My question is:

Can a convex/concave function have a saddle point?

My answer would be:

Convex and concave function do not have saddle points, because a saddle point is not a local extremum.

Is this answer correct? How could I explain it better?

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- May 31st 2012, 12:53 PMinfernalmichCan a convex/concave function have a saddle point?
My question is:

Can a convex/concave function have a saddle point?

My answer would be:

Convex and concave function do not have saddle points, because a saddle point is not a local extremum.

Is this answer correct? How could I explain it better? - Jun 1st 2012, 08:03 AMHallsofIvyRe: Can a convex/concave function have a saddle point?
Are you asserting that

**every**point on the graph of a convex or concave function must be a local extremum?

If not, the fact that a saddle point is not a local extremum does not immediately imply that it cannot be a saddle point. - Jun 1st 2012, 08:33 AMinfernalmichRe: Can a convex/concave function have a saddle point?
But the affermation that a convex function can`t have a saddle point is true, right? - So the problem is that my explaination of the why is not correct?

- Jun 1st 2012, 10:42 AMjj323Re: Can a convex/concave function have a saddle point?
Neither convex or concave functions can have saddle points.

Let's show this for convex functions, then a similar argument can be used for concave. First, we need some definitions.

1. denotes the -neighborhood of x.

2. Given a Frechet differentiable where X is a Hilbert space (which includes ), a stationary point is a point x where . I'm sure we could do this with Gatteaux (directional) derivatives, but it's simpler to assume we have a gradient.

3. A saddle point is a stationary point that is neither a local min or a local max.

4. Given , a local min is a point x such that there exists an -neighborhood such that for all .

5. A convex function is a function where for all and ,

Now, let us assume that f is Frechet differentiable and convex. Let us also assume that there exists a stationary point x that is a saddle point. Since x is a saddle point, x is not a local min. That means that for every -neighborhood of x there exists such that . Next, from convexity we have that

for . Then, since , we have that

Rearranging terms, we have

Taking the limit as , we have that

where denotes the inner product on X. In , we typically have that . In any case, since x is stationary, . Hene, we reduce the above inequality into

This contradicts the assumption that x is not a local min. Hence, if f is convex and x is stationary, then x is a local, and in fact global, min.