Let $x\in\mathbb{R}^N$ and $f(x),g(x)$ two smooth functions from $\mathbb{R}^N$ to $\mathbb{R}$. Define:
$$ h(x) = \max_x \{ f(x), g(x) \} $$
What are the conditions for a local minimum of $h(x)$?
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Related: https://math.stackexchange.com/questions/132974/minimizing-a-maximum – a06e May 07 '23 at 10:36
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For example, here are some conditions I can see:
- $f(x) > g(x)$ in some neighbourhood, and $x$ is a local minimum of $f$ (so $f'(x)$ vanishes and $f''(x)$ is posdef).
- Similarly if $f(x) < g(x)$ in some neighbourhood, etc ...
But this picture is incomplete. My question is: What happens if $f(x) = g(x)$ "cross" at some point or a surface? I have the intuition that a local minimum of $h(x)$ is possible provided that "$f(x)$ grows in the directions where $g(x)$ decreases", or something along those lines. But I cannot make this precise.
a06e
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I am not too sure, since self answering is emcouraged. But given that this is a partial solution/ attempt, wouldn't it be better to have this in the question itself, as some kind of context? People do tend to skip new questions which are already answered. – user0 May 06 '23 at 17:54
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@user0 Is it important where I put it? I don't intend to accept this answer since I think it's not complete. But it still a "partial answer". – a06e May 06 '23 at 17:55