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I encounter an optimization problem like this:

$$\min_{w(x)}{\int {w(x)f(x|e)dx}}$$ subject to $$\int {v(w(x))f(x|e)dx} - g(e)=u$$

$w(x)$ is a function and suppose it has desirable differentiability. $f(x|e)$ is the density function of x conditional on a variable $e$.

The book I have says that the first-order condition is $-f(x|e)+\lambda v'(w(x))f(x|e)=0$. I understand that this is analogous to the Lagrangian multiplier, but I am wondering what is the mathematical foundation of this? Is there a rigorous theory for this kind of optimization problem?

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Yes. Look at for a quick overview. There are some tweaks needed since you are no longer dealing with infinite dimensions, but the idea is the same. – copper.hat Aug 21 '12 at 6:29
Thank you! Could you recommend a reference for this? Should I look for books on calculus of variations? – Zariski Aug 21 '12 at 10:49
I have no reference for this that I strongly recommend. Possibly you could look at Luenberger's "Optimization by vector space methods" (see Theorem 2 in Section 7.7). The result is fairly standard, and is typically proved using the implicit function theorem. The finite dimensional proof carries through with appropriate allowances for infinite dimensional spaces. – copper.hat Aug 21 '12 at 14:58

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