Vol. 50, No. 1, 1974

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An extension of Fenchel’s duality theorem to saddle functions and dual minimax problems

Lynn McLinden

Vol. 50 (1974), No. 1, 135–158

Fenchel’s Duality Theorem (or more precisely, Rockafellar’s extension of it) is extended here from the context of convex functions and dual convex extremum problems to that of saddle functions and dual minimax problems. The paper is written in the spirit of mathematical programming. Inequalities between optimal values are established, stable optimal solutions are characterized, strong duality theorems proved, and an existence criterion given. An associated Lagrangian saddle point problem is introduced and an extension of the Kuhn-Tucker Theorem derived. The proofs, which are necessarily different from the purely convex case, rely on recently developed pairs of dual operations on saddle functions, as well as on more widely known facts about conjugate saddle functions.

Mathematical Subject Classification 2000
Primary: 90C25
Received: 29 September 1972
Published: 1 January 1974
Lynn McLinden