# Saddlepoint

### From Glossary

Let Then, is a saddle point of if minimizes on , and maximizes on . Equivalently,

Von Neumann (1928) proved this equivalent to:

A sufficient condition for a saddle point to exist is that and are non-empty, compact, convex sets, is convex on for each in , and is concave on for each in .

Saddle point equivalence underlies duality. See also Lagrangian saddlepoint equivalence as well as the associated supplement.