Scalarization of vector optimization problems

  • D. T. Luc
Contributed Papers


In this paper, we investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.

Key Words

Scalar representation vector optimization monotonic functions upper semicontinuity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chankong, V., andHaimes, Y. Y.,Multiobjective Decision Making: Theory and Methodology, North-Holland, Amsterdam, Holland, 1983.Google Scholar
  2. 2.
    Warburton, A. R.,Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives, Journal of Optimization Theory and Applications, Vol. 40, pp. 537–557, 1983.Google Scholar
  3. 3.
    Pascoletti, A., andSerafini, P.,Scalarizing Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 42, pp. 499–524, 1984.Google Scholar
  4. 4.
    Serafini, P.,A Unified Approach for Scalar and Vector Optimization, Proceedings of the Conference on Mathematics of Multiobjective Optimization, CISM, Udine, Italy, 1984.Google Scholar
  5. 5.
    Jahn, J.,Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203–218, 1984.Google Scholar
  6. 6.
    Luc, D. T.,Connectedness of Efficient Point Sets in Quasiconcave Vector Maximization, Journal of Mathematical Analysis and Applications (to appear).Google Scholar
  7. 7.
    Krabs, W.,Optimization and Approximation, John Wiley, Chichester, England, 1979.Google Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    Naccache, P. H.,Stability in Vector Optimization, Journal of Mathematical Analysis and Applications, Vol. 68, pp. 441–453, 1979.Google Scholar
  10. 10.
    Berge, C.,Topological Spaces, Macmillan, New York, New York, 1963.Google Scholar
  11. 11.
    Hiriart-Urruty, J. B.,Images of Connected Sets by Semicontinuous Multifunctions, Journal of Mathematical Analysis and Applications, Vol. 111, pp. 407–422, 1985.Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • D. T. Luc
    • 1
  1. 1.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations