how can i find min of max multivariable function

Question

I want to find: $min[y] max[x] f(x,y)$.(notation means that maximum of f on x and minimum on y). where is: $f(x,y) = h(x)+y*x$. i want to determine y such that decrease the maximum of $f(x,y)$. $h(x)$ is nonlinear. please help me with a matlab code or analytical approach.

Answer 1

If you want to find the points where $f$ is a maximum with respect to $x$ and a minimum with respect to $y,$ we call that point a saddle.

If $f$ is continuous and twice differential.

Find $f_x, f_y, f_{xx}, f_{yy}$ and $f_{x,y}$

Those are the partial derivatives with respect to x,y, and the second partial derivatives.

At a saddle: $f_x = 0, f_y = 0$ and $4f_{xx}f_{yy} < (f_{xy})^2$