Necessary and Sufficient Global Optimality Conditions for Convex Maximization Revisited

Abstract

Ž . convex maximization problem . Throughout this article it is assumed that Ž . the global maximum in 1 exists. For the state-of-the-art in convex maximization including various algorithms and abundant applications, we w x w x refer to the textbooks 10, 11 and to the excellent survey 1 . In recent years several interesting necessary and sufficient global optiŽ . mality conditions characterizing a point x g D satisfying 1 have been proposed. Other conditions can be derived from optimality criteria formulated for more general global optimization problems. In this article, four of Ž . Ž w x these criteria are revisited: the condition HU of Hiriart-Urruty cf. 5]9 , w x. Ž . Žw x with a short elegant proof in 7 , the condition S of Strekalovski 18]20 , . which we will reprove, generalize, and modify , the specialization of the Ž . Ž . Ž w x. Singer]Toland duality ST to problem 1 e.g., 13]17, 21, 22 , and a Ž . reformulation of an optimality condition CDC for so-called canonical Ž w x d.c.-problems originally given in 23 , modified as used here, and with a