Abstract

A generalization of the class of lower CkC^k−functions introduced by R.T. Rockafellar [28] called lower TkT^k−functions is proposed in the infinite dimensional case. Mappings of class TkT^k are studied for themselves as they seem to deserve some attention. Other classes of functions such as subconvex functions, subsmooth functions, semismooth functions are either introduced or extended to the case of an infinite dimensional Banach space.

Author information

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

Jean-Paul Penot

Département de Mathématiques Appliquéees, CNRS URA 1204, Université de Pau, 64000 Pau, France

References

  1. R. Abraham, J. Robbin: Transversal mappings and flows, W.A. Benjamin, New York, Amsterdam, 1967.Google Scholar
  2. J.-P. Aubin, I. Ekeland: Applied nonlinear analysis, John Wiley, New York, 1984.Google Scholar
  3. D. Aussel, J.-N. Corvellec and M. Lassonde: Mean Value Theorem and subdifferential criteria for lower semicontinuous functions, preprint, Univ. Clermont, April 1994.https://doi.org/10.2307/2155218Google Scholar
  4. V.T. Averbuh, O.G. Smolyanov: Different definitions of derivatives in linear topological spaces, Uspehi Mat. Nauk 23(4) (1968) (162) 67–116.https://doi.org/10.1070/RM1968v023n04ABEH003770Google Scholar
  5. J. Birge, L. Qi: Semiregularity and generalized subdifferentials with applications to optimization, Math. Oper. Research 18 (4) (1993) 982–1005.https://doi.org/10.1287/moor.18.4.982Google Scholar
  6. J. Borwein, S.P. Fitzpatrick and J. R. Giles: The differentiability of real functions on normed linear spaces using generalized subdifferentials, J. Math. Anal. Appl. 128 (1987) 512–538.https://doi.org/10.1016/0022-247X(87)90203-4Google Scholar
  7. N. Bourbaki: Eléments de mathématiques, espaces vectoriels topologiques, chap. III-V, Hermann, Paris, 1955.Google Scholar
  8. J.V. Burke, L. Qi: Weak directional closedness and generalized subdifferentials, J. Math. Anal. Appl. 159 (2) (1991) 485–499.https://doi.org/10.1016/0022-247X(91)90209-IGoogle Scholar
  9. F.H. Clarke: Optimization and nonsmooth analysis, Wiley Interscience, New York, 1983.Google Scholar
  10. F.H. Clarke, R.J. Stern and P.R. Wolinski: Subgradient criteria for monotone and Lipschitz behavior, Canad. J. Math. 45 (1993) 1167–1183.https://doi.org/10.4153/CJM-1993-065-xGoogle Scholar
  11. R. Correa, A. Jofre: Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory and Appl. 61 (1989) 1–21.https://doi.org/10.1007/BF00940840Google Scholar
  12. J. Danes: On continuous convergence of sequences of linear mappings, Rev. Roum. Math. Pures et Appl. 18 (7) (1973) 1045–1049.Google Scholar
  13. J.-B. Hiriart, Ph. Plazanet: Moreau’s theorem revisited, Analyse nonlinéaire (Perpignan, 1986) Ann. Inst. H. Poincaré 6 (1989) suppl., 325–338.https://doi.org/10.1016/s0294-1449(17)30028-8Google Scholar
  14. A.D. Ioffe: Proximal analysis and approximate subdifferentials, J. London Math. Soc. (2) 41 (1990) 175–192.https://doi.org/10.1112/jlms/s2-41.1.175Google Scholar
  15. J.L. Kelley: General topology, Van Nostrand, Princeton, 1955.Google Scholar
  16. A. Kruger: Properties of generalized subdifferentials, Siberian J. Math. 26 (1985) 822–832.https://doi.org/10.1007/BF00969103Google Scholar
  17. Ph. Loewen: A Mean Value Theorem for Fréchet subgradients, preprint, Nonlinear Anal. Th. Meth. Appl. 23 (11) (1994) 1365–1382.https://doi.org/10.1016/0362-546X(94)90132-5Google Scholar
  18. Ph. Michel, J.-P. Penot: Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes, C. R. Acad. Sc. Paris, 298 (1984) 269–272.Google Scholar
