Unified Representation of Proper Efficiency by Means of Dilating Cones

We reduce the definitions of proper efficiency due to Hartley, Henig, Borwein, and Zhuang to a unified form based on the notion of a dilating cone, i.e., an open cone containing the ordering cone. This new form enables us to obtain a comprehensive comparison among these and other kinds of proper efficiency. The most advanced results are obtained for a special class of proper efficiencies corresponding to one-parameter families of uniform dilations. This class is sufficiently wide and includes, for example, the Hartley and Henig proper efficiencies as well as superefficiency.     

Density theorems for generalized Henig proper efficiency

We develop a new, simple technique of proof for density theorems (i.e.,for the sufficient conditions to guarantee that the proper efficient points of a set are dense in the efficient frontier) in an ordered topological vector space. The results are the following: (i) the set of proper efficient points of any compact setQ is dense in the set of efficient points with respect to the original topology of the space whenever the ordering coneK is weakly closed and admits strictly positive functionals; moreover, ifK is not weakly closed, then there exists a compact set for which the density statement fails; (ii) ifQ is weakly compact, then we have only weak density, but ifK has a closed bounded base, then we can assert the density with respect to the original topology, (iii) there exists a similar possibility to assert the strong density for weakly compactQ if additional restrictions are placed onQ instead ofK. These three results are obtained in a unified way as corollaries of the same statement. In this paper, we use the concept of proper efficiency due to Henig. We extend his definition to the setting of a Hausdorff topological vector space.Research of the first author was supported by the Foundation of Fundamental Research of the Republic of Belarus. Authors are grateful to Professor Valentin V. Gorokhovik for suggesting the problem studied in this paper and for numerous fruitful conversations.     

Efficient Sets of Convex Compacta are Arcwise Connected

We prove that the efficient point set Max(Q|K) of a compact convex set QX in a Hausdorff topological vector space X ordered by a closed convex pointed cone KX with nonempty K ⁺ⁱ:={lK\{0}:l(x)>0} is arcwise connected.