More understanding about $E_u[\partial_x h(x,u)]$, $u$ is a random variable

Question

Consider the subdifferential "$\partial_x h(x,u)$", $u$ is a random variable. (Note: subdifferential is a set with the definition in subgradient method.)

How to understand $$E_u[\partial_x h(x,u)]$$ and $$\{E_u[g_u]\ \ \big| \ \ g_u\in \partial_x h(x,u)\}$$

1. $E_u$ is the expected value with respect to $u$.

Moreover, are both equal?

Answer 1

It seems $\{E_u[g_u]|g_u \in \partial_x h(x,u)\}$ is the definition of $E_u[\partial_x h(x,u)]$. If $h(\cdot,u)$ is a convex function, then $\partial_x E_u[h(x,u)] = E_u[\partial_x h(x,u)]$.

A good reference is Bertsekas's 1973 paper, "Stochastic Optimiation Problems with Nondifferentiable Cost Functionals"