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?

I thought saddles were by definition neither max nor min. – Randall Jan 04 '19 at 18:32 — Randall, Jan 04 '19 at 18:32
What game theory question? – cgiovanardi Jan 05 '19 at 00:02 — cgiovanardi, Jan 05 '19 at 00:02

score 1 · Answer 1 · answered Jan 04 '19 at 18:32

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...

answered Jan 04 '19 at 18:32

gt6989b

54,422

Thanks, So can we say that is is only boundary solutions? – user3425989 Jan 04 '19 at 18:40
@user3425989 if no relative extrema lie inside the region, then all candidates for the optimal solutions will come from the boundary. – gt6989b Jan 06 '19 at 05:29

score 1 · Answer 2 · answered Jan 04 '19 at 18:39

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.

Saddle Point Maximization

2 Answers2