Vector discrete problems: parametrization of an optimality principle and conditions of solvability in the class of algorithms involving linear convolution of criteria

An n–criteria problem with a finite set of vector valuations is considered. An optimality principle of this problem is given by an integer-valued parameter s, which is varied from 1 to n−1. At that, the majority and Pareto optimality principles correspond to the extreme values of the parameter. Sufficient conditions, under which the problem of finding efficient valuations corresponding to the parameter s is solvable by the linear convolution of criteria, are indicated. 1 Basic definitions and lemma As usually [1], let a vector function y = (y1(x), y2(x), ..., yn(x)) : X → R, n ≥ 2, be defined on a set of alternatives X. When choosing an optimal alternative from the set X it is enough to consider the set of feasible valuations Y = {y ∈ R : y = y(x), x ∈ X}. Here Rn is the n–dimentional criteria space. We consider a vector problem y → min y∈Y c ©2000 by V.A.Emelichev, A.V.Pashkevich