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An Approach to Optimality Conditions in Vector and Scalar Optimization

  • Alberto Cambini
  • Laura Martein

Abstract

The implication of concavity in economics have suggested in the scalar case several kinds of generalization starting from the pioneering work of Arrow-Enthoven (1961).

The aim of this paper is to point out the role played by generalized concavity and by the tangent cone to the feasible region at a point, in stating several necessary and/or sufficient optimality conditions for a vector and scalar optimization problem.

Furthermore, in deriving F.John optimality conditions, the role of separations theorems is analyzed in order to suggest suitable formulations of Kuhn-Tucker conditions and a way for studying regularity conditions.

Keywords

Scalar Case Tangent Cone Vector Optimization Problem Closed Convex Cone Separation Theorem 
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.

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References

  1. Arrow, K.J and Enthoven, A.C (1961) “Quasi concave Programming” Econometrica 29, 779–800Google Scholar
  2. Ben-Tal, A. and Zowe, J. (1985): “Directional derivatives in Nonsmooth Optimization” J.O.T.A. 47, 483–490CrossRefGoogle Scholar
  3. Cambini A., Martein, L. (1991): “Optimality conditions in vector and scalar optimization”, report 50, Dept of Statistics and Applied Mathematics, University of Pisa, 1991.Google Scholar
  4. Cambini, R. (1992): “Alcune condizioni di ottimalitk relative ad un insieme stellato”, report 54, Dept. of Statistics and Applied Mathematics, University of Pisa, 1992.Google Scholar
  5. Mangasarian, O. L. (1969): Nonlinear Programming, McGraw-Hill, New York.Google Scholar
  6. Martein, L.(1989): “Some results on regularity in vector optimization” Optimization, 20, 787–798.CrossRefGoogle Scholar
  7. Rockafellar, R. T. (1970): Convex Analysis, Princeton, New Jersey.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1993

Authors and Affiliations

  • Alberto Cambini
  • Laura Martein
    • 1
  1. 1.Department of Statistics and Applied MathematicsUniversity of PisaItaly

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