Conjugacy Correspondences : a Unified View

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel's correspondence among convex functions and Rockafellar's extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping. Introduction. Conjugacy correspondences are the fundamental vehicle for applying the powerful concept of duality to extremum problems involving convex functions or saddle functions. The most basic and probably most familiar of these correspondences, and the one from which the others derive, is Fenchel's conjugacy correspondence among convex functions [2], [3]. Other important conjugacy correspondences are Rockafellar's extension of Fenchel's correspondence to (equivalence classes of ) saddle functions [8] and the "partial" conjugacy correspondence relating convex functions to (equivalence classes of ) saddle functions (Rockafellar [9] ; see also Moreau [6] ). A thorough treatment of these three conjugacy correspondences in finite-dimensional spaces, together with some of their applications to duality in convex extremum problems, can be found in [10]. In the infinite-dimensional case, the convex conjugacy correspondence is developed in Brjóndsted [1] and Moreau [7], while the partial conjugacy and saddle conjugacy correspondences are presented in Rockafellar [11]. The literature contains many related papers by these and other researchers too numerous to mention here. In this paper we develop in finite-dimensional spaces a new conjugacy correspondence which includes the three mentioned above as degenerate special cases. Received by the editors November 7, 1972 and, in revised form, January 16, 1973. AMS (MOS) subject classifications (1970). Primary 90C25, 90C30, 90C99, 90D05; Secondary 26A51, 49A40, 49B40.