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Convergence of Approximate Solutions and Values in Parametric Vector Optimization

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Vector Variational Inequalities and Vector Equilibria

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 38))

Abstract

New concepts of approximate values and solutions for vector optimization problems are introduced. Then, under conditions of minimal character, we present convergence results involving the above mentioned concepts.

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References

  1. Attouch H. and Rihai H., “Stability results for Ekeland’s ε-variational principle and cone extremal solutions”. Mathematics of Operations Research, Vol. 18, 1993, pp. 173–201.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubin J.P. and Frankovska H., “Set valued Analysis”. Birkhauser, Systems and Control; Foundations and Algorithms, 1990.

    Google Scholar 

  3. Bednarczuk E., “Some stability results for vector optimization problems in partially ordered topological vector spaces”. Proceedings of the First Congress of “Nonlinear Analysis”, Tampa, 1992, pp. 37–48.

    Google Scholar 

  4. Bednarczuk E., “An approach to well-posedness in vector optimization: consequences to stability”. Control and Cybernetics, Vol. 23, 1994, pp. 107–122.

    MathSciNet  MATH  Google Scholar 

  5. Dentcheva D. and Helbig S., “On variational principles, level sets, well-posedness and ε-solutions in vector optimization”. Preprint 1994, University of Frankfurt am Main.

    Google Scholar 

  6. Dolecki S. and Malivert C., “Polarities and stability in vector optimization”. Lecture Notes in Econ. and Mathem. Systems, Vol. 294, 1987, pp. 96–113.

    Google Scholar 

  7. Dolecki S. and Malivert C., “Stability of efficient sets: continuity of mobile polarities”. Nonlinear Analysis: Theory, Methods and Applications, Vol. 12, 1988, pp. 1461–1486.

    Article  MathSciNet  MATH  Google Scholar 

  8. Helbig S. and Pateva D., “On several concepts for ε-efficiency”. Operation Research Spektrum, 1994.

    Google Scholar 

  9. Kuratowski C., Topology, Academic Press, New-York, 1966.

    Google Scholar 

  10. Lemaire B., “Approximation in multiobjective optimization”. Jour. of Global Optimiz., Vol. 2, 1992, pp. 117–132.

    Article  MathSciNet  MATH  Google Scholar 

  11. Lignola M.B. and Morgan J., “Semicontinuities of marginal functions in a sequential setting”. Optimization, Vol. 24, 1992, pp. 307–318.

    Article  MathSciNet  Google Scholar 

  12. Lignola M.B. and Morgan J., “Semicontinuity and episemicontinuity: equivalence and applications”. Bollettino Unione Matematica Italiana, Vol.7, 8-B, 1994, pp. 1–16.

    MathSciNet  Google Scholar 

  13. Lignola M.B. and Morgan J., “Topological existence and stability for Stackelberg problems”. Jou. of Optimiz. Theory and Appls., Vol. 84, 1995, pp. 145–169.

    Article  MathSciNet  MATH  Google Scholar 

  14. Loridan P., “ε-solutions in vector minimization problems”. Jour. of Optimiz. Theory and Appls., Vol. 43, 1984, pp. 265–276.

    Article  MathSciNet  MATH  Google Scholar 

  15. Loridan P., “Well-posedness in vector optimization”. In: Recent developments in well-posed variational problems, edited by R. Lucchetti and J. Revalski, Kluwer Academic Publishers, 1995, pp. 171–192.

    Google Scholar 

  16. Loridan P. and Morgan J., “New results on approximate solutions in two level optimization”. Optimization, Vol. 20, 1989, pp. 819–836.

    Article  MathSciNet  MATH  Google Scholar 

  17. Loridan P. and Morgan J., “On strict s-solutions for a two-level optimization problem”. In: Operational Research Proceedings, edited by W. Buhler et al., Springer-Verlag, Berlin, 1992, pp. 165–172.

