Suppose we have a function $f(x_1,x_2)$ with the following properties:
- Let $x^*=\arg \max_{x_1} f(x_1,x_2=x^*)$ and $x^*=\arg \min_{x_2}f(x_1=x^*,x_2)$.
- $f(x_1,x_2)$ is concave in $x_1$.
- $f(x_1,x_2)$ is convex in $x_2$
- $0 \le f(x_1,x_2) \le K$ non-negative and bounded
- Assume the set of $(x_1,x_2)$ over which we are optimizing is compact.
Here are the thing I am trying to answer:
- Is $(x_1,x_2)=(x^*,x^*)$ a saddle point?
- Suppose I am trying to find the maximum of the following function \begin{align} g(x_1,x_2)=f(x_1,x_2)+f(x_2,x_1) \text{ **note the flip**} \end{align} What can we say about maximum of $g$? Or anything else about $g$?
- Does max (at least one) of $g$ occur at the point $x_1=x_2$?
Does anyone know of any similar scenarios or any materials on this so I can study them? I feel similar case should have been studied before. Thank you very much.
If the concavity/convexity in (2, 3) are strict, then yes. From (1) it follows that $\partial f/\partial x_1=\partial f/\partial x_2=0$, so it is a critical point, but (2) and (3) together imply that it is neither a local maximum nor a local minimum.
All you can say is that $(x^*,x^*)$ is a critical point of $g$, but you don't know if it is a local maximum, a local minimum, or a saddle point. Examples of all three cases are provided by $f_1(x_1,x_2)=x_2^2-2x_1^2$, $f_2(x_1,x_2)=2x_2^2-x_1^2$, and $f_3(x_1,x_2)=x_2^2-x_1^2+x_1x_2$ respectively.
As above, $g$ could be convex, concave, or neither, so you can't guarantee whether a local maximum exists at all.