I am interested in methods to solve the following optimization problem:
\begin{align} \max_{x\in\mathbb{R}^n} \ (x-x_0)^TQ(x-x_0)\quad \text{subject to} \quad \mathbb{E}_\omega[g(x)]=g_0 \end{align} In the above $Q$ is assumed to be positive definite and $g(x)$ is a random variable which additionally depends on $x$. If $g$ were deterministic I could just apply regular Lagrangian multiplier theory. However, I don't know how to deal with the stochasticity of $g$. Is there some Robbins-Monro type algorithm for these kinds of problems?