Where can I solve a system of non-linear inequalities I have a problem on the form: find $x\in\mathbb{R}^n,y\in\mathbb{R}^m$ such that:
$$Ax\ge a$$
$$By\ge b$$
$$f(x)\ge 0$$
$$f_i(x)\cdot y_j=0$$
$$x_i,y_j\ge 0$$
In the later $A$ and $B$ are matrices, $a\in\mathbb{R}^n,b\in\mathbb{R}^m$, $f=(f_1\dots f_n)$ is a linear vectorial function of $x$ and $x=(x_1\dots x_n)$, $y=(y_1\dots y_m)$.
There is a solver (Matlab for example) or a method that could solve that kind of inequalities?
 A: One fairly generic way to find a point that satisfies all of the required inequalities is to model it as a nonlinear feasibility problem (optimization problem with trivial objective). One such formulation is below:
\begin{align}
\label{eqn:fp}
\underset{x \in \mathbb{R}^n, \: y \in \mathbb{R}^m}{\min} \:\: & 0 \tag{FP} \\
\text{s.t.} \quad\:\: & Ax \geq a, \nonumber \\
& By \geq b, \nonumber \\
& f(x) \geq 0, \nonumber \\
& f_i(x) \cdot y_j = 0, \nonumber \\
& x, y \geq 0 \nonumber.
\end{align}
Any feasible solution to Problem \eqref{eqn:fp}, if one exists, is also optimal. You can potentially throw Problem \eqref{eqn:fp} to fmincon in MATLAB to try and find a feasible solution.
It is noteworthy that Problem \eqref{eqn:fp} is a (special instance of) nonlinear program with complementarity constraints (see, for instance, this article) since it contains equations of the form $f_i(x) \cdot y_j = 0$ (whatever the notation means). Nonlinear programming solvers may face numerical difficulties on instances of problems in this general class, so you may have to use tailored solution techniques based on the form of your function $f$.
