SADDLE POINT
Saddle point - Let f:X × Y –> R. Then, (x*, y*) is a saddle point of f if x* minimizes f(x, y*) on X, and y* maximizes f(x*, y) on Y. Equivalently,
f(x*, y) <= f(x*, y*) <= f(x, y*) for all x in X, y in Y.
von Neumann (1928) proved this equivalent to:
Inf(Sup(f(x, y): y in Y): x in X) = Sup(Inf(f(x, y): x in X): y in Y) = f(x*, y*).
A sufficient condition for a saddle point to exist is that X and Y are non-empty, compact, convex sets, f(.,y) is convex on X for each y in Y, and f(x,.) is concave on Y for each x in X. Saddle point equivalence underlies duality .
