Optimizing the ratio of two integrals

[aetherane]

Frequently, I have encountered optimization problems which involve finding the optimal function to maximize the ratio of two integrals (bounds from 0 to infinity). e.g: Maximize $$\frac{\int_0^\infty (g(x))^3 g'(x) dx}{\int_0^\infty (g(x))^3(1-g(x)) dx}$$

I am aware that variational calculus works on a single integral, but is there a general approach that might work for these types of problems?

 

[arildno]

Yes, variational calculus!


Define:
$$F(\epsilon)=\frac{\int_{0}^{\infty}(g(x)+\epsilon\gamma(x))^{3}(g^{,}(x)+\epsilon\gamma^{,}(x))dx}{\int_{0}^{\infty}(g(x)+\epsilon\gamma(x))^{3}(1-(g(x)+\epsilon\gamma(x)))dx}$$
where $\gamma(x)$ is an arbitrary function that vanishes at the boundaries.

Now, a necessary condition in order to let $\epsilon=0$ be the maximum of F would be that $\frac{dF}{d\epsilon}_{\epsilon=0}=0$

This condition will yield the differential equation g must satisfy.

 

[arildno]

To treat this problem in a somewhat general manner, let us assume that y's boundary values are fixed, say
$$y(a)=y_{a},y(b)=y_{b}$$
We look then at the set of comparison functions:
$$Y(x,\epsilon)=y(x)+\epsilon\gamma(x),\gamma(a)=\gamma(b)=0$$
That is, the $\gamma$-function is arbitrary except for vanishing at the boundaries.

We have a functional,
$$F(\epsilon)=\frac{\int_{a}^{b}N(Y,Y^{,},x)dx}{\int_{a}^{b}D(Y,Y^{,},x)dx}$$
and also define the quantities:
$$n=\int_{a}^{b}N(y,y^{,},x)dx, d=\int_{a}^{b}D(y,y^{,},x)dx (*)$$

Now, differentiating F with respect to $\epsilon$ and then setting thederivative of F equal to 0 at $\epsilon=0$ yields, with some rearrangement:
$$\frac{\int_{a}^{b}(d(\frac{\partial{N}}{\partial{y}}-\frac{d}{dx}\frac{\partial{N}}{\partial{y^{,}}})-n(\frac{\partial{D}}{\partial{y}}-\frac{d}{dx}\frac{\partial{D}}{\partial{y^{,}}}))\gamma(x)dx}{d^{2}}=0$$

Thus, we get the following diff.eq problem to solve:
$$d(\frac{\partial{N}}{\partial{y}}-\frac{d}{dx}\frac{\partial{N}}{\partial{y^{,}}})-n(\frac{\partial{D}}{\partial{y}}-\frac{d}{dx}\frac{\partial{D}}{\partial{y^{,}}})=0, y(a)=y_{a},y(b)=y_{b}$$

The solution of this diff.eq problem will typically be a function of the two parameters d and n, (in addition of course, of being a function in x)!

In order to determine d and n, (*) represents a system of algebraic equations in d and n, so we solve this system to complete our solution.

 

[arildno]

In your case, the differential equation becomes exceedingly simple:
$$3g^{2}-4g^{3}=0$$
where d and n vanish as determining parameters of the equation, and we retain an algebraic equation in g.

Thus, the only acceptable solution for a stationary point for the functional is $$g(x)=\frac{3}{4}$$

However, since this yields a divergent integral in the denominator, that particular ratio cannot be said to have a maximizing (or minimizing) function.