Journal of Optimization Theory and Applications

, Volume 142, Issue 2, pp 323–342 | Cite as

Optimality Conditions for Vector Optimization Problems



In this paper, some necessary and sufficient optimality conditions for the weakly efficient solutions of vector optimization problems (VOP) with finite equality and inequality constraints are shown by using two kinds of constraints qualifications in terms of the MP subdifferential due to Ye. A partial calmness and a penalized problem for the (VOP) are introduced and then the equivalence between the weakly efficient solution of the (VOP) and the local minimum solution of its penalized problem is proved under the assumption of partial calmness.


Vector optimization problem Optimality condition Partial calmness Exact penalization MP subdifferential 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aghezzaf, B., Hachimi, M.: Generalized invexity and duality in multiobjective programming problems. J. Glob. Optim. 18, 91–101 (2000) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005) MATHGoogle Scholar
  3. 3.
    Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Deng, S.: Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. J. Optim. Theory Appl. 96, 123–131 (1998) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Flores-Bazãn, F.: Ideal, weakly efficient solutions for vector optimization problems. Math. Program. Ser. A 93, 453–475 (2002) MATHCrossRefGoogle Scholar
  6. 6.
    Liang, Z.A., Huang, H.X., Pardalos, P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Glob. Optim. 27, 447–471 (2003) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) Google Scholar
  8. 8.
    Yuan, D.H., Chinchuluun, A., Liu, X.L., Pardalos, P.M.: Optimality conditions and duality for multiobjective programming involving (C;α;ρ;d)-type I functions. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 73–87. Springer, Berlin (2007) CrossRefGoogle Scholar
  9. 9.
    Deng, S., Yang, X.Q.: Weak sharp minima in multicriteria linear programming. SIAM J. Optim. 15, 456–460 (2004) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983) MATHGoogle Scholar
  11. 11.
    Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12, 229–236 (1974) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res. 9, 159–189 (1984) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969). Reprinted as Classics. Applied Mathematics, vol. 10. SIAM, Philadelphia (1994) MATHGoogle Scholar
  14. 14.
    Mordukhovich, B.S.: On necessary conditions for an extremum in nonsmooth optimization. Sov. Math. Dokl. 283, 215–220 (1985) MathSciNetGoogle Scholar
  15. 15.
    Michel, P., Penot, J.-P.: Calcus sous-différentiel pour des fonctions Lipschitziennes et non Lipschitziennes. C. R. Acad. Sci. Paris Ser. I Math. 12, 269–272 (1984) MathSciNetGoogle Scholar
  16. 16.
    Michel, P., Penot, J.-P.: A generalized derivative for calm and stable functions. Diff. Integral Equ. 5, 433–454 (1992) MATHMathSciNetGoogle Scholar
  17. 17.
    Treiman, J.S.: Lagrange multipliers for nonconvex generalized gradients with equality, inequality, and set constraints. SIAM J. Control Optim. 37, 1313–1329 (1999) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ye, J.J.: Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15, 252–274 (2004) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rockafellar, R.T.: Proximal subgradient, marginal functions, and augmented Lagrangians in nonsmooth optimization. Math. Oper. Res. 6, 427–437 (1981) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). English translation, Wiley/Interscience MATHGoogle Scholar
  22. 22.
    Birge, J.R., Qi, L.: Semiregularity and generalized subdifferentials with applications to optimization. Math. Oper. Res. 18, 982–1005 (1993) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  24. 24.
    Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China
  2. 2.School of Mathematics and InformationChina West Normal UniversityNanchongPeople’s Republic of China
  3. 3.Institute of Applied MathematicsNational Cheng-Kung UniversityTainanTaiwan
  4. 4.National Center for Theoretical SciencesTainanTaiwan

Personalised recommendations