Advertisement

Well-posedness in Vector Optimization

  • P. Loridan
Part of the Mathematics and Its Applications book series (MAIA, volume 331)

Abstract

In this paper, we give a survey on some theoretical results in vector optimization mainly related to various notions of well-posedness, approximate solutions (or efficient points) and variational principles. We lay emphasis on papers published in the past decade.

Keywords

Variational Principle Multiobjective Optimization Vector Optimization Minimal Solution Vector Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Attouch, H. and Riahi, H.: Stability results for Ekeland’s ε-variational principle and cone extremal solutions, Math. Oper. Res. 18(1993), 173–201.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bednarczuk, E. and Penot, J.-P.: Metrically well-set optimization problems, Appl. Math. Optim. 26(1992), 273–285.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bednarczuk, E.: Some stability results for vector optimization problems in partially ordered topological vector spaces, Proceedings of the First World Congress of Nonlinear Analysis, Tampa, August 1992, pp.37–48.Google Scholar
  4. 4.
    Bednarczuk, E.: An approach to well-posedness in vector optimization: consequences to stability, Control and Cybern. 23(1994), 107–122.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bednarczuk, E.: Berge-type theorems for vector optimization problems, submitted for publication in A. Fiacco (eds.), Proceedings of the Seminar on Mathematical Programming with Data Perturbations, May 1993.Google Scholar
  6. 6.
    Dentcheva, D. and Helbig, S.: On variational principles, level sets, well-posedness and ε-solutions in vector optimization, Preprint 1994, Univ. Frankfurt am Main.Google Scholar
  7. 7.
    Dolecki, S. and Malivert, C.: Polarities and stability in vector optimization, Lecture Notes in Economics and Mathematical Systems Vol. 294, Springer Verlag, Berlin, 1987, pp.96–113.CrossRefGoogle Scholar
  8. 8.
    Dolecki, S. and Malivert, C.: Stability of effiicient sets: continuity of mobile polarities, Nonlinear Analysis, TMA 12(1988), 1461–1486.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dontchev, A.L. and Zolezzi, T. Well-posed optimization problems, Lecture Notes in Mathematics Vol. 1543, Springer Verlag, Berlin, 1993.Google Scholar
  10. 10.
    Gôpfert, A. and Tammer, C.: Approximately efficient solutions in multicriteria optimization, 7th French-German Conference on Optimization, Dijon, 1994.Google Scholar
  11. 11.
    Helbig, S. and Pateva (Dentcheva), D.: On several concepts for ε-efficiency, OR Spektrum 16(1994), 179–186.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Jahn, J.: Vector optimization: theory, methods and application to design problems in engineering, Lecture Notes in Economics and Mathematical Systems Vol. 378, Springer Verlag, Berlin, 1992, pp.127–150.Google Scholar
  13. 13.
    Lemaire, B.: Approximation in multiobjective optimization, J. Global Optin. 2(1992), 117–132.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Loridan, P.: ε-solutions in vector minimization problems, J. Optin. Th. App. 1 43(1984), 265–276.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Loridan, P.: An application of Ekeland’s variational principle to generalized Stackelberg problems, Ricerche di Mathematica XLII(1993), 159–178.MathSciNetGoogle Scholar
  16. 16.
    Luc, D.T.: Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems Vol. 319, Springer Verlag, Berlin, 1989.Google Scholar
  17. 17.
    Luc, D.T., Lucchetti, R. and Malivert, C.: Convergence of efficient sets, Set- Valued Analysis, special issue “Set-convergence in nonlinear analysis and optimization” (H. Attoncl, M. Théra eds.). 1994. 1–12.Google Scholar
  18. 18.
    Lucchetti, R: Well-posedness, towards vector optimization, Lecture Notes in Economics and Mathematical Systems Vol. 294, Springer Verlag, Berlin, 1987, pp.194–207.CrossRefGoogle Scholar
  19. 19.
    Nemeth, A.B.: A nonconvex vector optimization problem, Nonlinear Analysis, TMA 10(1986), 669–678.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Nemeth A.B.: Between Pareto efficiency and Pareto ε-efficiency, Optimization 20(1989), 615–637.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Patrone, F. and Tijs, S.H.: Unified approach to approximate solutions in games and multiobjective programming, J. Optim. Th. App. 1 52(1987), 273–278.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Penot, J.-P. and Sterna-Kravat, A.: Parametrized multicriteria optimization: continuity and closedness of optimal multifunctions, J. Math. Anal. App. 1 120(1986), 150–168.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Penot, J.-P. and Sterna-Kravat, A.: Parametrized multicriteria optimization: order continuity of the marginal multifunctions, J. Math. Anal. App. 1 144(1989), 1–15.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Sawaragi, Y., Nakayama, H. and Tanino, T.: Theory of Multiobjective Optimization, Academic Press Inc., 1985.zbMATHGoogle Scholar
  25. 25.
    Staib, T.: On two generalizations of Pareto minimality, J. Optim. Th. App. 1 59(1988), 289–306.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Tammer, C.: A generalization of Ekeland’s variational principle, Optimization 25(1992), 129–141.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Tammer, C.: Variational inequalities for approximately efficient element, 7th FrenchGerman Conference on Optimization, Dijon. 1994.Google Scholar
  28. 28.
    Todorov, M.: Generic uniqueness and well-posedness in linear vector semi-infinite optimization, Proceedings of the Nineteenth Spring Conference of the Union of Bulgarian Mathematicians, Sunny Beach, 1990, pp. 413–418.Google Scholar
  29. 29.
    Todorov, M.: Linear vector optimization. Properties of the efficient sets, Serdica Bulg. Math. Publ. 18(1992), 179–185.zbMATHGoogle Scholar
  30. 30.
    Todorov, M.: Well-posedness in the linear vector semi-infinite optimization, Proceedings of the tenth international Conference on multiple criteria decision making, Taiwan, Taipei, July 1992, Vol 4, pp. 1–10.Google Scholar
  31. 31.
    Todorov, M.: Kuratowski convergence of the efficient sets in the parametric linear vector semi-infinite optimization, Preprint no 15, Inst. Math., Sofia, 1993.Google Scholar
  32. 32.
    Todorov, M., Georgiev, P. and Helbig, S.: Well-defined efficient points in the vector optimization, Workshop on well-posedness and stability in optimization, September 1993, Sozopol, Bulgaria.Google Scholar
  33. 33.
    Valyi, I.: A general maximality principle and a fixed point theorem in uniform space, Per. Math. Hung. 16(1985), 127–134.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Valyi, I.: Epsilon solutions and duality in vector optimization, Working paper, WP 87-43. 1987, IASA. Laxembourg, Austria.Google Scholar
  35. 35.
    White, D.J.: Epsilon efficiency, J. Optin. Th. App. 1 49(1986), 319–337.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • P. Loridan
    • 1
  1. 1.CERMSEMUniversité de Paris 1Paris Cedex 13France

Personalised recommendations