The structure of admissible points with respect to cone dominance

Abstract

We study the set of admissible (pareto-optimal) points of a closed convex set X when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions of X, we (i) extend a representation theorem of Arrow, Barankin and Blackwell by showing that all admissible points are either limit points of certain "strictly admissible" alternatives or translations of such limit points by rays in the closure of the preference cone, and (ii) show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed upon X, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element of X to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.

Duality Based Characterizations of Efficient Facets

“…A direct consequence of proposition 2.6 and assumption 3 is: [21], and Yu and Zeleny [29] have obtained the same result. In fact, the result is true whenP is any closed convex and strictly supported cone (see [5]). …”

“…For example, Bitran and Magnanti (theorem 3.1 in [5]) proved that only mild conditions need be imposed upon the cone dominance problem to insure that any efficient point x can be written as…”

“…Bitran and Magnanti [5] have also shown that for strictly supported closed convex cones, a point x C X is strictly efficient if, and only if, x°is efficient in some conical support Lx°0 ) to X at x, i.e., L(x°) -{x°} is a closed cone and X c LCx°). However, to simplify notation, we will not adopt such a representation.…”

Meilleure approximation en norme vectorielle et théorie de la localisation

“…Il reste enfin à examiner le cas où Z Pi X 2 -{z 2 Les exemples considérés dans [14], et plus généralement les seuls exemples qu'on peut visualiser directement, entrent dans le cadre de la Proposition 3.…”

“…La structure de l'ensemble des points efficients dans le cadre général de l'optimisation multicritère ou de l'optimisation vectorielle a été étudiée notamment dans [2,10,12,20], mais avec la préoccupation essentielle de montrer sa connexité ; [21] est un ouvrage de synthèse. Pour ce qui concerne la théorie de la localisation des résultats liant l'ensemble des points efficients à des questions de convexité se trouvent dans [9, 16 et 18].…”