Download PDF - On relations among the generalized second-order directional derivativesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On relations among the generalized second-order directional derivatives

Authors 1

Affiliations

  1. Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tř. Svobody 26, 771 46 Olomouc, Czech Republic

Abstract

In the paper, we deal with the relations among several generalized second-order directional derivatives. The results partially solve the problem which of the second-order optimality conditions is more useful.

Keywords

ENG
generalized second-order directional derivative, convexity, second-order optimality conditions

Bibliography

  1. J.M. Borwein, M. Fabián, On generic second-order Gâteaux differentiability, Nonlinear Anal. T.M.A 20 (1993), 1373-1382.
  2. J.M. Borwein and D. Noll, Second-order differentiability of convex functions in Banach spaces, Trans. Amer. Math. Soc. 342 (1994), 43-81.
  3. A. Ben-Tal and J. Zowe, Directional derivatives in nonsmooth optimization, J. Optim. Theory Appl. 47 (1985), 483-490.
  4. F.H. Clarke, Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, Ph.D. thesis, Univ. of Washington 1973.
  5. F.H. Clarke, Optimization and nonsmooth analysis, J. Wiley, New York 1983.
  6. R. Cominetti, Equivalence between the classes of C1,1 and twice locally Lipschitzian functions, Ph.D. thesis, Université Blaise Pascal 1989.
  7. R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (1990), 789-809.
  8. W.L. Chan, L.R. Huang and K.F. Ng, On generalized second-order derivatives and Taylor expansions in nosmooth optimiyation, SIAM J. Control Optim. 32 (1994), 789-809.
  9. P.G. Georgiev and N.P. Zlateva, Second-order subdifferentials of C1,1 functions and optimality conditions, Set-Valued Analysis 4 (1996), 101-117.
  10. J.B. Hiriart-Urruty and C. Lemarchal, Convex Analysis nad Minimization Algorithms, Springer Verlag, Berlin 1993.
  11. L.R. Huang and K.F. Ng, On some relations between Chaney's generalized second-order directional derivative and that of Ben-Tal and Zowe, SIAM J. Control Optim. 34 (1996), 1220-1234.
  12. V. Jeyakumar and X.Q. Yang, Approximate generalized Hessians and Taylor's expansions for continously Gâteaux differentiable functions, Nonlinear Anal. T.M.A 36 (1999), 353-368.
  13. G. Lebourg, Generic differentiability of Lipschitz functions, Trans. Amer. Math. Soc. 256 (1979), 125-144.
  14. Y. Maruyama, Second-order necessary conditions for nonlinear problems in Banach spases and their applications to optimal control problems, Math. Programming 41 (1988), 73-96.
  15. P. Michel and J.P. Penot, Second-order moderate derivatives, Nonlinear Anal. T.M.A 22 (1994), 809-821.
  16. K. Pastor, Generalized second-order directional derivatives for locally Lipschitz functions, Preprint no. 17/2001 of Palacký Univerzity, Olomouc.
  17. R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497-509.
  18. R.T. Rockafellar, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. Oper. Res. 14 (1989), 462-484.
  19. X.Q. Yang, On relations and applications of generalized second-order directional derivatives, Non. Anal. T.M.A 36 (1999), 595-614.
  20. X.Q. Yang, On second-order directional derivatives, Non. Anal. T.M.A 26, 55-66.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2001/21/2
Export to PBN, Opens in new tab
Pages:
235-247
Main language of publication
English
Received
2001-09-10
Published
2001

DOI: 10.7151/dmgt.1026

Exact and natural sciences