Abstract

We provide several characterizations of compact epi-Lipschitzness for closed convex sets in normed vector spaces. In particular, we show that a closed convex set is compactly epi-Lipschitzian if and only if it has nonempty relative interior, finite codimension, and spans a closed subspace. Next, we establish that all boundary points of compactly epi-Lipschitzian sets are proper support points. We provide the corresponding results for functions by using inf-convolutions and the Legendre-Fenchel transform. We also give an application to constrained optimization with compactly epi-Lipschitzian data via a generalized Slater condition involving relative interiors

Author information

Contact details are reproduced from the original publication and may be historical.

Jonathan Borwein

Dept. of Mathematics and Statistics, Simon Fraser University, Burnaby BC, V5A 1S6, Canada

Yves Lucet

Dept. of Mathematics and Statistics, University of Victoria, Victoria BC, V8W 3P4, Canada

Boris Mordukhovich

Dept. of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.

Keywords

  • Compactly epi-Lipschitzian set
  • convex set

References

  1. J. Arias de Reyna: Dense hyperplanes of first category, Mathematische Annalen 249 (1980) 111–114.https://doi.org/10.1007/BF01351408Google Scholar
  2. E. Bishop, R. R. Phelps: The support functionals of a convex set, Proc. of Symposia in Pure Math., Amer. Math. Soc., Providence, R.I., Convexity VII (1963) 27–35.https://doi.org/10.1090/pspum/007/0154092Google Scholar
  3. J. M. Borwein: Epi-Lipschitz-like sets in Banach space: theorems and examples, Nonlinear Anal. 11 (1987) 1207–1217.https://doi.org/10.1016/0362-546X(87)90008-3Google Scholar
  4. J. M. Borwein, A. Jofré: A nonconvex separation property in Banach spaces, Math. Methods Oper. Res. 48 (1998) 169–179.https://doi.org/10.1007/s001860050019Google Scholar
  5. J. M. Borwein, A. S. Lewis: Partially finite convex programming, part 1: Quasi relative interiors and duality theory, Math. Programming 57 (1992) 15–48.https://doi.org/10.1007/BF01581072Google Scholar
  6. J. M. Borwein, H. M. Strójwas: Tangential approximations, Nonlinear Anal. 9 (1985) 1347–1366.https://doi.org/10.1016/0362-546X(85)90095-1Google Scholar
  7. J. M. Borwein, H. M. Strójwas: Proximal analysis and boundaries of closed sets in Banach space, I. Theory, Canad. J. Math. 38 (1986) 431–452.https://doi.org/10.4153/CJM-1986-022-4Google Scholar
  8. J. M. Borwein, H. M. Strójwas: Proximal analysis and boundaries of closed sets in Banach space, II. Applications, Canad. J. Math. 39 (1987) 428–472.https://doi.org/10.4153/CJM-1987-019-4Google Scholar
  9. J. M. Borwein, J. D. Vanderwerff: Banach spaces that admit support sets, Proc. Amer. Math. Soc. 124 (1996) 751–755.https://doi.org/10.1090/S0002-9939-96-03122-XGoogle Scholar
  10. J. M. Borwein, Q. J. Zhu: A survey of subdifferential calculus with applications, J. Nonlinear Analysis: Th. Meth. Appl. 38 (1999) 687–773.https://doi.org/10.1016/S0362-546X(98)00142-4Google Scholar
  11. F. H. Clarke: Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second ed., 1990.https://doi.org/10.1137/1.9781611971309Google Scholar
  12. M. Fabian, B. S. Mordukhovich: Nonsmooth characterizations of Asplund spaces and smooth variational principles, Set-Valued Anal. 6 (1998) 381–406.https://doi.org/10.1023/A:1008799412427Google Scholar
  13. V. P. Fonf: On supportless convex sets in incomplete normed spaces, Proc. Amer. Math. Soc. 120 (1994) 1173–1176.https://doi.org/10.1090/S0002-9939-1994-1216811-6Google Scholar
  14. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms, A Series of Comprehensive Studies in Mathematics, Springer-Verlag, 1993.https://doi.org/10.1007/978-3-662-06409-2Google Scholar
  15. A. D. Ioffe: Approximate subdifferentials and applications, III. The metric theory, Mathematika 36 (1989) 1–38.https://doi.org/10.1112/S0025579300013541Google Scholar
  16. A. D. Ioffe: Codirectional compactness, metric regularity and subdifferential calculus, Tech. Rep., In Press, 1997.Google Scholar
  17. A. Jourani: Compactly epi-Lipschitzian sets and A-subdifferentials in WT-spaces, Optimization 34 (1995) 1–17.https://doi.org/10.1080/02331939508844090Google Scholar
  18. A. Jourani: The role of locally compact cones in nonsmooth analysis, Comm. Appl. Nonlinear Anal. 5 (1998) 1–35.https://doi.org/10.1016/S0362-546X(98)00031-5Google Scholar
