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Journal of Global Optimization

, Volume 52, Issue 2, pp 243–252 | Cite as

Second-order differentiability of generalized perturbation maps

  • S. J. Li
  • C. M. Liao
Article

Abstract

In this paper, by using the second-order proto-differentiability and second-order lower semidifferentiability, second-order differential properties of a class of set-valued maps are investigated and an explicit expression of the second-order derivatives is obtained. Then, second-order sensitivity properties are discussed for generalized perturbation maps.

Keywords

Generalized equation Generalized perturbation map Second-order proto-differentiability Second-order semidifferentiability Second-order lower semidifferentiability 

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References

  1. 1.
    Aubin J.P., Frankowska H.: Set-Valued Analysis. Birkhauser, Boston (1990)Google Scholar
  2. 2.
    Bonnans J.F., Shapiro A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research, New York (2000)Google Scholar
  3. 3.
    Cambini A., Martein L., Vlach M.: Second order tangent sets and optimality conditions. Math. Japonica. 49, 451–461 (1999)Google Scholar
  4. 4.
    Huy N.Q., Lee G.M.: Sensitivity of solutions to a parametric generalized equation. Set-Valued Anal. 16, 805–820 (2008)CrossRefGoogle Scholar
  5. 5.
    Jahn J., Khan A.A., Zeilinger P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005)CrossRefGoogle Scholar
  6. 6.
    Kalashnikov, V., Jadamba, B., Khan, A.A.: First and second-order optimality conditions in set optimization. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, pp. 265–276. (2006)Google Scholar
  7. 7.
    Lee G.M., Huy N.Q.: On proto-differentiablility of generalized perturbation maps. J. Math. Anal. Appl. 324, 1297–1309 (2006)CrossRefGoogle Scholar
  8. 8.
    Lee G.M., Huy N.Q.: On sensitivity analysis in vector optimization. Taiwanese J. Math. 11, 945–958 (2007)Google Scholar
  9. 9.
    Levy A.B.: Implicit multifunction theorems for the sensitivity analysis of variational conditions. Math. Program. 74, 333–350 (1996)Google Scholar
  10. 10.
    Levy A.B., Rockafellar R.T.: Sensitivity analysis of solutions to generalized equations. Trans. Am. Math. Soc. 345, 661–671 (1994)CrossRefGoogle Scholar
  11. 11.
    Levy A.B.: Sensitivity of solutions to variational inequalities on Banach spaces. SIAM J. Control Optim. 38, 50–60 (1999)CrossRefGoogle Scholar
  12. 12.
    Li S.J., Teo K.L., Yang X.Q.: Higher-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137, 533–553 (2008)CrossRefGoogle Scholar
  13. 13.
    Li S.J., Meng K.W., Penot J.P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)CrossRefGoogle Scholar
  14. 14.
    Li  S.J.,  Zhao P.: A method of duality for a mixed vector equilibrium problem. Optim. Lett. 4, 85–96 (2010)CrossRefGoogle Scholar
  15. 15.
    Luc D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991)CrossRefGoogle Scholar
  16. 16.
    Penot J.P.: Differentiability of relations and differential stability of perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)CrossRefGoogle Scholar
  17. 17.
    Penot J.P.: Are dualities appropriate for duality theories in optimization?.   J. Global Optim. 47, 503–525 (2010)CrossRefGoogle Scholar
  18. 18.
    Pardalos, P.M., Rassias, T.M., Khan, A.A. (eds): Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, Vol 35. Springer, New York (2010)Google Scholar
  19. 19.
    Rockafellar R.T.: Proto-differentiablility of set-valued mappings and its applications in optimization. Ann. Inst. H. Poincare Anal. Non Lineaire. 6, 449–482 (1989)Google Scholar
  20. 20.
    Rockafellar R.T., Wets R.J.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
  21. 21.
    Shi D.S.: Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl. 70, 385–396 (1991)CrossRefGoogle Scholar
  22. 22.
    Tanino T.: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479–499 (1988)CrossRefGoogle Scholar
  23. 23.
    Yang X.Q.: Second-order global optimality conditions for optimization problems. J. Global Optim. 30, 271–284 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingChina

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