HW10_Solutions

# HW10_Solutions - ORIE 3300/5300 1 SOLUTIONS ASSIGNMENT 10...

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ORIE 3300/5300 SOLUTIONS: ASSIGNMENT 10 Fall 2008 1. (a) Assume that the LP has a feasible solution x . Thus Ax = b, x 0 Then we know that y T Ax = y T b = ( b T y ) T < 0 , y T Ax = ( A T y ) T x 0 . But y T Ax cannot be both negative and non-negative. Therefore this is a contradiction. Thus the LP cannot be feasible. (b) The auxiliary problem is clearly feasible since v = b, x = 0 is a feasible solution (we are assuming b 0). Also notice that for any feasible so- lution, v 0. Therefore the objective function is non-positive (optimal value 0). This linear program is feasible and bounded and therefore by the Fundamental Theorem of Linear Programming, this problem has an optimal solution. Thus we just have to argue that the objective value cannot be equal to 0. Assume that there is a solution to the auxiliary problem, ( x, v ), such that ( - e ) T v = 0. Then, it must be the case that v = 0. Thus, Ax = b, x 0. Therefore x is a feasible solution for the original LP, contradicting our assumption that the original LP was infeasible. Therefore, the auxiliary problem has an optimal solution with a nega- tive optimal value.

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