Convex subcones of the contingent cone in nonsmooth calculus and optimization
HTML articles powered by AMS MathViewer
- by Doug Ward PDF
- Trans. Amer. Math. Soc. 302 (1987), 661-682 Request permission
Corrigendum: Trans. Amer. Math. Soc. 311 (1989), 429-431.
Abstract:
The tangential approximants most useful in nonsmooth analysis and optimization are those which lie "between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone, there are such convex subcones which are more "accurate" approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is equally strong. In addition, a large class of convex subcones of the contingent cone can replace the Clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program. However, the Clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven. Overall, the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones.References
- 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. MR 749753
- J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9–52. MR 825383, DOI 10.1007/BF00938588
- Jonathan Michael Borwein, Epi-Lipschitz-like sets in Banach space: theorems and examples, Nonlinear Anal. 11 (1987), no. 10, 1207–1217. MR 913679, DOI 10.1016/0362-546X(87)90008-3 J. M. Borwein and H. M. Strojwas, The hypertangent cone study (submitted).
- 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. MR 709590
- 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
- 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). MR 736244
- Halina Frankowska, Necessary conditions for the Bolza problem, Math. Oper. Res. 10 (1985), no. 2, 361–366. MR 793890, DOI 10.1287/moor.10.2.361 E. Giner, Ensembles et fonctions étoilés. Application au calcul différentiel généralisé, Univ. Toulouse III, 1981, Manuscript.
- Monique Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space, SIAM J. Control 7 (1969), 232–241. MR 0252042
- Jean-Baptiste Hiriart-Urruty, Contributions à la programmation mathématique: cas déterministe et stochastique, Série E, No. 247, 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. MR 0489877
- 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
- Alexander D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61–69. MR 531969, DOI 10.1090/S0002-9947-1979-0531969-6
- 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. MR 525025, DOI 10.1137/0317019
- A. D. Ioffe, Approximate subdifferentials and applications. I. The finite-dimensional theory, Trans. Amer. Math. Soc. 281 (1984), no. 1, 389–416. MR 719677, DOI 10.1090/S0002-9947-1984-0719677-1 V. Jeyakumar, On optimality conditions in nonsmooth inequality constrained minimization, Univ. of Melbourne, Research Report No. 13, 1985. A. G. Kusraev and G. G. Kutateladze, Local convex analysis, J. Soviet Math. 26 (1984), 2048-2087.
- 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. MR 616568, DOI 10.1007/BF00935756
- 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. MR 812815, DOI 10.1287/moor.10.4.565
- 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). MR 745320
- Jean-Paul Penot, Calcul sous-différentiel et optimisation, J. Functional Analysis 27 (1978), no. 2, 248–276 (French). MR 481218, DOI 10.1016/0022-1236(78)90030-7
- D. Pallaschke (ed.), Nondifferentiable optimization: motivations and applications, Lecture Notes in Economics and Mathematical Systems, vol. 255, Springer-Verlag, Berlin, 1985. MR 821998, DOI 10.1007/978-1-4613-8268-3 B.N. Pshenichnyi and R. A. Khachatryan, Constraints of equality tupe in nonsmooth optimization problems, Soviet Math. Dokl. 26 (1982), 659-662.
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- 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
- 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. MR 623763
- R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming Stud. 17 (1982), 28–66. MR 654690, DOI 10.1007/bfb0120958
- R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), no. 7, 665–698. MR 796082, DOI 10.1016/0362-546X(85)90012-4 J. S. Treiman, Shrinking the set of generalized gradients (submitted).
- Corneliu Ursescu, Tangent sets’ calculus and necessary conditions for extremality, SIAM J. Control Optim. 20 (1982), no. 4, 563–574. MR 661033, DOI 10.1137/0320041
- M. Vlach, Approximation operators in optimization theory, Z. Oper. Res. Ser. A-B 25 (1981), no. 1, A15–A23 (English, with German summary). MR 627156, DOI 10.1007/bf01917339 D. E. Ward, Tangent cones, generalized subdifferential calculus, and optimization, Thesis, Dalhousie Univ. 1984.
- D. E. Ward and J. M. Borwein, Nonsmooth calculus in finite dimensions, SIAM J. Control Optim. 25 (1987), no. 5, 1312–1340. MR 905047, DOI 10.1137/0325072
- D. Ward, Isotone tangent cones and nonsmooth optimization, Optimization 18 (1987), no. 6, 769–783. MR 916209, DOI 10.1080/02331938708843290
- G. G. Watkins, Nonsmooth Milyutin-Dubovitskiĭ theory and Clarke’s tangent cone, Math. Oper. Res. 11 (1986), no. 1, 70–80. MR 830108, DOI 10.1287/moor.11.1.70
- Constantin Zălinescu, On convex sets in general position, Linear Algebra Appl. 64 (1985), 191–198. MR 776526, DOI 10.1016/0024-3795(85)90276-9
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 302 (1987), 661-682
- MSC: Primary 58C20; Secondary 46G05, 90C30
- DOI: https://doi.org/10.1090/S0002-9947-1987-0891640-2
- MathSciNet review: 891640