The Kuhn-Tucker theory
of Lagrange multipliers centers on a one-to-one correspondence between
nonlinear programs and minimax problems. This correspondence has been
extended by Dantzig, Eisenberg and Cottle to one in which every minimax
problem of a certain type gives rise to a pair of nonlinear programs dual to
each other. The aim here is to show how, by forming conjugates of convex
functions and saddle-functions (i.e. functions of two vector arguments which
are convex in one argument and concave in the other), one can set up a
more symmetric correspondence with even stronger duality properties. The
correspondence concerns problems in quartets, each quartet being comprised of
a dual pair of convex and concave programs and a dual pair of minimax
problems. The whole quartet can be generated directly from any one of its
members.
