1. Why is the polyhedron, P considered with the transposes A T p = 0 , b T p = 1 as opposed to p T A = 0 T 2. Why does theorem 4.7 imply that if x is a solution to the subsystem we must have 0 T x = 0 ≤ − 1? I also don't get how this is implied by p ^ T A = 0 T , p ^ T b = 1 , p ≥ 0. Theorem 4.7 is given as:Theorem 4.7 Suppose that the system of linear inequalities A x ≤ b has at least one solution, and let d be some scalar. Then, the following are equivalent:(a) Every feasible solution to the system A x ≤ b satisfes c ′ x ≤ d.(b) There exists some p ≥ 0 such that p ′ A = c ′ and p'bleq d.

Question

1. Why is the polyhedron,   P considered with the transposes       A    T    p  =  0  ,      b    T    p  =  1 as opposed to       p    T    A  =      0    T  2. Why does theorem 4.7 imply that if x is a solution to the subsystem we must have       0    T    x  =  0  ≤  −  1? I also don&#039;t get how this is implied by               p      ^        T    A  =      0    T    ,              p      ^        T    b  =  1  ,  p  ≥  0. Theorem 4.7 is given as:Theorem 4.7 Suppose that the system of linear inequalities   A  x  ≤  b has at least one solution, and let d be some scalar. Then, the following are equivalent:(a) Every feasible solution to the system   A  x  ≤  b satisfes       c    ′    x  ≤  d.(b) There exists some   p  ≥  0 such that       p    ′    A  =      c    ′   and p&#039;bleq d.

pulpasqsltl · Accepted Answer

Notice that       p    T    A  =      0    T      ⟺        A    T    p  =  0, these two conditions are equivalent, so it doesn&#039;t matter if we consider its transpose or not.To use theorem 4.7,Notice that               p      ^        T    (  −  A  )  =      0    T  , here in theorem 4.7,   c  =  0. We also have   p  ≥  0, and               p      ^        T    (  −  b  )  =  −  1  ≤  −  1, here   d  =  −  1. That is part (b) of condition 4.7 holds, then we conclude that result for part (a) holdsIf       x    ^   is a solution to the subsystem, then   (  −  A  )  x  ≤  (  −  b  ), hence we have       c    ′    x  =  0  ≤  d  =  −  1 which is a contradiction.

1. Why is the polyhedron, P considered with the transposes A T </msup> p = 0

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