  19. A. Jourani, L. Thibault: Metric regularity for strongly compactly Lipschitzian mappings, Nonlinear Anal. 24 (1995) 229–240.https://doi.org/10.1016/0362-546X(94)E0061-KGoogle Scholar
  20. A. Jourani, L. Thibault: Extensions of subdifferential calculus rules in Banach spaces, Canad. J. Math. 48 (1996) 834–848.https://doi.org/10.4153/CJM-1996-042-1Google Scholar
  21. A. Jourani, L. Thibault: Coderivatives of multivalued mappings, locally compact cones and metric regularity, Nonlinear Anal. 35 (1999) 925–945.https://doi.org/10.1016/S0362-546X(98)00031-5Google Scholar
  22. A. Y. Kruger, B. S. Mordukhovich: Extremal points and the Euler equation in nonsmooth optimization problems, Dokl. Akad. Nauk BSSR 24 (1980) 684–687, 763.Google Scholar
  23. P. D. Loewen: Limits of Fréchet normals in nonsmooth analysis, in: Optimization and Nonlinear Analysis (Haifa, 1990), vol. 244 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow (1992) 178–188.Google Scholar
  24. B. S. Mordukhovich: Maximum principle in the problem of time optimal response with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976) 960–969.https://doi.org/10.1016/0021-8928(76)90136-2Google Scholar
  25. B. S. Mordukhovich: Approximation Methods in Problems of Optimization and Control, “Nauka”, Moscow, 1988. (Russian). Second English ed. to appear.Google Scholar
  26. B. S. Mordukhovich: Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc. 343 (1994) 609–657.https://doi.org/10.1090/S0002-9947-1994-1242786-4Google Scholar
  27. B. S. Mordukhovich, Y. Shao: Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1996) 197–205.https://doi.org/10.1090/S0002-9939-96-03049-3Google Scholar
  28. B. S. Mordukhovich, Y. Shao: Nonconvex differential calculus for infinite-dimensional multifunctions, Set-Valued Anal. 4 (1996) 205–236.https://doi.org/10.1007/BF00419366Google Scholar
  29. B. S. Mordukhovich, Y. Shao: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996) 1235–1280.https://doi.org/10.1090/S0002-9947-96-01543-7Google Scholar
  30. B. S. Mordukhovich, Y. Shao: Stability of set-valued mappings in infinite dimensions: point criteria and applications, SIAM J. Control Optim. 35 (1997) 285–314.https://doi.org/10.1137/S0363012994278171Google Scholar
  31. J.-J. Moreau: Sur la fonction polaire d’une fonction semi-continue supérieurement, C. R. Acad. Sci. Paris, 258 (1964) 1128–1130.Google Scholar
  32. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, second ed. 1993.Google Scholar
  33. H. Rådström: An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169.https://doi.org/10.1090/S0002-9939-1952-0045938-2Google Scholar
  34. R. T. Rockafellar: Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966) 46–63.https://doi.org/10.1090/S0002-9947-1966-0192318-XGoogle Scholar
  35. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton, New York, 1970.Google Scholar
  36. R. T. Rockafellar: Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3) 39 (1979) 331–355.https://doi.org/10.1112/plms/s3-39.2.331Google Scholar
  37. R. T. Rockafellar: Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980) 257–280.https://doi.org/10.4153/CJM-1980-020-7Google Scholar
  38. R. T. Rockafellar: Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré. Anal. Non Linéaire 2 (1985) 167–184.https://doi.org/10.1016/s0294-1449(16)30401-2Google Scholar
  39. C. Swartz: An Introduction to Functional Analysis, Pure and Applied Mathematics, A Series of Monographs and Textbooks, Marcel Dekker Inc., New York, 1992.Google Scholar
  40. M. Valdivia: Topics in Locally Convex Spaces, vol. 67 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1982. Notas de Matemática [Mathematical Notes] 85.Google Scholar
  41. A. Wilansky: Modern Methods in Topological Vector Spaces, McGraw-Hill International Book Co., London, New York, 1978.Google Scholar
  42. J. Ye: Necessary optimality conditions for control of strongly monotone variational inequalities, in: Control of Distributed Parameter and Stochastic Systems, S. Chen et al. (ed.), Kluwer, Boston, (1999) 153–160.https://doi.org/10.1007/978-0-387-35359-3_19Google Scholar

Suggested citation

J. Borwein, Y. Lucet, B. Mordukhovich. “Compactly Epi-Lipschitzian Convex Sets and Functions in Normed Spaces.” Journal of Convex Analysis 7 (2000), No. 2, 375–393. https://doi.org/10.68381/jca07019

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2000. Rights now held by Banach Press.