doi:10.1016/0362-546X(87)90008-3
[4] J. M. Borwein and H. M. Strojwas, The hypertangent cone study (submitted). · Zbl 0697.49014
[5] Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. · Zbl 0582.49001
[6] Szymon Dolecki, Tangency and differentiation: some applications of convergence theory, Ann. Mat. Pura Appl. (4) 130 (1982), 223 – 255. · Zbl 0518.49009 · doi:10.1007/BF01761497
[7] Halina Frankowska, Inclusions adjointes associées aux trajectoires minimales d’inclusions différentielles, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 8, 461 – 464 (French, with English summary). · Zbl 0532.49024
[8] Halina Frankowska, Necessary conditions for the Bolza problem, Math. Oper. Res. 10 (1985), no. 2, 361 – 366. · Zbl 0589.49011 · doi:10.1287/moor.10.2.361
[9] E. Giner, Ensembles et fonctions étoilés. Application au calcul différentiel généralisé, Univ. Toulouse III, 1981, Manuscript.
[10] Monique Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space, SIAM J. Control 7 (1969), 232 – 241. · Zbl 0182.53101
[11] Jean-Baptiste Hiriart-Urruty, Contributions à la programmation mathématique: cas déterministe et stochastique, Université de Clermont-Ferrand II, Clermont-Ferrand, 1977 (French). Thèse présentée à l’Université de Clermont-Ferrand II pour obtenir le grade de Docteur ès Sciences Mathématiques; Série E, No. 247. · Zbl 0316.90052
[12] J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res. 4 (1979), no. 1, 79 – 97. · Zbl 0409.90086 · doi:10.1287/moor.4.1.79
[13] Alexander D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61 – 69. · Zbl 0427.58008
[14] A. D. Ioffe, Necessary and sufficient conditions for a local minimum. I. A reduction theorem and first order conditions, SIAM J. Control Optim. 17 (1979), no. 2, 245 – 250. , https://doi.org/10.1137/0317019 A. D. Ioffe, Necessary and sufficient conditions for a local minimum. II. Conditions of Levitin-Miljutin-Osmolovskiĭ type, SIAM J. Control Optim. 17 (1979), no. 2, 251 – 265. , https://doi.org/10.1137/0317020 A. D. Ioffe, Necessary and sufficient conditions for a local minimum. III. Second order conditions and augmented duality, SIAM J. Control Optim. 17 (1979), no. 2, 266 – 288. · Zbl 0417.49029 · doi:10.1137/0317021
[15] A. D. Ioffe, Approximate subdifferentials and applications. I. The finite-dimensional theory, Trans. Amer. Math. Soc. 281 (1984), no. 1, 389 – 416. · Zbl 0531.49014
[16] V. Jeyakumar, On optimality conditions in nonsmooth inequality constrained minimization, Univ. of Melbourne, Research Report No. 13, 1985. · Zbl 0611.90081
[17] A. G. Kusraev and G. G. Kutateladze, Local convex analysis, J. Soviet Math. 26 (1984), 2048-2087. · Zbl 0543.46025
[18] D. H. Martin, R. J. Gardner, and G. G. Watkins, Indicating cones and the intersection principle for tangential approximants in abstract multiplier rules, J. Optim. Theory Appl. 33 (1981), no. 4, 515 – 537. · Zbl 0427.49025 · doi:10.1007/BF00935756
[19] D. H. Martin and G. G. Watkins, Cores of tangent cones and Clarke’s tangent cone, Math. Oper. Res. 10 (1985), no. 4, 565 – 575. · Zbl 0584.49008 · doi:10.1287/moor.10.4.565
[20] Philippe Michel and Jean-Paul Penot, Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 12, 269 – 272 (French, with English summary). · Zbl 0567.49008
[21] Jean-Paul Penot, Calcul sous-différentiel et optimisation, J. Funct. Anal. 27 (1978), no. 2, 248 – 276 (French). · Zbl 0404.90078 · doi:10.1016/0022-1236(78)90030-7
[22] D. Pallaschke , Nondifferentiable optimization: motivations and applications, Lecture Notes in Economics and Mathematical Systems, vol. 255, Springer-Verlag, Berlin, 1985. · Zbl 0572.00008
[23] B.N. Pshenichnyi and R. A. Khachatryan, Constraints of equality tupe in nonsmooth optimization problems, Soviet Math. Dokl. 26 (1982), 659-662. · Zbl 0527.49016
[24] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0193.18401
[25] R. T. Rockafellar, Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3) 39 (1979), no. 2, 331 – 355. · Zbl 0413.49015 · doi:10.1112/plms/s3-39.2.331
[26] Ralph T. Rockafellar, The theory of subgradients and its applications to problems of optimization, R & E, vol. 1, Heldermann Verlag, Berlin, 1981. Convex and nonconvex functions. · Zbl 0462.90052
[27] R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming Stud. 17 (1982), 28 – 66. Nondifferential and variational techniques in optimization (Lexington, Ky., 1980). · Zbl 0478.90060
[28] R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), no. 7, 665 – 698. · Zbl 0593.49013 · doi:10.1016/0362-546X(85)90012-4
[29] J. S. Treiman, Shrinking the set of generalized gradients (submitted). · Zbl 0674.49016
[30] Corneliu Ursescu, Tangent sets’ calculus and necessary conditions for extremality, SIAM J. Control Optim. 20 (1982), no. 4, 563 – 574. · Zbl 0488.49009 · doi:10.1137/0320041
[31] M. Vlach, Approximation operators in optimization theory, Z. Oper. Res. Ser. A-B 25 (1981), no. 1, A15 – A23 (English, with German summary). · Zbl 0448.90061
[32] D. E. Ward, Tangent cones, generalized subdifferential calculus, and optimization, Thesis, Dalhousie Univ. 1984.
[33] D. E. Ward and J. M. Borwein, Nonsmooth calculus in finite dimensions, SIAM J. Control Optim. 25 (1987), no. 5, 1312 – 1340. · Zbl 0633.46043 · doi:10.1137/0325072
[34] D. Ward, Isotone tangent cones and nonsmooth optimization, Optimization 18 (1987), no. 6, 769 – 783. · Zbl 0633.49012 · doi:10.1080/02331938708843290
[35] G. G. Watkins, Nonsmooth Milyutin-Dubovitskiĭ theory and Clarke’s tangent cone, Math. Oper. Res. 11 (1986), no. 1, 70 – 80. · Zbl 0596.49013 · doi:10.1287/moor.11.1.70
[36] Constantin Zălinescu, On convex sets in general position, Linear Algebra Appl. 64 (1985), 191 – 198. · Zbl 0581.52001 · doi:10.1016/0024-3795(85)90276-9
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