How to check satisfiability of a large number of "lorenzian" quadratic inequalities

by cplusplusguru   Last Updated May 16, 2019 04:20 AM

Given a list of $m$ vectors $x^i=(x^i_t,\textbf{x}^i)\in\mathbb{R}^{n+1}$, $i\in \mathbb{Z}_m$ and two disjoint sets of vector pairs $A,B\subset \mathbb{Z}_m\times\mathbb{Z}_m$ as well as a set $C\subseteq \mathbb{Z}_m$ is there an easy way (computational or analytical) to check if the following set of inequalities is satisfiable?

$\forall(i,j)\in A\quad (x_t^i-x_t^j)^2-(\textbf{x}^i-\textbf{x}^j)^2>0$

$\forall(i,j)\in B\quad (x_t^i-x_t^j)^2-(\textbf{x}^i-\textbf{x}^j)^2<0$

$\forall i\in C\quad x_t^i-x_t^j>0$

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