Advertisement

Optimality Conditions for C1,1 Vector Optimization Problems

  • A. Guerraggio
  • D. T. Luc
Article

Abstract

The main purpose of this paper is to make use of the second-order subdifferential of vector functions to establish necessary and sufficient optimality conditions for vector optimization problems.

efficient solutions second-order subdifferentials convex vector functions optimality conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hiriart-Urruty, J. B., Contributions a la Programmation Mathematique: Deterministe et Stocastique, Doctoral Thesis, Université de Clemont-Ferrand, 1977.Google Scholar
  2. 2.
    Hiriart-Urruty, J. B., Strodiot, J. J., and Hien Nguyen, V., Generalized Hessian Matrix and Second-Order Optimality Conditions for Problems with C 1,1 Data, Applied Mathematics and Optimization, Vol. 11, pp. 43-56, 1984.Google Scholar
  3. 3.
    Klatte, D., and Tammer, K., On the Second-Order Sufficient Optimality Conditions for C 1,1 Optimization Problems, Optimization, Vol. 19, pp. 169-180, 1988.Google Scholar
  4. 4.
    Luc, D. T., Taylor's Formula for C k,1 Functions, SIAM Journal on Optimization, Vol. 5, pp. 659-669, 1995.Google Scholar
  5. 5.
    Luc, D. T., and Schaible, S., Generalized Monotone Nonsmooth Maps, Journal of Convex Analysis, Vol. 3, pp. 195-205, 1996.Google Scholar
  6. 6.
    Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley, New York, NY, 1983.Google Scholar
  7. 7.
    Yu, P. L., Multicriteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, NY, 1985.Google Scholar
  8. 8.
    Sawaragi, Y., Nakayama, H., and Tanino, T., Theory of Multiobjective Optimization, Academic Press, New York, NY, 1985.Google Scholar
  9. 9.
    Luc, D. T., Theory of Vector Optimization, Springer Verlag, Berlin, Germany, 1989.Google Scholar
  10. 10.
    Luc, D. T., Tan, N. X., and Tinh, P. N., Convex Vector Functions and Their Subdifferentials, Acta Mathematica Vietnamica, Vol. 23, pp. 107-127, 1998.Google Scholar
  11. 11.
    Henig, M. I., Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 6, pp. 387-407, 1982.Google Scholar
  12. 12.
    Giannessi, F., Mastroeni, G., and Pellegrini, L., On the Theory of Vector Optimization and Variational Inequalities: Image Space Analysis and Separation, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, England, pp. 153-216, 2000.Google Scholar
  13. 13.
    Guerraggio, A., Molho, E., and Zaffaroni, A., On the Notion of Proper Efficiency in Vector Optimization, Journal of Optimization Theory and Applications, Vol. 82, pp. 1-21, 1994.Google Scholar
  14. 14.
    Khanh, P. Q., Proper Solutions of Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 74, pp. 105-130, 1992.Google Scholar
  15. 15.
    Luc, D. T., and Schaible, S., On Efficiency and Generalized Concavity, Journal of Optimization Theory and Applications, Vol. 94, pp. 147-153, 1997.Google Scholar
  16. 16.
    Luc, D. T., On the Properly Efficient Points of Nonconvex Sets, European Journal of Operations Research, Vol. 86, pp. 332-336, 1995.Google Scholar
  17. 17.
    Craven, B. D., Nonsmooth Multiobjective Programming, Numerical Functional Analysis and Optimization, Vol. 10, pp. 49-64, 1989.Google Scholar
  18. 18.
    El Abdouni, B., and Thibault, L., Lagrange Multipliers for Pareto Nonsmooth Programming Problems in Banach Spaces, Optimization, Vol. 26, pp. 277-285, 1992.Google Scholar
  19. 19.
    Minami, M., Weak Pareto-Optimal Necessary Conditions in a Nondifferentiable Multiobjective Program on a Banach Space, Journal of Optimization Theory and Applications, Vol. 41, pp. 451-461, 1983.Google Scholar
  20. 20.
    Cambini, A., and Martein, L., Second-Order Necessary Optimality Conditions in the Image Space: Preliminary Results, Proceedings of the Workshop on Scalar and Vector Optimization in Economic and Financial Problems, Edited by E. Castagnoli and G. Giorgi, Milan, Italy, pp. 27-38, 1995.Google Scholar
  21. 21.
    Aghezzaf, B., and Hachimi, M., Second-Order Optimality Conditions in Multiobjective Optimization Problems, Journal of Optimization Theory and Applications, Vol. 102, pp. 37-50, 1999.Google Scholar
  22. 22.
    Wang, S., Second-Order Necessary and Sufficient Conditions in Multiobjective Programming, Numerical Functional Analysis and Optimization, Vol. 12, pp. 237-252, 1991.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • A. Guerraggio
    • 1
  • D. T. Luc
    • 2
    • 3
  1. 1.University of PaviaVareseItaly
  2. 2.University of AvignonAvignonFrance
  3. 3.Institute of MathematicsHanoiVietnam

Personalised recommendations