On Well-Posedness and Hausdorff Convergence of Solution Sets of Vector Optimization Problems

  • Laura J. Kettner
  • Sien DengEmail author


In this paper, we refine and improve the results established in a 2003 paper by Deng in a number of directions. Specifically, we establish a well-posedness result for convex vector optimization problems under a condition which is weaker than that used in the paper. Among other things, we also obtain a characterization of well-posedness in terms of Hausdorff distance of associated sets.


Well-posedness Weakly efficient solution Hausdorff distance Convexity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Deng, S.: Coercivity properties and well-posedness in vector optimization. RAIRO. Rech. Opér. 3, 195–208 (2003) CrossRefGoogle Scholar
  2. 2.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Springer, Berlin (1993) zbMATHGoogle Scholar
  3. 3.
    Deng, S., Yang, X.Q.: Weak sharp minima in multicriteria linear programming. SIAM J. Control Optim. 2, 456–460 (2004) MathSciNetGoogle Scholar
  4. 4.
    Zheng, X.Y., Yang, X.Q.: Weak sharp minima for piecewise linear multiobjective optimization in normed spaces. Nonlinear Anal. 3771–3779 (2008) Google Scholar
  5. 5.
    Xu, S., Li, S.J.: Optimality conditions for weak ψ-sharp minima in vector optimization problems. Preprint (2011) Google Scholar
  6. 6.
    Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 2, 391–409 (2005) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Luo, H.L., Huang, X.X., Peng, J.W.: Generalized well-posedness of vector optimization problems. Preprint Google Scholar
  8. 8.
    Crespi, C.P., Guerraggio, A., Rocca, M.: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 1, 213–226 (2007) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Deng, S.: Well-posed problems and error bounds in optimization. In: Kukushima, M., Qi, L. (eds.) Reformulations: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, vol. 22, pp. 117–126. Kluwer, Dordrecht (1998) Google Scholar
  10. 10.
    Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization, Set-Valued and Variational Analysis. Springer, Berlin (2005) zbMATHGoogle Scholar
  11. 11.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1991) Google Scholar
  12. 12.
    Deng, S.: On approximate solutions in convex vector optimization. SIAM J. Control Optim. 6, 2128–2136 (1997) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

Personalised recommendations