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The Quantificational Tangent Cones

Published online by Cambridge University Press:  20 November 2018

Doug Ward
Doug Ward*
Affiliation:
Miami University, Oxford, Ohio
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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.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 666 - 694
Copyright
Copyright © Canadian Mathematical Society 1988

References

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. Aubin, J.-P. and Ekeland, I., Applied nonlinear analysis (Wiley, New York, 1984).Google Scholar
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. Borwein, J. M., Proper efficient points for maximizations with respect to cones, SIAM J. Cont. Opt. 75 (1977), 57–63.Google Scholar
5. Borwein, J. M., Weak tangent cones and optimization in a Banach space, SIAM J. Cont. Opt. 16 (1978), 512–522.Google Scholar
6. Borwein, J. M., Stability and regular points of inequality systems, Journal of Optimization Theory and Applications 48 (1986), 9–52.Google Scholar
7. Borwein, J. M. and Strojwas, H. M., Directionally Lipschitzian mappings on Baire spaces, Can. J. Math. 36 (1984), 95–130.Google Scholar
8. Borwein, J. M. and Strojwas, H. M., Tangential approximations, Nonlinear Analysis: Theory, Methods, and Applications 9 (1985), 1247–1266.Google Scholar
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. 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. Borwein, J. M. and Strojwas, H. M., The hypertangent cone study, submitted.Google Scholar
12. Clarke, F. H., Optimization and nonsmooth analysis (Wiley, New York, 1983).Google Scholar
13. Dolecki, S., Tangency and differentiation: some applications of convergence theory, Annali di Matematica pura ed applicata 130 (1982), 223–255.Google Scholar
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. Guignard, M., Generalized Kuhn-Tucker conditions for mathematical programming in a Banach space, SIAM J. Control 7 (1969), 232–241.Google Scholar
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. Ioffe, A. D., On the theory of subdifferential, in Fermat Days 85: Mathematics for optimization (North Holland, Amsterdam, 1986).Google Scholar
18. Kutateladze, S. S., Infinitesimal tangent cones, Siberian Math. J. 26 (1985), 833–840.Google Scholar
19. Kutateladze, S. S., Nonstandard analysis of tangent cones, Soviet Math. Dokl. 32 (1985), 437–439.Google Scholar
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. 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. Penot, J.-P., Calcul sous-différentiel et optimisation, Journal of Functional Analysis 27 (1978), 248–276.Google Scholar
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. Rockafellar, R. T., Convex analysis (Princeton University Press, Princetown, N.J., 1970).CrossRefGoogle Scholar
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. Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331–355.Google Scholar
27. Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math. 32 (1980), 157–180.Google Scholar
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. Rosenberg, E., Exact penalties and stability in locally Lipschitz programming, Mathematical Programming 30 (1984), 340–356.Google Scholar
30. Studniarski, M., Mean value theorems and sufficient optimality conditions for nonsmooth functions, J. Math. Anal. Appl. 111 (1985), 313–326.Google Scholar
31. Thibault, L., Subdifferentials of nonconvex vector-valued functions, J. Math. Anal. Appl. 86 (1982), 319–344.Google Scholar
32. Thibault, L., Tangent cones and quasi-interiorly tangent cones to multifunctions, Trans. Am. Math. Soc. 277 (1983), 601–621.Google Scholar
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. Treiman, J. S., Shrinking generalized gradients, to appear in Nonlinear Analysis: Theory, Methods, and Applications.Google Scholar
35. Ursescu, C., Tangent sets’ calculus and necessary conditions for extremality, SIAM J. Cont. Opt. 20 (1982), 563–574.Google Scholar
36. Vlach, M., On necessary conditions of optimality in linear spaces, Commentationes Mathematicae Universitatis Carolinae 11 (1970), 501–513.Google Scholar
37. Vlach, M., On the cones of tangents, Methods of Operations Research 37 (1980), 251–256.Google Scholar
38. Vlach, M., Approximation operators in optimization theory, Zeitschrift fur O.R. 25 (1981), 15–23.Google Scholar
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.CrossRefGoogle Scholar
40. Ward, D. E., Tangent cones, generalized subdifferential calculus, and optimization, Ph.D. thesis, Dalhousie University, (1984).Google Scholar
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. 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. Ward, D. E., Isotone tangent cones and nonsmooth optimization, Optimization 18 (1987), 769–783.Google Scholar
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. Ward, D. E., Which subgradients have sum formulas'!', to appear in Nonlinear Analysis.Google Scholar
46. Ward, D. E. and Borwein, J. M., Nonsmooth calculus in finite dimensions, SIAM J. Cont. Opt. 25 (1987), 1312–1340.Google Scholar
47. Watkins, G. G., Nonsmooth Milyutin-Dubovitskii theory and Clarke's tangent cone, Mathematics of Operations Research 11 (1986), 70–80.Google Scholar