Consider the optimization problem with $x,y\in\mathbb{R}^n$, with $f:\mathbb{R}^{2n}\to \mathbb{R}$:
$$ \min_{x}\min_{y} - f(x,y). $$
If I want to exchange the second minimum with a maximum is this the correct way to do it?
$$ \min_{x}\max_{y}f(x,y),$$
or I need to change also the second minimum into a maximum? Can you explain the reasoning behind these change of min/max? Thanks!
Since $\min_{y} - f(x,y) = -\max_{y} f(x,y)$:
$$ \min_{x}\min_{y} - f(x,y) = \min_{x} - \max_{y} f(x,y). $$ Applying the same rule again, you get $-\max_{x} \max_{y} f(x,y)$. Alternatively, you can do it in one step:
$$ \min_x \min_y - f(x,y) = \min_{x,y} - f(x,y) = - \max_{x,y} f(x,y). $$