Vol. 25, No. 3, 1968

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A general correspondence between dual minimax problems and convex programs

Ralph Tyrrell Rockafellar

Vol. 25 (1968), No. 3, 597–611

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.

Mathematical Subject Classification
Primary: 90.60
Received: 3 January 1966
Published: 1 June 1968
Ralph Tyrrell Rockafellar