$F(x,y)=\int_a^{b} \int_c^{d} G(x,y,s,t)dsdt$. I wish to find $x$ and $y$ (subject to some constraints) that minimize $F$. $F$ is not expressible in closed form. Is the only way to solve this problem as follows: "Replace different $x$ and $y$ values according to the constraints and evaluate the integral numerically each time"?

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    $\begingroup$ What form are the constraints? Are they linear, quadratic form, non-linear? And similarly, what form is $G$? $\endgroup$
    – Daryl
    Apr 14 '13 at 4:34
  • $\begingroup$ Additionally, is $G$ expressible in closed form? Is $G$ convex? Are $x$ and $y$ independent of $s$ and $t$? $\endgroup$ Apr 14 '13 at 4:37
  • $\begingroup$ @John Yes, G is expressible in closed form - I need to check if it is convex. Yes, x and y are indpt of s and t. Also, I tried to simplify the situation. The $x$ and $y$ are actually $k$-dimensional vectors with $-1\leq x_{1}<x_{2}<\dots <x_{k}\leq 1$ and $0 < y_{i} < 1$ with $\Sigma y_{i} = 1$ $\endgroup$
    – Ken Dunn
    Apr 14 '13 at 4:42

Since $G$ is in closed form (as per your comment above), you can express derivatives of $F$ as

$$ \frac{\partial F}{\partial x_i} = \int_a^b \int_c^d \frac{\partial}{\partial x_i}G(x,y,s,t)\ ds\ dt $$

The same goes for $\frac{\partial F}{\partial y_i}$, so you can perform gradient descent if your $F$ is convex. (I don't recall what the conditions are for an integral of a function to be convex. It may be enough to know whether $G$ is convex just in $x$ and $y$.)

If $F$ is concave, then your solution will be at one of the vertices of your bounding region, since your bounding region is a convex polytope.

  • $\begingroup$ Thanks John. Will give this a try, $\endgroup$
    – Ken Dunn
    Apr 14 '13 at 5:56

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