Skip to main content
SpringerLink
Log in

First and Second Order Optimality Conditions Using Approximations for Nonsmooth Vector Optimization in Banach Spaces

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We use the first and second order approximations of mappings to establish both necessary and sufficient optimality conditions for unconstrained and constrained nonsmooth vector optimization problems. Ideal solutions, efficient solutions, and weakly efficient solutions are considered. The data of the problems need not even be continuous. Some often imposed compactness assumptions are also relaxed. Examples are provided to compare our results and some known recent results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. JOURANI, A., and THIBAULT, L., Appoximations and Metric Regularity in Mathematical Programming in Banach Spaces, Mathematics of Operations Research, Vol. 18, pp. 390–400, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  2. ALLALI, K., and AMAHROQ, T., Second-Order Approximations and Primal and Dual Necessary Optimality Conditions, Optimization, Vol. 40, pp. 229–246, 1997.

    MathSciNet  MATH  Google Scholar 

  3. PENOT, J. P., Second-Order Generalized Derivatives: Comparison of Two Types of Epiderivatives, Advances in Optimization, Edited by W. Oettli and D. Pallaschke, Springer Verlag, Berlin, Germany, pp. 52–76, 1992.

    Google Scholar 

  4. HIRIART-URRUTY, J. B., STRODIOT, J. J., and NGUYEN, V. H., 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.

    Article  MathSciNet  MATH  Google Scholar 

  5. 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.

  6. LIU, L. P., The Second-Order Conditions of Nondominated Solutions for C 1,1 Generalized Multiobjective Mathematical Programming, Systems Sciences and Mathematical Sciences, Vol. 4, pp. 128–138, 1991.

    MATH  Google Scholar 

  7. GUERRAGGIO, A., and LUC, D. T., Optimality Conditions for C 1,1 Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 109, pp. 615–629, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  8. GUERRAGGIO, A., and LUC, D. T., Optimality Conditions for C 1,1 Constrained Multiobjective Problems, Journal of Optimization Theory and Applications, Vol. 116, pp. 117–129, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  9. DEMYANOV, V. F., and RUBINOV, A. M., Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main, Germany, 1995.

    MATH  Google Scholar 

  10. STUDNIARSKI, M., Second-Order Necessary Conditions for Optimality in Nonsmooth Nonlinear Programming, Journal of Mathematical Analysis and Applications, Vol. 154, pp. 303–317, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  11. GINCHEV, I., and GUERRAGGIO, A., Second-Order Optimality Conditions in Nonsmooth Unconstrained Optimization, Pliska Studia Mathematica Bulgarica, Vol. 12, pp. 39–50, 1998.

    MathSciNet  MATH  Google Scholar 

  12. GINCHEV, I., Higher-Order Optimality Conditions in Nonsmooth, Optimization, Optimization, Vol. 51, pp. 47–72, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  13. GINCHEV, I., GUERRAGGIO, A., and ROCCA., M., First-Order Conditions for C 0,1 Constrained Vector Optimization, Variational Analysis and Applications, Nonconvex Optimization Application, Springer, New York, Vol. 79, pp. 427– 450, 2005.

    Google Scholar 

  14. GINCHEV, I., GUERRAGGIO, A., and ROCCA., M., From Scalar to Vector Optimization, Applications of Mathematics, Vol. 51, pp. 5–36, 2006.

  15. KHANH, P. Q., LUC, D. T., and TUAN, N. D., Local Uniqueness of Solutions for Equilibrium Problems, Advances in Nonlinear Variational Inequalities, Vol. 9, pp. 13–27, 2006.

  16. JEYAKUMAR, V., and LUC, D. T., Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C 1 -Optimization, SIAM Journal on Control and Optimization, Vol. 36, pp. 1815–1832, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  17. JEYAKUMAR, V., and WANG, X., Approximate Hessian Matrices and Second-Order Optimality Conditions for Nonlinear Programming Problem with C 1 Data, Journal of the Australian Mathematical Society, Vol. 40B, pp. 403–420, 1999.

    MathSciNet  Google Scholar 

  18. GUERRAGGIO, A., LUC., D. T., and MINH., N. B., Second-Order Optimality Conditions for C 1 Mulliobjective Programming Problems, Acta Mathematica Vietnamica, Vol. 26, pp. 257–268, 2001.

    MathSciNet  MATH  Google Scholar 

  19. LUC, D. T., Second-Order Optimality Conditions for Problems with Continuously Differentiable Data, Optimization, Vol. 51, pp. 497–510, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  20. LUC, D., T., The Fréchet Approximate Jacobian and Local Uniqueness in Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 268, pp. 629–646, 2002.

  21. WANG, X., and JEYAKUMAR, V., A Sharp Lagrange Multiplier Rule for Nonsmooth Mathematical Programming Problems Involving Equality Constraints, SIAM Journal on Optimization, Vol. 10, pp. 1136–1148, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  22. LUC, D. T., A Multiplier Rule in Multiobjective Programming Problems with Continuous Data, SIAM Journal on Optimization, Vol. 13, pp. 168–178, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  23. AMAHROQ, T., and TAA, A., On Lagrange - Kuhn - Tucker Multipliers for Multiobjective Optimization Problems, Optimization, Vol. 41, pp. 159–172, 1997.

    MATH  MathSciNet  Google Scholar 

  24. KHANH, P. Q., and LUU, L. M., Necessary Optimality Conditions in Problems Involving Set - Valued Maps with Parameters, Acta Mathematica Vietnamica, Vol. 26, pp. 279–295, 2001.

    MathSciNet  MATH  Google Scholar 

  25. KHANH, P. Q., and LUU, L. M., Multifunction Optimization Problems Involving Parameters: Necessary Optimality Conditions, Optimization, Vol. 51, pp. 577– 595, 2002.

  26. AMAHROQ, T., and GADHI, N., Second-Order Optimality Conditions for the Extremal Problem under Inclusion Constraints, Journal of Mathematical Analysis and Applications, Vol. 285, pp. 74–85, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  27. MORDUKHOVICH, B. S., Maximum Principle in the Optimal Time Control Problem with Nonsmooth Constraints, Soviet Journal of Applied Mathematics and Mechanics, Vol. 40, pp. 960–968, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  28. MORDUKHOVICH, B. S., Metric Approximations and Necessary Optimality Conditions for General Classes of Nonsmooth Extremal Problems, Soviet Mathematics Doklady, Vol. 22, pp. 526–530, 1980.

    MATH  Google Scholar 

  29. IOFFE, A. D., Nonsmooth Analysis: Differential Calculus of Nondifferentiable Mappings, Transactions of American Mathematical Society, Vol. 266, pp. 1–56, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  30. CLARKE, F. H., Optimization and Nonsmooth Analysis, Wiley Interscience, New York, NY, 1983.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, International University of Hochiminh City, Hochiminh City, Vietnam

    P. Q. Khanh (Professor)

  2. Department of Mathematics, University of Natural Sciences of Hochiminh City, Hochiminh City, Vietnam

    N. D. Tuan (Lecturer)

Authors
  1. P. Q. Khanh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. D. Tuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by F. Giannessi

This work was partially supported by the National Basic Research Program in Natural Sciences of Vietnam.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khanh, P.Q., Tuan, N.D. First and Second Order Optimality Conditions Using Approximations for Nonsmooth Vector Optimization in Banach Spaces. J Optim Theory Appl 130, 289–308 (2006). https://doi.org/10.1007/s10957-006-9103-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-006-9103-y

Keywords

Search

Navigation