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?

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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?

Brendon Fernandez · Accepted Answer

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.

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

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