Recent Results on Duality in Vector Optimization

Abstract

In connection with the remarkable and rapid development of linear and nonlinear vector optimization, an increasing number of papers were devoted to the construction and investigation of dual problems. Several concepts of duality were treated, especially Lagrange duality and Fenchel duality (cf. Elster /1/, Elster/Iwanow /3/, Jahn /8/). Apart of linear problems we can find extensive investigations of convex problems and more and more of nonconvex problems. In the last case some difficulties can arise concerning p.e. separation theorems which are sufficiently general. In the present paper a direct duality theorem is derived for rather general (nonconvex) vector optimization problems. For the proof a separation theorem is needed where the separating functional is nonlinear (more precisely: convex). The results are formulated in finite dimensional spaces, but an extension to infinite dimensions is possible.