We are working on a combinatorial optimization problem. In order to solve it using CPLEX, we need to linearize the non-linear constraint stated in the following.
Let $p_i, i \in I$ denotes a set of positive continuous decision variables. $y_i, i \in I$, $x_{ji}, j \in J, i \in I$ are two sets of binary decision variables. How to linearize the following constraint:$$p_i y_i - \sum_{j \in J}b_{j} x_{ji} \le \sqrt{\sum_{j \in J} x_{ji}^2 \sigma_j^2}, \quad\quad\forall i \in I$$
where $b_{j}$ and $\sigma_j$ are positive known parameters of the problem.