I'm doing research in Optimization and I have found this obstacle in the way.



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> If we have set of half planes like $c\_ix\leq b\_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better if it is polynomial time) such that we can certify $\{\cap \_{i\neq p}\{c\_ix\leq b\_i\}\}\subset \{c\_px\leq b\_p\}$
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If you give me some hint or some references I really appreciate it.



Does a half plane contain intersection of some other half planes?
Find the maximum of $c\_{p}x$ subject to the constraints $c\_{i}x\le b\_i$ for $i\ne p$. If the maximum is at most $b\_p$, the subset relation holds.


