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Mathematical Modelling in Economics pp 345–358Cite as

An Approach to Optimality Conditions in Vector and Scalar Optimization

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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

  • Arrow, K.J and Enthoven, A.C (1961) “Quasi concave Programming” Econometrica 29, 779–800

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  • Ben-Tal, A. and Zowe, J. (1985): “Directional derivatives in Nonsmooth Optimization” J.O.T.A. 47, 483–490

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  • Cambini A., Martein, L. (1991): “Optimality conditions in vector and scalar optimization”, report 50, Dept of Statistics and Applied Mathematics, University of Pisa, 1991.

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  • Cambini, R. (1992): “Alcune condizioni di ottimalitk relative ad un insieme stellato”, report 54, Dept. of Statistics and Applied Mathematics, University of Pisa, 1992.

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  • Mangasarian, O. L. (1969): Nonlinear Programming, McGraw-Hill, New York.

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  • Martein, L.(1989): “Some results on regularity in vector optimization” Optimization, 20, 787–798.

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  • Rockafellar, R. T. (1970): Convex Analysis, Princeton, New Jersey.

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Author information

Authors and Affiliations

  1. Department of Statistics and Applied Mathematics, University of Pisa, Italy

    Laura Martein

Authors
  1. Alberto Cambini
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  2. Laura Martein
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Editor information

Editors and Affiliations

  1. Department of Economics, University of British Columbia, #997-1873 East Mall, V6T 1W5, Vancouver, B.C., Canada

    Prof. Dr. W. Erwin Diewert

  2. University of Hong Kong, K.K. Leung Building, Hong Kong

    Prof. Dr. Klaus Spremann

  3. Department of Economics, University of Ulm, Helmholtzstraße 18, D-89069, Ulm/Donau, Germany

    Prof. Dr. Frank Stehling

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© 1993 Springer-Verlag Berlin · Heidelberg

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Cambini, A., Martein, L. (1993). An Approach to Optimality Conditions in Vector and Scalar Optimization. In: Diewert, W.E., Spremann, K., Stehling, F. (eds) Mathematical Modelling in Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78508-5_34

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  • DOI: https://doi.org/10.1007/978-3-642-78508-5_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-78510-8

  • Online ISBN: 978-3-642-78508-5

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