I am trying to solve a problem of the following form: pick a continuous random vector $(X_1,X_2)$ with support within $[0,\infty)^2$ to maximise $$\mathbb E\left[\sqrt{X_1+X_2}\right]$$ subject to the following constraint: for any $i,j$ with $\{i,j\} =\{1,2\}$ and almost all $x \ge 0$, $$\frac {f_i(x)}{1-F_i(x)} \ge \frac{1}{\mathbb E[1-e^{-X_j}|X_i = x]}$$ where $F_i$ is the marginal CDF of $X_i$ and $f_i$ is the density of $F_i$.
I only took an introductory course in functional optimisation and I am wondering whether this problem can be solved with standard techniques? In particular, can this be thought as an optimal control problem?
Thank you