Conjugate Duality in Vector Optimization

In this paper we develop a conjugate duality theory in vector optimiza- tion. Conjugate duality was fully developed in scalar optimization by Rockafellar [3 ] and provides a unified framework for the duality theory. The author and Sawaragi extended it to the case of multiobjective optimization by introducing some new concepts such as conjugate maps and subgradients for vector-valued, set-valued maps [4,6]. Their results are based on the efficiency (Pareto maximality). Kawasaki refined their results and obtained a reciprocal duality by introducing the concept of “supremum set” in an extended Euclidean space [ 1,2]. His supremum is based on the weak efficiency (weak Pareto maximality) of the closure of a set in an extended sense. The author [S] newly defined the supremum of a set in the extended multi-dimensional Euclidean space on the basis of weak ehiciency through several discussions on supremum. The definition satisfies some desirable fundamental properties. It can be extended to the case of general partially ordered linear topological spaces in a straightforward manner. Based on this definition of supremum, some useful concepts such as conjugate maps and subgradients are introduced for vector-valued, set-valued maps in this paper. These concepts enable us to develop the conjugate duality in vector optimization. Although the results obtained are quite similar to the earlier works by the author and Sawaragi [4] or Kawasaki [2], our new approach makes the proofs much easier and more understandable, and provides a better version of conjugate duality.