Advertisement

New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization

  • P. H. Sach
Article

Abstract

In this paper, we introduce a new generalized convexity notion for set-valued maps, called ic-cone-convexlikeness, and use it as the main tool to derive an alternative theorem and necessary conditions for efficient, weakly efficient, and Benson properly efficient solutions of the problem of minimizing a set-valued map subject to set-valued constraints. Our results are valid for a class of optimization problems broader than that of the problems considered in Refs. 1--6 and generalize the corresponding results of these references.

Keywords

Convexlikeness set-valued maps alternative theorems efficiency Benson proper efficiency 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Li, Z. F., Wang, S. Y. 1994Lagrange Multipliers and Saddle Points in Multiobjective ProgrammingJournal of Optimization Theory and Applications836381Google Scholar
  2. 2.
    Li, Z. F. 1998Benson Proper Efficiency in the Vector Optimization of Set-Valued MapsJournal of Optimization Theory and Applications98623649Google Scholar
  3. 3.
    Chen, G.Y., Rong, D.W. 1998Characterization of the Benson Proper Efficiency for Nonconvex Vector OptimizationJournal of Optimization Theory and Applications98365384Google Scholar
  4. 4.
    Yang, X. M., Yang, X. Q., Chen, G. Y. 2000Theorems of the Alternative and Optimization with Set-Valued MapsJournal of Optimization Theory and Applications107627640Google Scholar
  5. 5.
    Yang, X. M., Li, D., Wang, S. Y. 2001Near-Subconvexlikeness in Vector Optimization with Set-Valued FunctionsJournal of Optimization Theory and Applications110413427Google Scholar
  6. 6.
    Huang, Y. W. 2002Optimality Conditions for Vector Optimization with Set-Valued MapsBulletin of the Australian Mathematical Society66317330Google Scholar
  7. 7.
    Fan, K. 1953Minimax TheoremsProceedings of the National Academy of Sciences of the USA394247Google Scholar
  8. 8.
    Jeyakumar, V. 1985Convexlike Alternative Theorems and Mathematical ProgrammingOptimization16643652Google Scholar
  9. 9.
    Paeck, S. 1992Convexlike and Concavelike Conditions in Alternative, Minimax, and Minimization TheoremsJournal of Optimization Theory and Applications14317332Google Scholar
  10. 10.
    Li, Z. 1999A Theorem of the Alternative and Its Application to the Optimization of Set-Valued MapsJournal of Optimization Theory and Application100365375Google Scholar
  11. 11.
    Yang, X. M. 1992Alternative Theorems and Optimality Conditions with Weakened ConvexityOpsearch29125135Google Scholar
  12. 12.
    Sach, P. H. 2003Nearly Subconvexlike Set-Valued Maps and Vector Optimization ProblemsJournal of Optimization Theory and Applications119335356Google Scholar
  13. 13.
    Dauer, J. P., Saleh, O. A. 1993A Characterization of Proper Minimal Points as Solutions of Sublinear Optimization ProblemsJournal of Mathematical Analysis and Applications178227246Google Scholar
  14. 14.
    Song, W. 1998Duality in Set-Valued OptimizationDissertations Mathematicae375167Google Scholar
  15. 15.
    Brecker, W. W., Kassay, G. 1997A Systematization of Convexity Concepts for Sets and FunctionsJournal of Convex Analysis4109127Google Scholar
  16. 16.
    Jeyakumar, V., Gwinner, J. 1991Inequality Systems and OptimizationJournal of Mathematical Analysis and Applications1595157Google Scholar
  17. 17.
    Aleman, A. 1985On Some Generalizations of Convex Sets and Convex FunctionsMathematica: Revue d’Analyse Numérique et de Théorie de l’Approximation1416Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • P. H. Sach
    • 1
  1. 1.Professor, Institute of MathematicsHanoiVietnam

Personalised recommendations