    Google Scholar 

  18. Loridan P., Morgan J. and Raucci R., “Convergence of minimal and approximate minimal elements of sets in partially ordered vector spaces”. To appear in Jou. of Mathem. Analysis and Appls.

    Google Scholar 

  19. Luc D.T., Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 319, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  20. Luc D.T., Lucchetti R. and Malivert C., “Convergence of efficient sets”. Set-Valued Analysis, Vol. 2, 1994, pp. 207–218.

    Article  MathSciNet  MATH  Google Scholar 

  21. Nemeth A.B., “A non convex vector optimization problem”. Nonlinear Analysis: Theory, Methods and Applications, Vol. 10, 1986, pp. 669–678.

    Article  MathSciNet  MATH  Google Scholar 

  22. Nemeth A.B., “Between Pareto efficiency and Pareto ε-efficiency”. Optimization, Vol. 20, 1989, pp. 615–637.

    Article  MathSciNet  MATH  Google Scholar 

  23. Patrone F. and Tijs S.H., “Unified approach to approximate solutions in games and multiobjective programming”. Jou. of Optimiz. Theory and Appls., Vol. 52, 1987, pp. 273–278.

    Article  MathSciNet  MATH  Google Scholar 

  24. Penot J.P. and Sterna-Karvat A., “Parametrized multi-criteria optimization: continuity and closedness of optimal multifunctions”. Jou. of Mathem. Analysis and Appls., Vol. 120, 1986, pp. 150–168.

    Article  MATH  Google Scholar 

  25. Penot J.P. and Sterna-Karvat A., “Parametrized multicriteria optimization: order continuity of the marginal multifunction”. Jou. of Mathem. Analysis and Appls., Vol. 144, 1989, pp. 1–15.

    Article  MATH  Google Scholar 

  26. Penot J.P. and Théra M., “Semi-continuité des applications et des multiapplications”. Comptes Rendus de l’Académie des Sciences de Paris, tome 288, Série A, 1979, pp. 241–244.

    Google Scholar 

  27. Raucci R., “Esistenza per problemi di Stackelberg con funzioni a valori in spazi parzialmente ordinati”. Rendiconti dell’Accademia delle Scienze Fisiche e Matematiche dell’Università di Napoli, Serie IV, Vol. LXI, 1994.

    Google Scholar 

  28. Sawaragi Y., Nakayama H. and Tanino T., “Theory of multiobjective optimization”. Academic Press Inc., 1985.

    Google Scholar 

  29. Staib T., “On two generalizations of Pareto minimality”. Jou. of Optimiz. Theory and Appls., Vol. 59, 1988, pp. 289–306.

    MathSciNet  MATH  Google Scholar 

  30. Tammer C., “A generalization of Ekeland’s variational principle”. Optimization, Vol. 25, 1992, pp. 129–141.

    Article  MathSciNet  MATH  Google Scholar 

  31. Tammer C., “Variational inequalities for approximately efficient elements”. 7th French-German Conference on Optimization, Dijon, 1994.

    Google Scholar 

  32. Valyi I., “A general maximality principle and a fixed point theorem in uniform space”. Periodica Mathematica Hungarica, Vol. 16, 1985, pp. 127–134.

    Article  MathSciNet  MATH  Google Scholar 

  33. White D.J., “Epsilon efficiency”. Jou. of Optimiz. Theory and Appls., Vol. 49, 1986, pp. 319–337.

    Article  MATH  Google Scholar 

  34. Yu P.L., “Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Problems with multiobjectives”. Jou. of Optimiz. Theory and Appls., Vol. 14, 1974, pp. 319–337.

    Article  MATH  Google Scholar 

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© 2000 Kluwer Academic Publishers

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Loridan, P., Morgan, J. (2000). Convergence of Approximate Solutions and Values in Parametric Vector Optimization. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0299-5_19

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  • DOI: https://doi.org/10.1007/978-1-4613-0299-5_19

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7985-0

  • Online ISBN: 978-1-4613-0299-5

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