Duality and stability in extremum problems involving convex functions

Abstract

In the theory of minimizing or maximizing functions subjectto constraints, a given problem sometimes leads to a certain4“dual” problem. The two problems are bound together like thestrategy problems of the opposing players in a two-persongame: neither can be solved without implicitly solving theother. The duality correspondence between linear programsis the best known example of this phenomenon. In the early1950's Fenchel came up with a general theory of convex andconcave functions on Rn which was capable of predicting andexplaining the duality in many problems. This paper attemptsa further development of Fenche's theory, in both finite- andinfinite-dimensional spaces. Fenches model problems arebroadened by building a linear transformation into them.The stability of the extrema in these problems is investigatedand shown to be a necessary and sufficient condition for theduality to manifest itself in full force. New light is therebythrown on the “duality gaps” which are known to occurin some finite-dimensional convex programs and infinitedimensionallinear programs. © 1967 by Pacific Journal of Mathematics.