Duality theory for maximizations with respect to cones

Abstract

Generalizations of Pareto optimality have been studied by a number of authors. In finite dimensions such work is exemplified by Corley [S 1, DaCunha and Polak [lo], Goeffrion [ 131, Hartley [ 151. Lin [ 171, Tanino and Sawaragi 1211, Wendell and Lee [22], and Yu [23]. The optimization of functions into possibly infinite dimensions has been considered by Borwein 121, Cesari and Suryanarayana [3], Christopeit [4], Corley [6,7], Craven [S, 91, Hurwicz [ 16 1, Neustadt [ 19 1. and Ritter [20]. An extensive bibliography on Pareto optimality, its extensions, and applications is given in 111. In this paper a duality theory is developed using the concept of saddlepoints for a problem in which the maximization of a function into possibly infinite dimensions is defined in terms of a cone. The results here extend the work of Tanino and Sawaragi [21] to infinite dimensions. A distinction is also made here between the notions of weak and strong optimality. Distinguishing between the two concepts allows the removal of the assumption of properness in [ 211 in establishing a relationship between the primal and dual problems, as well as permits additional duality relationships to be proved.