Minmax and duality in nonlinear programming

Abstract Recently in an important paper Stoer gave a necessary and sufficient condition for the existence of a saddle point of a function when certain assumptions are satisfied, and derived some duality relations from this result. The purpose of this paper is (1) to weaken Stoer's hypotheses and simplify his crucial condition, the so-called B -property, and thereby strengthen his main theorem, (2) to avoid a number of implicit assumptions concerning the existence of maxima and minima over unbounded sets, and (3) to apply the results of (1) and (2) more explicitly and obtain two types of duality relations that subsume many previously obtained duality results.