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assign3.sol - CO 350 Assignment 3 Winter 2010 Solutions Due...

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CO 350 Assignment 3 – Winter 2010 Solutions Due: Friday January 29 at 10:25 a.m. Question # Max. marks Part marks 1 4 2 4 3 8 2 2 2 2 4 8 4 4 5 6 6(bonus) (3) Total 30 + (3) 1. Write the dual of the LP problem: Max 2 x 1 + 3 x 2 + 4 x 3 + x 4 - x 1 + 2 x 2 + x 3 + 3 x 4 = 3 x 1 - x 2 - x 3 + 2 x 4 7 x 1 , x 3 , 0 . Solution: Min 3 y 1 + 7 y 2 - y 1 + y 2 2 2 y 1 - y 2 = 3 y 1 - y 2 4 3 y 1 + 2 y 2 = 1 y 2 0 . 2. Prove the following version of the Duality Theorem. Let ( P ) be an LP problem in standard equality form and suppose that ( P ) and its dual both have feasible solutions. Then ( P ) has an optimal solution x * and its dual has an optimal solution y * having equal objective values. You may use the Fundamental Theorem of LP and the results of Sections 4.1 and 4.2. Solution: We are given that ( P ) has a feasible solution. By Corollary 4.2 of weak duality, since the dual has a feasible solution, we know that ( P ) is not unbounded. Therefore, by the Fundamental Theorem, ( P ) must have an optimal solution. We can now apply Theorem 4.5 to obtain the desired conclusion.
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