Vol. 21, No. 1, 1967

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Duality and stability in extremum problems involving convex functions

Ralph Tyrrell Rockafellar

Vol. 21 (1967), No. 1, 167–187

In the theory of minimizing or maximizing functions subject to constraints, a given problem sometimes leads to a certain “dual” problem. The two problems are bound together like the strategy problems of the opposing players in a two-person game: neither can be solved without implicitly solving the other. The duality correspondence between linear programs is the best known example of this phenomenon. In the early 1950’s Fenchel came up with a general theory of convex and concave functions on Rn which was capable of predicting and explaining the duality in many problems. This paper attempts a further development of Fenchel’s theory, in both finite- and infinite-dimensional spaces. Fenchel’s model problems are broadened by building a linear transformation into them. The stability of the extrema in these problems is investigated and shown to be a necessary and sufficient condition for the duality to manifest itself in full force. New light is thereby thrown on the “duality gaps” which are known to occur in some finite-dimensional convex programs and infinitedimensional linear programs.

Mathematical Subject Classification
Primary: 90.60
Received: 13 August 1965
Published: 1 April 1967
Ralph Tyrrell Rockafellar