 Semaj Christian

2022-06-24

I am trying to solve functional maximization problems. They are typically of the following form (where support of $\theta$ is [0,1]):
$\int \left[v\left(\theta ,x\left(\theta \right)\right)+u\left(\theta ,x\left(\theta \right)\right)-{u}_{1}\left(\theta ,x\left(\theta \right)\right)\left(\frac{1-F\left(\theta \right)}{f\left(\theta \right)}\right)\right]f\left(\theta \right)d\theta$
Now one way that was proposed to me was of point-wise maximization. That is you fix a $\theta$ and then solve:
$argma{x}_{x\left(\theta \right)}v\left(\theta ,x\left(\theta \right)\right)+u\left(\theta ,x\left(\theta \right)\right)-{u}_{1}\left(\theta ,x\left(\theta \right)\right)\left(\frac{1-F\left(\theta \right)}{f\left(\theta \right)}\right)$
Solving this problem would give me a number $x$ for each $\theta$ and I will recover a function $x\left(\theta \right)$ that will maximize the original objective function.

I have two questions related to this:
1) Does such point-wise maximization always work?
2) What happens if rather than doing point-wise maximization I try and take the derivative of the objective function with respect to x(θ) and equating the first order condition to 0? Is this a legitimate way of solving the problem? Can someone show exactly what such a derivative would look like and how to compute it? Brendon Fernandez

Expert

Regarding your first question, pointwise maximization works as long as the boundaries of the integral are not a function of your control.
Regarding your second question, no you cannot, precisely because you are solving a functional maximization problem. You do not know if $x\left(\theta \right)$ is continuous, differentiable, etc. For instance, imagine $x\left(\theta \right)=1,\mathrm{\forall }d$. Then taking the derivative of the objective function with respect to $x\left(\theta \right)$ does not make any sense.

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