a4 - Consider the following linear program. ( P ) max 2 x 1...

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CO350 Assignment 4 Due: June 1 at 2pm. Question 1: Consider the linear program max( c T x : Ax = b, x 0) where A = 1 0 1 1 - 2 1 - 2 1 0 2 0 1 1 - 1 0 , b = 1 2 2 , and c = [1 , 1 , 0 , 1 , 0] T . (a) Note that x = (1 , 1 , 2 , 1 , 1) T is a feasible solution. Using the method described in class Fnd a basic feasible solution x . (b) Let B be the basis corresponding to your basic feasible solution x . ±ind an equiv- alent LP max( c T x : A x = b , x 0) where A B is an identity matrix and c B = 0. (c) ±ind an invertible matrix M such that MA = A . Question 2: Let A R m × n , b R m , c R n , and let M R m × m be an invertible matrix. Now let A = MA and b = Mb . Consider the following linear programs. ( P ) max( c T x : Ax = b, x 0), and ( P ) max( c T x : A x = b , x 0). (a) Show that ( P ) and ( P ) have the same set of feasible solutions. (b) Write the duals ( D ) and ( D ) of ( P ) and ( P ) respectively. (c) Given a feasible solution y of ( D ) show how to obtain a feasible solution of ( D ). Question 3:
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Unformatted text preview: Consider the following linear program. ( P ) max 2 x 1 + 3 x 2 Subject to x 1 + x 2-x 3 + 2 x 4 5 x 2 + 3 x 3-3 x 4 = 3 x 1 ,x 3 ,x 4 , x 2 free (a) Write the dual (D) of (P). (b) Write the complementary slackness conditions for (P) and (D). (c) Show that, if x is feasible for ( P ) and y is feasible for (D), then x is optimal for ( P ) and y is optimal for (D) if and only if x and y satisfy the complementary slackness conditions. (d) Use the complementary slackness conditions to determine if there is an optimal solution to (P) with x 3 > 0. (e) Use the complementary slackness conditions to determine if there is an optimal solution to (P) with x 1 > 0....
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