Abstract

This paper studies the weak efficient set (WEff P) of a minimization problem P with k objectives defined on a convex set X of ℝⁿ. These objectives are continuous and belong to the class of so-called strictly quasiconvex functions, which contains, in particular, convex as well as linear fractional functions. When k is greater than n, it is of interest to replace the original problem by several subproblems, having at most n objectives. We show that if WEff P is bounded, the knowledge of the efficient sets of such subproblems, completely determines WEff P.

Author information

Contact details are reproduced from the original publication and may be historical.

C. Malivert

LACO (URA 1586) - Département de Mathématiques, Faculté des Sciences, 123 Avenue Albert Thomas, F-87060 Limoges Cedex, France.

N. Boissard

LACO (URA 1586) - Département de Mathématiques, Faculté des Sciences, 123 Avenue Albert Thomas, F-87060 Limoges Cedex, France.

Keywords

  • Multiple objectives
  • Efficient sets
  • Quasiconvexity
  • Fractional programming

References

  1. J.P. Crouzeix: Contribution à l’étude des fonctions quasi-convexes, Thèse de doctorat, U.E.R. des Sciences, Université de Clermont-Ferrand II, 1977, France.Google Scholar
  2. N. Popovici: Sur une notion de quasi-convexité pour des fonctions vectorielles, (préprint)Google Scholar
  3. T.J. Lowe, J.-F. Thisse, J.E. Ward and R.E. Wendell: On efficient solutions to multiple objective mathematical programs. Management Sci. 30, 1984, 1346–1349.https://doi.org/10.1287/mnsc.30.11.1346Google Scholar
  4. R.T. Rockafellar: Convex Analysis. Princeton University Press, 1970, Princeton N.J.Google Scholar
  5. S. Schaible: Bibliography in fractional programming. Zeitschrift für Operations Research 26, 1982, 211–241.https://doi.org/10.1007/BF01917115Google Scholar
  6. S. Schaible: A survey of fractional programming in : S. Schaible and W.T. Ziemba eds., Generalized Concavity in Optimization and Economics, (Acad. Press, New York 1981), 417–440.Google Scholar
  7. J. Ward: Structure of efficient sets for convex objectives. Math. of Oper. Res. 14, 1989, 249–257.https://doi.org/10.1287/moor.14.2.249Google Scholar

Suggested citation

C. Malivert, N. Boissard. “Structure of Efficient Sets for Strictly Quasi Convex Objectives.” Journal of Convex Analysis 1 (1994), No. 2, 143–150. https://doi.org/10.68381/jca01011

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1994. Rights now held by Banach Press LLC.