Application of a Vector-Valued Ekeland-Type Variational Principle for Deriving Optimality Conditions

  • G. Isac
  • C. Tammer
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 35)

Abstract

In order to show necessary conditions for approximate solutions of vector-valued optimization problems in general spaces, we introduce an axiomatic approach for a scalarization scheme. Several examples illustrate this scalarization scheme. Using an Ekeland-type variational principle by Isac [12] and suitable scalarization techniques, we prove the optimality conditions under different assumptions concerning the ordering cone and under certain differentiability assumptions for the objective function.

Keywords

Manifold 

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References

  1. 1.
    Artzner, P., Delbean, F., Eber, J.-M. and Heath, D., 1999, Coherent measures of risk. Math. Finance, 9 (3) , 203–228.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Borwein, J.M., 1982, Continuity and Differentiability Properties of Convex Operators, Proc. London Math. Soc., 44, 420–444.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Clarke, F. H., 1983 Optimization and Nonsmooth Analysis, Wiley Interscience, New York.MATHGoogle Scholar
  4. 4.
    Da Silva, A.R., 1987, Evaluation functionals are the extreme points of a basis for the dual of C 1+[a,b]. In: Jahn, J. and Krabs, W.: Recent Advances and Historical Development of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 294, 86–95.Google Scholar
  5. 5.
    Durea, M., Tammer, C., 2009,Fuzzy necessary optimality conditions for vector optimization problems, Reports of the Institute for Mathematics, Martin-Luther-University Halle-Wittenberg, Report 08. To appear in Optimization 58, 449–467.Google Scholar
  6. 6.
    Dutta, J. and Tammer, Chr., 2006, Lagrangian conditions for vector optimization in Banach spaces. Mathematical Methods of Operations Research, 64, 521–541.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Föllmer, H., Schied, A., 2004, Stochastic Finance. Walter de Gruyter, Berlin.MATHCrossRefGoogle Scholar
  8. 8.
    Gerth, C., and Weidner, P., 1990, Nonconvex Separation Theorems and Some Applications in Vector Optimization. Journal of Optimization Theory and Applications, 67, 2, 297–320.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Gæpfert, A., Riahi, H., Tammer, Chr. and Zălinescu, C., 2003, Variational Methods in Partially Ordered Spaces, Springer, New York.Google Scholar
  10. 10.
    Heyde, F., 2006, Coherent risk measures and vector optimization. In: Küfer, K.-H., Rommelfanger, H., Tammer, Chr., Winkler, K. Multicriteria Decision Making and Fuzzy Systems, SHAKER Verlag, 3–12; Reports of the Institute of Optimization and Stochastics, Martin-Luther-University Halle-Wittenberg, No. 19 (2004), 23–29.Google Scholar
  11. 11.
    Isac, G., 1983, Sur l’existence de l’optimum de Pareto, Riv. Mat. Univ. Parma (4) 9, 303–325.MathSciNetGoogle Scholar
  12. 12.
    Isac, G., 1996, The Ekeland’s principle and the Pareto ε-efficiency, Multiobjective Programming and Goal Programming: Theories and applications (M. Tamiez ed.) Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 432, 148–163.Google Scholar
  13. 13.
    Isac, G., 2004, Nuclear cones in product spaces, Pareto efficiency and Ekeland-type variational principles in locally convex spaces, Optimization, 53, 3, 253–268.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Isac, G., Bulavsky, A.V., Kalashnikov, V.V., 2002, Complementarity, Equilibrium, Efficiency and Economics. Kluwer Academic Publishers, Dordrecht.MATHGoogle Scholar
  15. 15.
    J.-B. Hiriart-Urruty, 1979, Tangent cones, generalized gradients and mathematical programmming in Banach spaces, Math. Oper. Res., 4, 79–97.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hyers, D. H., and Isac, G., and Rassias, T. M., 1997, Topics in Nonlinear Analysis and Applications, World Scientific, Singapore.MATHCrossRefGoogle Scholar
  17. 17.
    Jahn, J., 2004, Vector Optimization. Theory, Applications, and Extensions. Springer-Verlag, Berlin.MATHGoogle Scholar
  18. 18.
    B. S. Mordukhovich and Y. Shao, 1996, Nonsmooth sequential analysis in Asplund spaces, Transactions of American Mathematical Society, 348, 1235–1280.MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Nemeth, A.B., 2004, Ordered uniform spaces and variational problems, Italian Journal of Pure and Applied Mathematics, 16, 183–192.MATHMathSciNetGoogle Scholar
  20. 20.
    Ng, K. F. and Zheng, X. Y., 2005, The Fermat rule for multifunctions on Banach spaces. Mathematical Programming, Ser. A, 104, 69–90.MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Pascoletti, A. and Serafini, P., 1984, Scalarizing Vector Optimization Problems. J. Opt. Theory Appl., 42, 499–524.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Peressini, A.L., 1967, Ordered Topological Vector Space, Harper and Row, New York.Google Scholar
  23. 23.
    Phelps, R.R., 1993, Convex Functions, Monotone Operators and Differentiability (2nd ed.). Lect. Notes Math., 1364, Springer, Berlin.MATHGoogle Scholar
  24. 24.
    Rockafellar, R. T., Uryasev, S. and Zabarankin, M., 2006, Deviation measures in risk analysis and optimization, Finance Stochastics, 10, 51–74.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Schaefer, H.H., 1986, Topological Vector Spaces, Springer-Verlag, Berlin.Google Scholar
  26. 26.
    Staib, T., 1989, Notwendige Optimalitätsbedingungen in der mehrkriteriellen Optimierung mit Anwendung auf Steuerungsprobleme, Friedrich-Alexander-Universität Erlangen-Nürnberg.Google Scholar
  27. 27.
    Tammer, Chr., 1994, A Variational Principle and a Fixed Point Theorem, System Modelling and Optimization, Henry, J., and Yven, J.-P., 197, Lecture Notes in Control and Informatics Sciences, Springer, Berlin, 248–257.CrossRefGoogle Scholar
  28. 28.
    Turinici, M., 1981, Maximality Principles and Mean-Value Theorems, Anais Acad. Brasil. Ciencias 53, 653–655.MATHMathSciNetGoogle Scholar
  29. 29.
    A. Zaffaroni, 2003, Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization, 42, 1071–1086.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • G. Isac
    • 1
  • C. Tammer
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaSTN Forces KingstonCanada
  2. 2.Institute of MathematicsMartin-Luther-University Halle-WittenbergHalleGermany

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