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Optimality conditions for maximizations of set-valued functions

  • H. W. Corley
Contributed Papers

Abstract

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined, and necessary and sufficient optimality conditions are established. In particular, an analogue of the Fritz John necessary optimality conditions is proved using a notion of derivative defined in terms of tangent cones.

Key Words

Optimality conditions set-valued functions cones tangent cones 

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References

  1. 1.
    Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529, 1979.Google Scholar
  2. 2.
    Corley, H. W.,Duality Theory for Maximizations with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 84, pp. 560–568, 1981.Google Scholar
  3. 3.
    Corley, H. W.,Existence and Lagrangian Duality for Maximizations of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 54, pp. 489–501, 1987.Google Scholar
  4. 4.
    Klein, K., andThompson, A. C.,Theory of Correspondences, John Wiley, New York, New York, 1984.Google Scholar
  5. 5.
    Zangwill, W. I.,Nonlinear Programming: a Unified Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.Google Scholar
  6. 6.
    Hogan, W. W.,Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591–603, 1973.Google Scholar
  7. 7.
    Robinson, S. M.,Generalized Equations and Their Solutions, Part 1: Basic Theory, Mathematical Programming Study, Vol. 10, pp. 128–141, 1979.Google Scholar
  8. 8.
    Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley, New York, New York, 1983.Google Scholar
  9. 9.
    Corley, H. W.,On Optimality Conditions for Maximizations with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 46, pp. 67–68, 1985.Google Scholar
  10. 10.
    Bazaraa, M. S., Goode, J. J., andNashed, M. Z.,On the Cones of Tangents with Applications to Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 389–426, 1974.Google Scholar
  11. 11.
    Borwein, J.,Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.Google Scholar
  12. 12.
    Borwein, J.,Multivalued Convexity and Optimization: a Unified Approach to Inequality and Equality Constraints, Mathematical Programming, Vol. 13, pp. 183–199, 1977.Google Scholar
  13. 13.
    Guignard, M.,Generalized Kuhn-Tucker Conditions for Mathematical Programming Problems in a Banach Space, SIAM Journal on Control, Vol. 7, pp. 232–241, 1969.Google Scholar
  14. 14.
    Varaiya, P. P.,Nonlinear Programming in Banach Spaces, SIAM Journal on Applied Mathematics, Vol. 15, pp. 284–293, 1967.Google Scholar
  15. 15.
    Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, John Wiley, New York, New York, 1984.Google Scholar
  16. 16.
    Hiriart-Urruty, J. B.,Tangent Cones, Generalized Gradients, and Mathematical Programming in Banach Spaces, Mathematics of Operations Research, Vol. 4, pp. 79–97, 1979.Google Scholar
  17. 17.
    Laurent, P. J.,Approximation et Optimisation, Hermann, Paris, France, 1972.Google Scholar
  18. 18.
    Edwards, R.,Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York, New York, 1965.Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • H. W. Corley
    • 1
  1. 1.Department of Industrial EngineeringUniversity of Texas at ArlingtonArlington

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