I have some constraints which are in the form $$ \dfrac{x_{1}-x_{2}+x_{3}+\cdots+x_{n}}{(x_{i}-x_{j})^{2}+\cdots+(x_{l}-x_{k})^{2}}+\cdots+\dfrac{x_{1}+x_{2}-x_{3}+\cdots+x_{n}}{(x_{j}-x_{i})^{2}+\cdots+(x_{l}-x_{k})^{2}} < 0$$
i.e., linear in the top and quadratic in the diameters. I am wondering is there any way for simplifying or linearizing of these kinds of constraints?
For example in $R^{5}$ we have: $$(2ak-2bk+k+1)(c_{0}(\dfrac{1}{(c_{0}+a(c_{1}+c_{4})-b(c_{2}+c_{3}))^{2}}+\dfrac{1}{((e(c_{1}-c_{4})+d(c_{2}-c_{3})))^{2}})+(5k+1-c_{0})(\dfrac{2a}{((e(c_{1}-c_{4})+d(c_{2}-c_{3})))^{2}}-\dfrac{b}{(c_{0}+a(c_{1}+c_{4})-b(c_{2}+c_{3}))^{2}}))<0$$ where a, b, c k, are some parameter in $R^{5}$ and $c_{0},...,c_{4}$ are variables.