The Quantificational Tangent Cones

Ward, Doug

Canadian journal of mathematics, Tome 40 (1988) no. 3, pp. 666-694

Résumé

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones. Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

Détail

DOI : 10.4153/CJM-1988-029-6

Bibliographie

[1] 1. Aubin, J.-P., Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Math. Anal. Appl., Advances in Math., Suppl. Studies 7A (Academic Press, New York, 1981). Google Scholar

[2] 2. Aubin, J.-P. and Ekeland, I., Applied nonlinear analysis (Wiley, New York, 1984). Google Scholar

[3] 3. Bazaraa, M. S., Goode, J. J., Nashed, M. F. and Shetty, C. M., Nonlinear programming without differentiability in Banach spaces: Necessary and sufficient constraint qualifications, Applicable Analysis 5 (1976), 165–173. Google Scholar

[4] 4. Borwein, J. M., Proper efficient points for maximizations with respect to cones, SIAM J. Cont. Opt. 75 (1977), 57–63. Google Scholar

[5] 5. Borwein, J. M., Weak tangent cones and optimization in a Banach space, SIAM J. Cont. Opt. 16 (1978), 512–522. Google Scholar

[6] 6. Borwein, J. M., Stability and regular points of inequality systems, Journal of Optimization Theory and Applications 48 (1986), 9–52. Google Scholar

[7] 7. Borwein, J. M. and Strojwas, H. M., Directionally Lipschitzian mappings on Baire spaces, Can. J. Math. 36 (1984), 95–130. Google Scholar

[8] 8. Borwein, J. M. and Strojwas, H. M., Tangential approximations, Nonlinear Analysis: Theory, Methods, and Applications 9 (1985), 1247–1266. Google Scholar

[9] 9. Borwein, J. M. and Strojwas, H. M., Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory, Can. J. Math. 38 (1986), 431–452. Google Scholar

[10] 10. Borwein, J. M. and Strojwas, H. M., Proximal analysis and boundaries of closed sets in Banach space, Part II : Applications, Can. J. Math. 39 (1987), 428–472. Google Scholar

[11] 11. Borwein, J. M. and Strojwas, H. M., The hypertangent cone study, submitted. Google Scholar

[12] 12. Clarke, F. H., Optimization and nonsmooth analysis (Wiley, New York, 1983). Google Scholar

[13] 13. Dolecki, S., Tangency and differentiation: some applications of convergence theory, Annali di Matematica pura ed applicata 130 (1982), 223–255. Google Scholar

[14] 14. Elster, K.-H. and Thierfelder, J., The general concept of cone approximations in nondifferentiable opiimization, in Nondifferentiable optimization: Motivations and applications, Lecture Notes in Economics and Mathematical Systems 255 (Springer-Verlag, Berlin, 1985). Google Scholar

[15] 15. Guignard, M., Generalized Kuhn-Tucker conditions for mathematical programming in a Banach space, SIAM J. Control 7 (1969), 232–241. Google Scholar

[16] 16. Hiriart-Urruty, J.-B., Tangent cones generalized gradients and mathematical programming in Banach spaces, Mathematics of Operations Research 4 (1979), 79–97. Google Scholar

[17] 17. Ioffe, A. D., On the theory of subdifferential, in Fermat Days 85: Mathematics for optimization (North Holland, Amsterdam, 1986). Google Scholar

[18] 18. Kutateladze, S. S., Infinitesimal tangent cones, Siberian Math. J. 26 (1985), 833–840. Google Scholar

[19] 19. Kutateladze, S. S., Nonstandard analysis of tangent cones, Soviet Math. Dokl. 32 (1985), 437–439. Google Scholar

[20] 20. Martin, D. H., Gardner, R. J. and Watkins, G. G., Indicating cones and the intersection principle for tangential approximants in abstract multiplier rules, Journal of Optimization Theory and Applications 33 (1981), 515–537. Google Scholar

[21] 21. Martin, D. H. and Watkins, G. G., Cores of tangent cones and Clarke's tangent cone, Mathematics of Operations Research 10 (1985), 565–575. Google Scholar

[22] 22. Penot, J.-P., Calcul sous-différentiel et optimisation, Journal of Functional Analysis 27 (1978), 248–276. Google Scholar

