Mixed Integer Formulation of Nonconvex program

I have the following optimization problem:

\begin{align} \min_{x,s,\lambda,\gamma} \quad J(x)\\ f(x)=0\\ h(x,\gamma,s)\leq0\\ \lambda^\top Gx \leq s\\ \lambda^\top e=1\\ \lambda\geq 0\\ \|G^\top \lambda\|\leq \gamma \end{align}

where $$x \in \mathcal{X} \subset \mathbb{R}^n$$, $$s \in \mathbb{R}$$, $$\mathcal{X}$$ is a convex set, $$f(x)$$ and $$h(x)$$ are affine functions, $$J(x)$$ is a quadratic function, $$e$$ is a vector consisting of all ones.

I know that the second constraint is a bilinear one. Is there any method to reformulate this problem into a mixed integer program? I know that constraints (3)-(5) could be reformulated into constraint on pointwise minimum, if there was no last constraint.