A polyhedral is defined as the intersection of finite many closed hyperplane half-spaces.That is,
$P=\{ {\bf a}_i^{\text T}{\bf x} \le b_i, i=1,...,n\}$
I am puzzled about how to show that $P$ must be non-empty? Thanks!
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2 
- 1$\begingroup$ Such a set may or may not be empty. $\endgroup$Angina Seng– Angina Seng2017-11-11 15:24:41 +00:00Commented Nov 11, 2017 at 15:24
- 1$\begingroup$ Determining if $P$ is nonempty is often referred to as a linear feasibility problem. Many algorithms have been developed for them. $\endgroup$Kajelad– Kajelad2017-11-11 18:41:36 +00:00Commented Nov 11, 2017 at 18:41
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1 Answer
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The expression you have used above can also show that the polyhedron is empty like when A = [1 -1] and b = [1 -1]. Therefore just this expression cannot help to tell you whether the polyhedral is empty or not.
A nonempty set is also a polyhedron and the above expression is for a polyhedron. Therefore you need more information to prove that it's nonempty.
