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I'm trying to solve for the optimal policy function $h^*(x)$ over the domain of all policy functions $u_t = h(x_t)$ for the following problem. \begin{equation} \min_{h(x)} \, \mathbb{E}[\sum_{t=0}^{\infty} \, f(x_t,u_t)] \end{equation} where $f(x_t,u_t) = x_t^2u_t^2$ and subject to the following constraints \begin{equation} x_{k+1} = g(x_k,u_k,\epsilon_k) \forall k \end{equation} where $\epsilon_k$ is the stochastic noise that can attain values from a finite discrete set \begin{equation} \sum_{r=0}^{N}u_r \geq M \end{equation} Can someone suggest methods to solve for the optimal policy function $u_t = h^*(x_t)$ with the given constraints as described above?

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