I struggling with the attached constraints as I want to find a suitable linear approximation to be able to implement it using linear solvers. Here are the constraints:
$$h_s^\top x_k+\eta_s^\top \mu_k^{\hat w}+\alpha_s(\eta_s^\top \Sigma_k^{\hat w}\eta_s)^{1/2}\leq b_s,$$ with $\alpha_s=F_N^{-1}(1-\varepsilon_s)$, where $F_N^{-1}(\cdot)$ is the standard normal inverse cumulative distribution function (CDF).
where $x_k$ is the system states at time $k$ in $\Bbb R^2$. $\mu$ and $\Sigma$ are also depending on $x_k$.
Thanks in advance