Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?

Question

Let $H$ be a $\mathbb R$-Hilbert space and $F:H^2\to\mathbb R$. Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?

Since the question is rather abstract, feel free to impose any addition assumption you like. For example, $F(\;\cdot\;,y)$ being sufficiently regular. If this is too hard, I'd also be interested in finding and minimizing suitable upper bounds.

Remark: This is the concrete instance of the problem I'm interested in: Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?.

Answer 1

The problem you are dealing with is of the form $$ \inf_{x\in H}\sup_{y\in H} F(x,y). $$ If $F$ is convex in $x$ and concave in $y$, this is a saddle point problem and you can find a lot of information under this buzzword. Note that there is the very important special case of "linear saddle point problems" which arise if you minimize a quadratic function over a linear constraint, but also non-linear saddle point problems are a thing.