I am trying to solve functional maximization problems. They are typically of the following form...

Semaj Christian

Semaj Christian

Answered

2022-06-24

I am trying to solve functional maximization problems. They are typically of the following form (where support of θ is [0,1]):
[ v ( θ , x ( θ ) ) + u ( θ , x ( θ ) ) u 1 ( θ , x ( θ ) ) ( 1 F ( θ ) f ( θ ) ) ] f ( θ ) d θ
Now one way that was proposed to me was of point-wise maximization. That is you fix a θ and then solve:
a r g m a x x ( θ ) v ( θ , x ( θ ) ) + u ( θ , x ( θ ) ) u 1 ( θ , x ( θ ) ) ( 1 F ( θ ) f ( θ ) )
Solving this problem would give me a number x for each θ and I will recover a function x ( θ ) 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?

Answer & Explanation

Brendon Fernandez

Brendon Fernandez

Expert

2022-06-25Added 14 answers

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 ( θ ) is continuous, differentiable, etc. For instance, imagine x ( θ ) = 1 , d. Then taking the derivative of the objective function with respect to x ( θ ) does not make any sense.

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