Test if a system of linear inequalities has a solution.system of linear inequalities A x &gt; 0 where A is a m × n matrix with m ≥ n. Is there a simple way to check whether it has a solution?What if one of the columns of A is parametrized? In a more general case, I have one of the columns of A ( t ) as a ( t ) = t v + w, where t ∈ [ 0 , 1 ]. Is there a way to check whether all of the systems A ( t ) x &gt; 0 , t ∈ [ 0 , 1 ]have a solution?

Question

Test if a system of linear inequalities has a solution.system of linear inequalities   A  x  &amp;gt;  0 where   A is a   m  ×  n matrix with   m  ≥  n. Is there a simple way to check whether it has a solution?What if one of the columns of   A is parametrized? In a more general case, I have one of the columns of   A  (  t  ) as   a  (  t  )  =  t  v  +  w, where   t  ∈  [  0  ,  1  ]. Is there a way to check whether all of the systems   A  (  t  )  x  &amp;gt;  0  ,  t  ∈  [  0  ,  1  ]have a solution?

humbast2 · Accepted Answer

Numerically solving the following linear program                    minimize                    0                            subject to                          A              x        ≥        ε                  1          m                    should produce a solution or tell you that none exists. Pick a small   ε  &amp;gt;  0. If   m and   n are small enough, you could even use quantifier elimination.