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Transactions of the American Mathematical Society

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Subsmooth sets: Functional characterizations and related concepts

Authors: D. Aussel, A. Daniilidis and L. Thibault
Journal: Trans. Amer. Math. Soc. 357 (2005), 1275-1301
MSC (2000): Primary 26B25; Secondary 49J52, 47H04
Published electronically: November 23, 2004
MathSciNet review: 2115366
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Abstract: Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function $\mbox{dist}\,(C;\cdot)$, or the local uniqueness of the projection mapping, but also because in the case where $C$is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C$^{2}$ property) of the function. In this paper we provide an adapted geometrical concept, called subsmoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C$^{1}$ property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.

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  • 1. AUSSEL, D., CORVELLEC, J.-N. & LASSONDE, M., Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal. 1 (1994), 195-201. MR 1363111 (97e:49012)
  • 2. BERNARD, F. & THIBAULT, L., Uniform Prox-regularity of functions and epigraphs in Hilbert spaces, Nonlinear Anal. TMA, to appear.
  • 3. BOUNKHEL, M. & THIBAULT, L., On various notions of regularity of sets in non-smooth analysis, Nonlinear Anal. TMA 48 (2002), 223-246. MR 1870754 (2002j:46048)
  • 4. CLARKE, F., Optimization and non-smooth analysis, Wiley Interscience, New York 1983 (Re-published in 1990: Vol. 5, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa.).MR 1058436 (91e:49001)
  • 5. CLARKE, F., STERN, R. & WOLENSKI, P., Proximal smoothness and the lower-C$^{2}$ property, J. Convex Anal. 2 (1995), 117-144.MR 1363364 (96j:49014)
  • 6. COLOMBO, G. & GONCHAROV, V., Variational inequalities and regularity properties of closed sets in Hilbert spaces, J. Convex Anal. 8 (2001), 197-221. MR 1829062 (2002f:49031)
  • 7. CORREA, R. & JOFRÉ, A., Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory Appl. 61 (1989), 1-21.MR 0993912 (90h:49009)
  • 8. CORREA, R., JOFRÉ, A. & THIBAULT, L., Subdifferential monotonicity as characterization of convex functions, Num. Func. Anal. Optim. 15 (1994), 531-535. MR 1281560 (95d:49029)
  • 9. DANIILIDIS, A. & GEORGIEV, P., Approximate convexity and submonotonicity, J. Math. Anal. Appl. 291 (2004), 292-301.MR 2034075 (2004m:49048)
  • 10. DANIILIDIS, A., GEORGIEV, P. & PENOT, J.-P., Integration of multivalued operators and cyclic submonotonicity, Trans. Amer. Math. Soc. 355 (2003), 177-195. MR 1928084 (2003h:49030)
  • 11. DEVILLE, R. GODEFROY, G. & ZIZLER, V., Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64. Longman Scientific & Technical, Harlow; co-published in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634 (94d:46012)
  • 12. EKELAND, I., Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474. MR 0526967 (80h:49007)
  • 13. FABIÁN, M., Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolin. Math. Phys. 30 (1989), 51-56. MR 1046445 (91c:49024)
  • 14. FEDERER, H., Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. MR 0110078 (22:961)
  • 15. HIRIART-URRUTY, J.-B., ``Generalized differentiability, duality and optimization for problems dealing with differences of convex functions'', in: Convexity and Duality in Optimization'', Lecture Notes in Econom. Math. Systems 256 (1984), 37-70. MR 0873269 (88b:90101)
  • 16. KRUGER, A.Y., $\varepsilon$-semidifferentials and $\varepsilon$-normal elements, Depon. VINITI, No. 1331-81, Moscow, 1981 (in Russian).
  • 17. LEBOURG, G., Generic differentiability of Lipschitzian functions, Trans. Amer. Math. Soc. 256 (1979), 125-144. MR 0546911 (80i:58012)
  • 18. LEWIS, A., Robust regularization, preprint CECM, 17p, 2002.
  • 19. MARCELLIN, S., Initiation à l'analyse séquentielle non lisse dans les espaces d'Asplund, Master Thesis, 142p, (Montpellier, 2002).
  • 20. MIFFLIN, R., Semismooth and semi-convex functions in constrained optimization, SIAM J. Control Optim. 15 (1977), 959-972. MR 0461556 (57:1541)
  • 21. MORDUKHOVICH, B.S., SHAO, Y., Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235-1280.MR 1333396 (96h:49036)
  • 22. NGAI, H.V., THÉRA, M.,``On $\varepsilon $-convexity and $\varepsilon$-monotonicity'', in: Calculus of Variations and Differential Equations, A. Ioffe, S. Reich, and I. Shafrir (Eds.), Research Notes in Mathematical Series (Chapman & Hall, 82-100, 1999). MR 1713840 (2000h:49027)
  • 23. NGAI, H.V., LUC, D.T. & THERA, M., Approximate convex functions, J. Nonlinear Convex Anal. 1 (2000), 155-176.MR 1777137 (2001e:49032)
  • 24. PENOT, J.-P., Favorable classes of mappings and multimappings in nonlinear analysis and optimization, J. Convex Anal. 3 (1996), 97-116.MR 1422755 (97i:90110)
  • 25. POLIQUIN, R., ROCKAFELLAR, R.T. & THIBAULT, L., Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000), 5231-5249.MR 1694378 (2001b:49024)
  • 26. ROCKAFELLAR, R.T., Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R} ^{n}$, Non-Linear Anal. TMA 3 (1979), 145-154.MR 0520481 (80d:49032)
  • 27. ROCKAFELLAR, R.T., Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257-280.MR 0571922 (81f:49006)
  • 28. ROCKAFELLAR, R.T., ``Favorable Classes of Lipschitz Continuous Functions in Subgradient Optimization'' in Nondifferentiable Optimization (1982), Nurminski E. (eds.), Pergamon Press, New York.MR 0704977 (85e:90069)
  • 29. ROLEWICZ, S., On the coincidence of some subdifferentials in the class of $\alpha(.)$-paraconvex functions, Optimization 50 (2001), 353-363. MR 1892909 (2003e:49033)
  • 30. SHAPIRO, A., Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optimization 4 (1994), 130-141.MR 1260410 (94m:90111)
  • 31. SPINGARN, J.E., Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc. 264 (1981), 77-89.MR 0597868 (82g:26016)
  • 32. THIBAULT, L., On the subdifferentials of optimal value functions, SIAM J. Control Optim. 29 (1991), 1019-1036. MR 1110085 (92e:49027)
  • 33. VIAL, J.-P., Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983), 231-259. MR 0707055 (84m:90107)

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Additional Information

D. Aussel
Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan Cedex, France

A. Daniilidis
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Cerdanyola del Vallès), Spain

L. Thibault
Affiliation: Université Montpellier II, Département de Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 05, France

Keywords: Variational analysis, subsmooth sets, submonotone operator, approximately convex functions
Received by editor(s): February 24, 2003
Published electronically: November 23, 2004
Additional Notes: The research of the second author has been supported by the Spanish Ministry of Education Program: “Ayudas para estancias de profesores e investigadores extranjeros en España” (Grant No SB2000-0369).
Article copyright: © Copyright 2004 American Mathematical Society

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