I would like to optimize a certain moment of a probability distribution by finding the optimal mixture $f(\theta)$ of a class of density functions $g(x;\theta)$ under some given constraints: $$ \max_{f(\theta)} \int_0^\infty x^{m+1} \int_\Theta f(\theta) \, g(x;\theta) \,{\rm d}\theta \,{\rm d}x $$ subject to some moment conditions and two density definitions $$ \int_0^\infty x^{m-2} \int_\Theta f(\theta) \, g(x;\theta) \,{\rm d}\theta \,{\rm d}x = \mu_{m-2} \\ \int_0^\infty x^{m-1} \int_\Theta f(\theta) \, g(x;\theta) \,{\rm d}\theta \,{\rm d}x = \mu_{m-1} \\ \int_0^\infty x^{m} \int_\Theta f(\theta) \, g(x;\theta) \,{\rm d}\theta \,{\rm d}x = \mu_{m} \\ \int_\Theta f(\theta) \,{\rm d}\theta = 1 \\ \int_0^\infty g(x;\theta) \,{\rm d}x = 1 $$ where $\Theta$ is some bounded interval.
To simplify matters, one may choose $g(x;\theta) := \delta(x-\theta)$ (such that $x$ can be substituted for $\theta$) and restrict $\Theta$ to positive numbers.
Numerical experiments (using a grid rather than a dense set $\Theta$) indicate that the optimum can be a smooth $f(\theta)$ on $\Theta$.
Any help is appreciated!