I am looking for an algorithm that, given a set of linear inequalities in $m$ variables, returns a simplified set. "Simplified" may mean an equivalent set with a smallest number of inequalities. If finding an optimal solution is computationally-hard, then a heuristic solution, that applies some reasonable simplification rules, would also be fine. As an example, given the inequalities:
$$ x + y \geq 1$$ $$ x + y \leq 0$$
the algorithm should return the empty set. Given the inequalities:
$$ x + y \geq 1$$ $$ 2 x + 2 y \geq 3$$
the algorithm should return e.g.
$$ 2 x + 2 y \geq 3$$
A Google search for "simplifying linear inequalities" yielded graphical solutions for two variables, e.g. here and here. I am not looking for a graphical solution but for a logical solution in the form of a small set of equivalent inequalities.