A Methodological Approach to Comparing Parametric Characterizations of Efficient Solutions

Abstract

The vector optimization problem considered here is to minimize a continuous vector-valued function f: S→R^m on a constraint set C ⊂ S. Let F= f(C) be a compact set (though much weaker assumptions are sufficient for the existence of optimal solutions — see Benson, 1978). While keeping in mind that the set F is usually defined implicitely and that an attainable decision outcome y F means that y= f(x) for some admissible decision x∈ C, we can restrict the discussion to the outcome or objective space only. We assume that all objectives are minimized and use the notation D= — R^m\{0} while int D denotes the interior of -R^m Thus, y′ ∈ $y' \in \mathop {y''}\limits^ + + D$ + D denotes⁺ here that y′.≦ y″, for all i=1,.. m, while y′ ∈ ⁺ y″+ D̃, D̃= D\{0} denotes y′_i ≦ y″. for all i=1,.. m and y′ < y″. for some j=1,.. m, and y′∈ y″+int D denotes y′ _i. < y″. for all i=1;.. m wlere y + D is the cone D shifted by y. The problem of vector minimization of y= f(x) over C can be equivalently stated as the problem of finding D-optimal elements of F. The set of all such elements, defined by:

$$ \bar F = \left\{ {\bar y \in F:\;F \cap \left( {\bar y + \tilde D} \right) = \emptyset } \right\} $$

((1))

is called the efficient set (D-optimal set, Pareto set) in objective or outcome space