The vector optimization problem considered here is to minimize a continuous vector-valued function f: S→Rm 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<sup>m</sup>\{0} while int D denotes the interior of -Rm Thus, y′ ∈ y′∈y′′++D + D denotes<sup>+</sup> 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:
F¯={y¯∈F:F∩(y¯+D~)=∅}
(1)
is called the efficient set (D-optimal set, Pareto set) in objective or outcome space