Connectedness of the efficient point sets in quasiconcave vector maximization

Abstract

One of the key questions of vector optimization theory is to investigate the structure of efficient point sets. Among the topological properties of these sets the connectedness and contractibility are of interest as they provide a possibility of continuous moving from one optimal solution to any other along optimal alternatives only, and they have a close relation to the fixed point property that is a useful argument in economic equilibrium theory (see [ 1, 21 and there cited references). Motivated by this in [2] the structure of the sets of points which are efficient with respect to cones has been studied for some classes of convex sets. Several topological properties as closedness, compactness, contractibility, etc., have been obtained, although the research has been carried out in the space of outcomes rather than in the space of alternatives. In [ 11 the author has pointed out his investigation in the space of alternatives for quasiconcave maximization problems, only the preference orders are assumed to be defined by the nonnegative orthant of the space. As noted in [ 1 ] the connectivity results for quasiconcave criteria do not extend to more general preference order. However, they may be expected under a modified quasiconcavity assumption. In the present paper we shall establish the connectedness of the set of weakly efficient alternatives for cone-quasiconcave criteria and the contractibility of the set of efficient alternatives for the strictly conequasiconcave case under a weakened assumption on the compactness of the upper level sets. The results then will be carried into the space of outcomes. Suppose that X is a nonempty closed convex set in the n-dimensional Euclidean space R, C is a pointed closed convex cone in R with a nonempty interior int C andf is a map from X into R”. We say that x E X is an efficient alternative (with respect to C) if there is no y E X such that