  19. Ph. Michel, J.-P. Penot: A generalized derivative for calm and stable functions, Diff. Int. Equations, 5(2) (1992) 433–454.https://doi.org/10.57262/die/1371043981Google Scholar
  20. R. Mifflin: Semismooth and semiconvex functions in constrained optimization, SIAM J. Control and Opt. 15(6) (1977) 959–972.https://doi.org/10.1137/0315061Google Scholar
  21. R. Mifflin: An algorithm for constrained optimization with semismooth functions, Math. Oper. Research, 2 (2) (1977) 191–207.https://doi.org/10.1287/moor.2.2.191Google Scholar
  22. B. Sh. Mordukhovich: Approximation methods in problems of optimization and control, Nauka, Moscow, 1988 (Russian; English translation to appear in Wiley-Interscience).Google Scholar
  23. J.-P. Penot: Continuité et différentiabilité des opérateurs de Nemytskii, Publications mathématiques de Pau (1976) VIII 1–45.Google Scholar
  24. J.-P. Penot: On the Mean Value Theorem, Optimization 19 (1988) 147–156.https://doi.org/10.1080/02331938808843330Google Scholar
  25. J.-P. Penot: A Mean Value Theorem with small subdifferentials, preprint, Univ. of Pau, Jan. 1994, revised July 1994.Google Scholar
  26. J.-P. Penot, M.L. Bougeard: Approximation and decomposition properties of some classes of locally d.c. functions, Math. Prog. 41 (1988) 195- 227.https://doi.org/10.1007/BF01580764Google Scholar
  27. R.T. Rockafellar: Local boundedness of nonlinear monotone operators, Michigan Math. J. 16 (1970) 397- 407.https://doi.org/10.1307/mmj/1029000324Google Scholar
  28. R.T. Rockafellar: Favorable classes of Lipschitz continuous functions in subgradient optimization, in “Progress in nondifferentiable optimization”, E. Nurminski, ed. IIASA Collaborative Proc. Series, Intern. Institute of Applied Analysis, Laxenburg, Austria, 1982, 125–144.Google Scholar
  29. R.T. Rockafellar: First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1) (1988) 75–108.https://doi.org/10.1090/S0002-9947-1988-0936806-9Google Scholar
  30. I.V. Skripnik: The application of Morse’s methods to nonlinear elliptic equations, (Russian) Dokl. Akad. Nauk SSSR 202 (1972) 769–771, MR 45# 1206.Google Scholar
  31. I.V. Skripnik: The differentiability of integral functionals, (Ukrainian) Dopividi Akad. Nauk Ukrain RSR ser. A (1972) 1086–1089. MR 2749.Google Scholar
  32. J.E. Spingarn: Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc. 264 (1) (1981) 77–89.https://doi.org/10.1090/S0002-9947-1981-0597868-8Google Scholar
  33. J.E. Spingarn: Submonotone mappings and the proximal point algorithm, Num. Funct. Anal. Opt. 4(2) (1981–1982) 123–150.https://doi.org/10.1080/01630568208816109Google Scholar
  34. L. Thibault, D. Zagrodny: Integration of subdifferentials of lower semicontinuous functions on Banach spaces, to appear J. Math. Anal. Appl.Google Scholar
  35. J. Treiman: Clarke’s generalized gradients and ε-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986) 65–78.https://doi.org/10.1090/S0002-9947-1986-0819935-8Google Scholar
  36. D. Zagrodny: Approximate Mean Value Theorem for upper subderivatives, Nonlinear Anal., Th., Methods, Appl. 12 (1988) 1413–1428.https://doi.org/10.1016/0362-546X(88)90088-0Google Scholar
  37. D. Zagrodny: A note on the equivalence between the Mean Value Theorem for the Dini derivative and the Clarke-Rockafellar derivative, Optimization 21 (1990) 179–183.https://doi.org/10.1080/02331939008843532Google Scholar

Suggested citation

J.-P. Penot. “Favorable Classes of Mappings and Multimappings in Nonlinear Analysis and Optimization.” Journal of Convex Analysis 3 (1996), No. 1, 97–116. https://doi.org/10.68381/jca03008

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.