Say that I am given a cone, as specified by the extremal rays whose facets form its convex hull, what is the most efficient algorithm that finds the linear-rank inequalities whose intersection defines the interior of the cone? Also, what is its run time? I have a situation where I have on the order of several thousand extremal rays organizable into 16 distinct families related by S_6 symmetry, and I am wondering about whether there exists a feasible algorithm that addresses this problem.
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