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.
