Advertisement

Journal of Global Optimization

, Volume 59, Issue 1, pp 191–205 | Cite as

Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems

  • C. S. Lalitha
  • Prashanto ChatterjeeEmail author
Article

Abstract

In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.

Keywords

Levitin–Polyak well-posedness Quasiconvexity Efficiency  Upper semicontinuity Hausdorff convergence 

Mathematics Subject Classification

49K40 90C26 90C29 

Notes

Acknowledgments

The authors are really grateful to the reviewers for the valuable comments and suggestions which helped in improving the paper.

References

  1. 1.
    Bednarczuk, E.: Well-posedness of vector optimization problems. In: Jahn, J., Krabs, W. (eds.), Recent Advances and Historical Development of Vector Optimization Problems, Lecture Notes in Econom. and Math. Systems, vol. 294, pp. 51–61. Springer, Berlin (1987)Google Scholar
  2. 2.
    Bednarczuk, E.: An approach to well-posedness in vector optimization consequences to stability. Parametric optimization. Control Cybern. 23, 107–122 (1994)Google Scholar
  3. 3.
    Crespi, G.P., Guerraggio, A., Rocca, M.: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213–226 (2007)CrossRefGoogle Scholar
  4. 4.
    Deng, S.: Coercivity properties and well-posedness in vector optimization. RAIRO Oper. Res. 37, 195–208 (2003)CrossRefGoogle Scholar
  5. 5.
    Dentcheva, D., Helbig, S.: On variational principles, level sets, well-posedness, and \(\varepsilon \text{-solutions }\) in vector optimization. J. Optim. Theory Appl. 89, 325–349 (1996)CrossRefGoogle Scholar
  6. 6.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems, vol. 1543. Springer, Berlin (1993)Google Scholar
  7. 7.
    Huang, X.X., Yang, X.Q.: Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J. Optim. 17, 243–258 (2006)CrossRefGoogle Scholar
  8. 8.
    Huang, X.X., Yang, X.Q.: Levitin–Polyak well-posedness of constrained vector optimization problems. J. Global Optim. 37, 287–304 (2007)CrossRefGoogle Scholar
  9. 9.
    Huang, X.X., Yang, X.Q.: Further study on the Levitin–Polyak well-posedness of constrained convex vector optimization problems. Nonlinear Anal. 75, 1341–1347 (2012)CrossRefGoogle Scholar
  10. 10.
    Ioffe, A.D., Lucchetti, R.E., Revalski, J.P.: Almost every convex or quadratic programming problem is well posed. Math. Oper. Res. 29, 369–382 (2004)CrossRefGoogle Scholar
  11. 11.
    Kettner, L.J., Deng, S.: On well-posedness and Hausdorff convergence of solution sets of vector optimization problems. J. Optim. Theory Appl. 153, 619–632 (2012)CrossRefGoogle Scholar
  12. 12.
    Konsulova, A.S., Revalski, J.P.: Constrained convex optimization problems-well-posedness and stability. Numer. Funct. Anal. Optim. 15, 889–907 (1994)CrossRefGoogle Scholar
  13. 13.
    Lalitha, C.S., Chatterjee, P.: Well-posedness and stability in vector optimization using Henig proper efficiency. Optimization. 62, 155–165 (2013)Google Scholar
  14. 14.
    Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 7, 764–767 (1966)Google Scholar
  15. 15.
    Loridan, P. (1995) Well-posedness in vector optimization. In: Lucchetti, R., Revalski, J. (eds.) Recent Developments in Well-Posed Variational Problems. Math. Appl. vol. 331, pp. 171–192. Kluwer, DordrechtGoogle Scholar
  16. 16.
    Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Econom, and Math, Systems, vol. 319. Springer, Berlin (1989)Google Scholar
  17. 17.
    Lucchetti, R.: Convexity and Well-Posed Problems. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. vol. 22. Springer, New York (2006)Google Scholar
  18. 18.
    Lucchetti, R.: Well-posedness, towards vector optimization. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization Problems, Lecture Notes in Econom, and Math. Systems, vol. 294, pp. 194–207. Springer, Berlin (1987)CrossRefGoogle Scholar
  19. 19.
    Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58, 375–385 (2003)CrossRefGoogle Scholar
  20. 20.
    Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)CrossRefGoogle Scholar
  21. 21.
    Tanaka, T.: Generalized semicontinuity and existence theorems for cone saddle points. Appl. Math Optim. 36, 313–322 (1997)CrossRefGoogle Scholar
  22. 22.
    Todorov, M. et al.: Well-posedness in the linear vector semi-infinite optimization. Multiple criteria decision making expand and enrich the domains of thinking and applications. In: Yu, P.L. (ed.) Proceedings of the Tenth International Conference, pp. 141–150. Springer, New York (1994)Google Scholar
  23. 23.
    Todorov, M.: Kuratowski convergence of the efficient sets in the parametric linear vector semi-infinite optimization. Eur. J. Oper. Res. 94, 610–617 (1996)CrossRefGoogle Scholar
  24. 24.
    Tykhonov, A.N.: On the stability of the functional optimization problem. U.S.S.R. Comput. Math. Math. Phys. 6, 28–33 (1966)CrossRefGoogle Scholar
  25. 25.
    Zolezzi, T.: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. 25, 437–453 (1995)CrossRefGoogle Scholar
  26. 26.
    Zolezzi, T.: Well-posedness and optimization under perturbations. Ann. Oper. Res. 101, 351–361 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Delhi South CampusNew DelhiIndia
  2. 2.Department of Mathematics, St. Stephen’s CollegeUniversity of DelhiDelhiIndia

Personalised recommendations