[23] 23. Penot, J.-P., Variations on the theme of nonsmooth analysis: another subdifferential, in Nondifferentiable optimization: Motivations and applications, Lecture Notes in Economics and Mathematical Systems 255 (Springer-Verlag, Berlin, 1985). Google Scholar

[24] 24. Rockafellar, R. T., Convex analysis (Princeton University Press, Princetown, N.J., 1970). Google Scholar | DOI

[25] 25. Rockafellar, R. T., Clarke's tangent cone and the boundaries of closed sets in Rn , Nonlinear Analysis: Theory, Methods, and Applications 3 (1979), 145–154. Google Scholar

[26] 26. Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331–355. Google Scholar

[27] 27. Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32 (1980), 157–180. Google Scholar

[28] 28. Rockafellar, R. T., The theory of subgradients and its applications to problems of optimization: Convex and nonconvex functions (Helderman Verlag, Berlin, 1981). Google Scholar

[29] 29. Rosenberg, E., Exact penalties and stability in locally Lipschitz programming, Mathematical Programming 30 (1984), 340–356. Google Scholar

[30] 30. Studniarski, M., Mean value theorems and sufficient optimality conditions for nonsmooth functions, J. Math. Anal. Appl. 111 (1985), 313–326. Google Scholar

[31] 31. Thibault, L., Subdifferentials of nonconvex vector-valued functions, J. Math. Anal. Appl. 86 (1982), 319–344. Google Scholar

[32] 32. Thibault, L., Tangent cones and quasi-interiorly tangent cones to multifunctions, Trans. Am. Math. Soc. 277 (1983), 601–621. Google Scholar

[33] 33. Treiman, J. S., Characterization of Clarke's tangent and normal cones in finite and infinite dimensions, Nonlinear Analysis: Theory, Methods, and Applications 7 (1983), 771–783. Google Scholar

[34] 34. Treiman, J. S., Shrinking generalized gradients, to appear in Nonlinear Analysis: Theory, Methods, and Applications. Google Scholar

[35] 35. Ursescu, C., Tangent sets’ calculus and necessary conditions for extremality, SIAM J. Cont. Opt. 20 (1982), 563–574. Google Scholar

[36] 36. Vlach, M., On necessary conditions of optimality in linear spaces, Commentationes Mathematicae Universitatis Carolinae 11 (1970), 501–513. Google Scholar

[37] 37. Vlach, M., On the cones of tangents, Methods of Operations Research 37 (1980), 251–256. Google Scholar

[38] 38. Vlach, M., Approximation operators in optimization theory, Zeitschrift fur O.R. 25 (1981), 15–23. Google Scholar

[39] 39. Vlach, M., Closures and neighborhoods properties induced by tangential approximations, in Selected topics in operations research and mathematical economics (Springer-Verlag, Berlin, 1984)., 119–127. Google Scholar | DOI

[40] 40. Ward, D. E., Tangent cones, generalized subdifferential calculus, and optimization, Ph.D. thesis, Dalhousie University, (1984). Google Scholar

[41] 41. Ward, D. E., Exact penalties and sufficient conditions for optimality in nonsmooth optimization, Journal of Optimization Theory and Applications 57 (1988), 485–499. Google Scholar

[42] 42. Ward, D. E., Convex subcones of the contingent cone in nonsmooth calculus and optimization, Trans. Am. Math. Soc. 302 (1987), 661–682. Google Scholar

[43] 43. Ward, D. E., Isotone tangent cones and nonsmooth optimization, Optimization 18 (1987), 769–783. Google Scholar

[44] 44. Ward, D. E., Subdifferential calculus and optimality conditions in nonsmooth mathematical programming, to appear in J. of Information and Optimization Sciences. Google Scholar

[45] 45. Ward, D. E., Which subgradients have sum formulas'!', to appear in Nonlinear Analysis. Google Scholar

[46] 46. Ward, D. E. and Borwein, J. M., Nonsmooth calculus in finite dimensions, SIAM J. Cont. Opt. 25 (1987), 1312–1340. Google Scholar

[47] 47. Watkins, G. G., Nonsmooth Milyutin-Dubovitskii theory and Clarke's tangent cone, Mathematics of Operations Research 11 (1986), 70–80. Google Scholar