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Say I have a functional of the form,

$$ F(x, f(x); c) = \max_{f(x),c} \left\{ f(x)c - 1/2 c^2 \right\} $$

with $c$ a parameter. Can I take first-order conditions with respect to c and substitute $c^* = f(x)$ back into the functional and still get the same solution for $f^*(x)$?

If only so under certain conditions, which conditions would suffice?

hrrrrrr5602
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  • I haven't found a definitive answer but since the functional I am dealing with is separately (but not jointly) convex, I think I could use something like here https://math.stackexchange.com/questions/453831/optimization-of-a-function-of-two-variables – hrrrrrr5602 Jun 19 '22 at 11:25
  • Also, it seems like a version of the envelope theorem holds in Banach space https://link.springer.com/article/10.1007/s11579-015-0145-5 – hrrrrrr5602 Jun 19 '22 at 11:31

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