Journal of Optimization Theory and Applications

, Volume 134, Issue 3, pp 433–443

# On Optimization over the Efficient Set in Linear Multicriteria Programming

• R. Horst
• N. V. Thoai
• Y. Yamamoto
• D. Zenke
Article

## Abstract

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example.

## Keywords

Multicriteria optimization Optimization over the efficient set Global optimization Reverse convex constraint Minimum maximal flow problem

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