Super Efficiency in Vector Optimization

We introduce a new concept of efficiency in vector optimization. This concept, super efficiency, is shown to have many desirable properties. In particular, we show that in reasonable settings the super efficient points of a set are norm-dense in the efficient frontier. We also provide a Chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex. 0. Introduction Decision-making problems appearing in economics, management science and operations research require frequently that decision making be based on optimizing several criteria. Vector optimization has provided an organized constructive approach to these problems. Throughout this note, we consider minimization problems. Efficient decisions are those decisions not minorized by any others. As observed by Kuhn and Tucker and later Geoffrion, a subset of an efficient decision set may not be satisfactorily characterized by a scalar minimization problem, so the concept of proper efficiency was introduced by Kuhn-Tucker, Geoffrion, and modified and formulated in a more general framework by Borwein, Benson, Henig, and Hartley among many other authors [Kuhn 1], [Benson 1], [Borwein 1, 2, 4, 6], [Geoffrion 1], [Henig 1], [Hartley 1] and the references therein. The motivation for introducing proper efficiency is that it enables one to eliminate certain anomalous efficient decisions and to prove the existence of equivalent scalar problems whose solutions produce at least most of the efficient decisions, namely the proper ones. It has been amply demonstrated that proper efficiency is a natural concept in vector optimization. In this note, we introduce a new kind of proper efficiency, namely super efficiency. Super efficiency refines the notions of efficiency and other kinds of proper efficiency, and provides a concise (and equivalent) scalar characterization and duality results when the underlying decision problem is convex. We also Received by the editors March 23, 1990 and, in revised form, March 6, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 49A27, 90C31; Secondary 46A40, 52A07.