a3 - (1) Write the dual (D) of (P). How many variables does...

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CO350 LINEAR OPTIMIZATION - HW 3 Due Date: Friday May 25th, at 2pm, in the drop box outside the Tutorial Center. Recall, late assignments will not be graded. Exercise 1. Write the dual of the following linear program, min x 1 - 2 x 3 + 3 x 4 - x 4 subject to x 1 + x 2 - x 3 + 3 x 4 2 - 3 x 2 + x 3 + x 4 0 - 3 x 1 - 2 x 4 = 8 2 x 1 - 2 x 2 1 x 1 , x 3 0 Exercise 2. