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.
