Saddle Point Maximization

Question

My question is that in general, is there a case where saddle point be the global max of a function? I am solving a game theory question which the optimal solution is the saddle point. Can I conclude that the optimal solution is at the boundaries?

Answer 1

By definition, a saddle point is

• a local min in some directions and
• a local max in other directions at the same time.

Since it is a local min in at least one direction, there are more optimal points for maximization. Ditto minimization from the other directions...

Answer 2

It is a common misunderstanding in optimization how a saddle point may be the optimal point. The saddle point that is the optimal solution is the saddle point for the Lagrange function $L(x,u,v)$, not the saddle point for the objective function itself.