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# assign4.sol - CO 350 Assignment 4 Winter 2010 Solutions...

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CO 350 Assignment 4 – Winter 2010 Solutions Question # Max. marks Part marks 1 6 3 3 2 6 3 7 3 2 2 4 5 3 2 5 6 1 1 4 6 (bonus) 2 Total 30 + (2) 1. Prove the following statements. (a) The set of optimal solutions of a linear programming problem is a convex set. (b) Either an LP problem has no optimal solution, or has exactly one optimal solution, or has inﬁnitely many diﬀerent optimal solutions. Solution: (a) Let x 1 ,x 2 be two optimal solutions with optimal value K . Then c T x 1 = c T x 2 = K . Let 0 λ 1, and x 0 = λx 1 + (1 - λ ) x 2 . Since the feasible region of a linear programming problem is convex by Proposition 5.2 of the Course Notes, x 0 is a feasible solution. Moreover, c T x 0 = c T [ λx 1 + (1 - λ ) x 2 ] = λ ( c T x 1 ) + c T x 2 - λ ( c T x 2 ) = λK + K - λK = K. Hence x 0 is also optimal, and the set of optimal solutions is convex. (b) It suﬃces to show that if an LP has at least two optimal solutions, then it has inﬁnitely many diﬀerent optimal solutions. This follows immediately from convexity: Let x 1 ,x 2 be two distinct optimal solutions. Then for any real λ with 0 < λ < 1, x ( λ ) = λx 1 +(1 - λ ) x 2 is also optimal. There are inﬁnitely many possibilities for λ , and all corresponding x ( λ ) are distinct as well, so there are inﬁnitely many optimal solutions. 2. Let

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## This note was uploaded on 05/28/2011 for the course CO 250 taught by Professor Guenin during the Fall '10 term at Waterloo.

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assign4.sol - CO 350 Assignment 4 Winter 2010 Solutions...

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