A random variable X follows a discrete distribution D(P,I), where $P = \{p_1,p_2,...,p_n\}$ is the the list of the probability, and $I=\{i_1, i_2, \cdots, i_n\}$ is the list of values. $T=X_1+X_2+\cdots+X_r$ is the sum of r independent identically distributed X, saying $T\sim S(P,r)$.
The objective function is
$$ g(P) = \frac{m-\beta P^TI}{1-\int_{-\infty}^{\infty} f(X) S(X|P,r) dX} $$
- $r \in \{1,2,\cdots\}$ is given, for example $r = 11$
- $m > 0$, is given
- $\beta \gt 0$, is given
- $I=\{i_1, i_2, \cdots, i_n\}$, is given, and each element $i_j > 0$.
- $f(x)$ is an function which is solved by a machine learning algorithm, but the closed-form of $f(x)$ is not available.
My optimization problem is,
$$ \begin{equation*} \begin{aligned} & \underset{x}{\text{maxmize}} & & g(P) \\ & \text{subject to} & & \sum_{i=1}^n p_i = 1 \\ & & & 0 \le p1 \le a_1 \\ & & & \cdots \\ & & & 0 \le p_n \le a_n \\ \end{aligned} \end{equation*} $$ where $\{a_1,\cdots,a_n\}$ is given.
Given P, I can use monte carlo method to calculate the value of the objective function. However, the key problem is how to efficiently search the feasible region of P?