Stability of efficient sets: continuity of mobile polarities

TLDR: In this article, the authors introduce the notion of strongly efficient points for antisymmetric transitive relations and define the set of efficient points of an image A := f(D) of the constraint D. This relation is frequently referred to as preference.

Abstract: whenever, for every x E D such that A(x) afi(xO) for i = 1,2, . . . , n, one hash(x) = fi(x,,) for each i. Pareto optimization is also referred to as uecror, multi-objecfiue or multi-criteria optimization. Multi-objective maximization problems (0.1) have been nowadays extended to the situations where f is a mapping from X to a set Y equipped with a transitive relation 2. This relation is frequently called a preference. Consider first the image A := f(D) of the constraint D. An element y. of A is called (3_)-efficient (up to indifference or in the broad sense) if, for every y E A, y z y. implies y 5 yo; it is said to be strongly z-eficient (or efficient in the narrow sense) if, for each y E A, y 2 y. implies y = y,,. The two notions coincide for antisymmetric transitive relations. We shall be primarily concerned with efficient points in the broad sense. The set of efficient points of A (with respect to 7, ) will be denoted by max A (max Z A) and that of strongly efficient points, by max, A. An element of Y is called an efficient value of (0.1) if it is an efficient point of f(D). An element x0 of D is a solution of (0.1) if f(xo) is an efficient point of f(D). We shall use the latter S to denote the set of solutions of (0.1):