Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Necessary and sufficient conditions for local and global nondominated solutions in decision problems with multi-objectives

  • 102 Accesses

  • 16 Citations

Abstract

In this paper, the decision problem with multi-objectives is considered, and the nondominated solutions associated with a polyhedral domination cone are discussed. The necessary and sufficient conditions for the solutions are given in the decision space rather than the objective space. The similarity of the solution conditions obtained in this article to the Kuhn-Tucker condition of a convex programming problem is examined. As an application of the solution condition, an algorithm to locate the set of all nondominated solutions is shown for the linear multi-objective decision problem.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Zadeh, L. A.,Optimality and Non-Scalar-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-8, pp. 59–61, 1963.

  2. 2.

    Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, 1951.

  3. 3.

    Tamura, K.,A Method for Constructing the Polar Cone of a Polyhedral Cone, with Application to Linear Multicriteria Decision Problems, Journal of Optimization Theory and Applications, Vol. 19, pp. 547–564, 1976.

  4. 4.

    Vincent, T. L., andLeitmann, G.,Control-Space Properties of Cooperative Games, Journal of Optimization Theory and Applications, Vol. 6, pp. 91–113, 1970.

  5. 5.

    Leitmann, G., Rocklin, S., and Vincent, T. L.,A Note on Control Space Properties of Cooperative Games, Journal of Optimization Theory and Applications, Vol. 9, pp. 379–390, 1972.

  6. 6.

    Reis, R. W., andCitron, S. J.,On Noninferior Performance Index Vectors, Journal of Optimization Theory and Applications, Vol. 7, pp. 11–28, 1971.

  7. 7.

    Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.

  8. 8.

    Yu, P. L., andZeleny, M.,The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method, Journal of Mathematical Analysis and Applications, Vol. 49, pp. 430–468, 1975.

  9. 9.

    Mangasarian, O.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.

  10. 10.

    Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions, I, Springer-Verlag, Berlin, Germany, 1970.

  11. 11.

    Canon, M. D., Cullum, C. D., andPolak, E.,Theory of Optimal Control and Mathematical Programming, McGraw-Hill Book Company, New York, New York, 1970.

Download references

Author information

Additional information

The author would like to thank the reviewer for his helpful comments.

Communicated by G. Leitmann

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tamura, K., Miura, S. Necessary and sufficient conditions for local and global nondominated solutions in decision problems with multi-objectives. J Optim Theory Appl 28, 501–523 (1979). https://doi.org/10.1007/BF00932220

Download citation

Key Words

  • Decision problems with multi-objectives
  • nondominated solutions
  • domination cones
  • polar cones
  • edge vectors