# given a system of inequalities expressed in the following way: A x &gt; <munder>

given a system of inequalities expressed in the following way:
$Ax>\underset{_}{k}$
where $A\in {\mathbb{M}}_{n,m\left(\mathbb{R}\right)}$, with n>m, and $\underset{_}{k}=\left(k,k,\dots ,k\right)\in {\mathbb{R}}^{n}$
In general, the system might or might not admit solutions. I would like to find a solution $x\in {\mathbb{R}}^{m}$ that minimizes the number of violated inequalities.
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garcialdariamcy4q
$Ax>k$ is a difficult concept in optimization. Usually we deal with $Ax\ge k$. (If needed you can add a small number $\epsilon >0$ to $k$; note also that solvers typically employ a feasibility tolerance, so ε should be larger than that).
So, assume we have $Ax\ge k$. Next we can do
$\begin{array}{rl}min\phantom{\rule{mediummathspace}{0ex}}& \sum _{i}{\delta }_{i}\\ & \sum _{j}{a}_{i,j}{x}_{j}\ge k-{\delta }_{i}M\\ & {\delta }_{i}\in \left\{0,1\right\}\end{array}$
Here $M$ is a large enough number.
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