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Let me define each half space i as:

$${H_i}:{c_i}{\bf{x}} \le {b_i}$$ The intersection of all such ${H_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H_i}$ is active (corresponding to a facet) in such polyhedron and if so, what is all of its vertices and rays, since we know that such each facet in a polyhedron must be in the form of $$Conv(V) + Rays(R)$$ for some set of vertices $V$ and some sets of $Rays$. Is there any well established codes (like qhull) that handles this type of problems well? Thank you.

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I've had luck with polymake and I think its basic functions can easily do all you ask for in your question. I think this tutorial page covers all you want (except getting the rays -- for that this page should help).

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  • $\begingroup$ Thank you very much, I am currently using qhull. Do you think we have some benchmarks, comparing the speed of qhull and polymake? Thank you very much. I am currently hitting the speed limit of qhull~~ $\endgroup$
    – user40780
    Commented Mar 24, 2017 at 17:08
  • $\begingroup$ @user40780, I'm not sure since I have never used qhull. As you may know, these problems are inherently hard as the dimension gets large. Others might have some ideas if you edit some more details about your cases (typical dimension, number of inequalities, where they come from, etc.) into your question. $\endgroup$
    – j.c.
    Commented Mar 24, 2017 at 17:16
  • $\begingroup$ I see. Thank you very much for suggesting polymake to me. I will give it a try :D $\endgroup$
    – user40780
    Commented Mar 24, 2017 at 17:19
  • $\begingroup$ These notes by Madeline Brandt make the good point that polymake might also be slower since it uses rational arithmetic rather than floating-point arithmetic, as qhull does: math.berkeley.edu/~brandtm/talks/polymake.pdf $\endgroup$
    – j.c.
    Commented Mar 24, 2017 at 17:20

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