I am struggling with the following real-valued optimization problem: \begin{align*} \max_{\Omega \subseteq [0,1]} \int_\Omega 2s^2 (1-s) \left( \frac{\int_\Omega 2s^2 ds}{\int_\Omega 2s ds}\right)\left(1- \frac{\int_\Omega 2s^2 ds}{\int_\Omega 2s ds}\right) ds + \int_{[0,1]\setminus \Omega} 2s^3 (1-s)^2 ds, \end{align*} where $\Omega$ is any subset of $[0,1]$.
As a non-mathematician, I do not even know which branch of optimization this kind of problem belongs to. Therefore, any sort of help (the solution, references to books or lecture notes, etc.) is highly appreciated. Thank you.
To put some context to the problem: I am studying a communication game in microeconomics. Essentially, the set $[0,1]$ are possible posterior beliefs that a specific event occurred (i.e. the posterior belief is just the success probability of a Bernoulli random variable). The pdf of the posterior beliefs is $2s$. Now, the objective function above is a profit function for the "message sender" for a pre-defined set of $\Omega$. All $s\in [0,1]\setminus \Omega$ will be revealed, that is, the sender just says "the posterior is $s$" to the receiver, which results in a profit for the sender of $s (1-s) s' (1-s')$ where $s$ is the sender's and $s'$ the receiver's belief. As the sender told the receiver the true posterior $s'=s$. Thus, the right integral captures the density-weighted profit for all $s$ where the sender reveals his posterior. The left integral also captures the density-weighted profit for the sender, but here the communication is different. Instead of telling the receiver his posterior, the sender "conceals" and just keeps silent. That is, the belief of the receiver does not coincide with the sender anymore, i.e. $s \neq s'$. Instead, the posterior belief of the receiver is formed "rationally" by Bayesian updating and is captured in the brackets in the left integral where the first is $s'$ and the latter $1-s'$. Essentially, the optimization problem now captures the question of the sender which communication strategy, i.e. which $s$ should be revealed (i.e. $s \in [0,1] \setminus \Omega$) versus concealed (i.e. $s \in \Omega$) , result in the largest profit.