I have an integer optimization problem with a non-linear function ($F(X)$) in the objective and one of the constraints. $F(X)=x_n^{i,j}\Big[\sum\limits_{\forall s\neq i}\sum\limits_{\forall m\in\mathcal{S}_s}x^{s,m}_{n}+x_n^{i,j}\Big]$, where $X$ is binary.

If I solve this integer problem, easily I can linearize $F(X$be) due to the binary nature of $X$. On the other hand, since the integer problems are NP-hard, I should relax this problem and solve a continuous problem. But when I relax $X$, $F(X)$ could not be linearized by the same way in integer problem. Therefore, the relaxed optimization problem becomes a non-linear problem and this increases the computation complexity. I was wondering if there is any way to linearize or approximate $F(X)$? As another question, if I use Taylor first order approximation, should be the problem feasible at the initial point?