I have a Mixed Integer Model. In order to solve it using CPLEX, I need to linearize the non-linear constraint stated in the following.
Let $X_{i,j,t}$ is binary variable. How to linearized the following constraint:
$m\geq \sum_{i,j \in A}\mu _{i,j}X_{i,j,t}+z_\alpha \sqrt{\sum_{i,j \in A}\sigma_{i,j}^2X_{i,j,t}}$
Where $\mu _{i,j},z_\alpha,m$ and $\sigma_{i,j}$ are positive known parameters of the problem
One approach is to rewrite as $$m - \sum_{i,j \in A}\mu _{i,j}X_{i,j,t} \ge z_\alpha \sqrt{\sum_{i,j \in A}\sigma_{i,j}^2X_{i,j,t}},$$ square both sides, and linearize the resulting products of binary variables in the usual way.
You might need to additionally impose $$m \ge \sum_{i,j \in A}\mu _{i,j}X_{i,j,t}$$ to avoid extraneous solutions.