# Monetary Policy under commitment. How to solve the optimization problem?

Under commitment the CB might follow this problem as Monetary Policy strategy: $$\min_{\pi_t,x_t}=E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( \pi_t^2 + \alpha x_t^2 )\right)$$ $$\text{s.t. }\text{ } \pi_t=\beta E_t[\pi_{t+1}]+kx_t+u_t$$ where $$u_t$$ follows AR(1): $$u_t=\rho u_{t-1}+w_t$$ ($$w_t$$ is white noise). $$\pi$$ is inflation and $$x$$ is output gap . I need help to solve this mathematically. I assume the Lagrangian looks something like: $$L(x_t,y_t)= E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( x_t^2 + \alpha y_t^2 ) -\lambda_t(\beta E_t[x_{t+1}]+ky_t+u_t-x_t) \right)$$ Am I right? How can I find the derivatives wrt. $$x_t,y_t,\lambda_t$$?

I am confused by how to differentiate with expectations. also I don't know how to deal with $$\partial E_t[\pi_{t+1}] / \partial \pi_t$$

All the parameters are positive.