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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?.

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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.

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  • $\begingroup$ Thank you for your answer. Do you think the instance of the problem in the link is of this form? (At least if $r$ is chosen such that the dependence on $(w_i)_{i\in I}$ is nice.) $\endgroup$
    – 0xbadf00d
    Commented Dec 6, 2019 at 10:41
  • $\begingroup$ Honestly, I don't know and the problem looks too involved that I would start to invest my time, sorry. $\endgroup$
    – Dirk
    Commented Dec 6, 2019 at 11:32
  • $\begingroup$ Thanks anyway. I've tried to reformulate the problem in a simplified manner: mathoverflow.net/q/347894/91890. Maybe you can take a look. $\endgroup$
    – 0xbadf00d
    Commented Dec 8, 2019 at 20:12

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