Quasiconcave vector maximization: Connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives

We study the sets of Pareto-optimal and weakly Pareto-optimal solutions to a vector maximization problem defined by a continuous vector-valued quasiconcave criterion functionf and a closed convex set of alternativesS. IfS is compact, it is shown that the set of weakly Pareto-optimal alternatives is connected, but that the set of Pareto-optimal alternatives is not necessarily connected. However, the set of Pareto optima is shown to be connected for some important subclasses of quasiconcave functions. We also provide some reasonable conditions under which the compactness assumption onS may be relaxed and connectedness maintained.