Let P = (xR IAx = b, x20) be a nonempty polyhedron, and let m be...

  

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Let P = (x€R" IAx = b, x20) be a nonempty polyhedron, and let m be the dimension of the vector b. We call x; a null variable if x; = 0 whenever r € P. %3D ONLY PART B AND C (a) Suppose that there exists some p € Rm for which p' A20', p'b = 0, and such that the jth component of p'A is positive. Prove that x; is a null variable. (b) Prove the converse of (a) : if xj is a null variable, then there exists some pERm with the properties stated in part (a). (c) If xj is not a null variable, then by definition, there exists some y €P for which yj > 0. Use the results in parts (a) and (b) to prove that there exist x € Pand p ERM such that: p'A20', p'b = 0, x+ A'p > 0. Linear Programming Exercise 4.19 Let P = (x€R" IAx = b, x20) be a nonempty polyhedron, and let m be the dimension of the vector b. We call x; a null variable if x; = 0 whenever r € P. %3D ONLY PART B AND C (a) Suppose that there exists some p € Rm for which p' A20', p'b = 0, and such that the jth component of p'A is positive. Prove that x; is a null variable. (b) Prove the converse of (a) : if xj is a null variable, then there exists some pERm with the properties stated in part (a). (c) If xj is not a null variable, then by definition, there exists some y €P for which yj > 0. Use the results in parts (a) and (b) to prove that there exist x € Pand p ERM such that: p'A20', p'b = 0, x+ A'p > 0. Linear Programming Exercise 4.19

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