The structure of admissible points with respect to cone dominance

We study the set of admissible (Pareto-optimal) points of a closed, convex setX 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 ofX, we extend a representation theorem of Arrow, Barankin, and Blackwell by showing that all admissible points are either limit points of certainstrictly admissible alternatives or translations of such limit points by rays in the closure of the preference cone. We also show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed uponX, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element ofX to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.Several results from this paper were presented in less general form at the National ORSA/TIMS Meeting, Chicago, Illinois, 1975.This research was supported, in part, by the United States Army Research Office (Durham), Grant No. DAAG-29-76-C-0064, and by the Office of Naval Research, Grant No. N00014-67-A-0244-0076. The research of the second author was partially conducted at the Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, Heverlee, Belgium.The authors are indebted to A. Assad for several helpful discussions and to A. Weiczorek for his careful reading of an earlier version of this paper.