I am trying to solve the following optimization problem:
$$\max\limits_{x \in [0,\infty)} {\bf E} \left[ x \left( Z - a x \right)_+ - b \left( Z - ax \right)_+^2 \right],$$
where $a$ and $b$ are positive constants, $(\cdot)_+$ denotes the positive part, ${\bf E} [ \cdot ]$ is the expectation, and $Z$ is a random variable with a generic distribution function such that ${\bf E} [ Z^2 ] < \infty$ (if needed, we may assume higher-order moments of $Z$ exist).
Any idea to tackle this problem is highly appreciated.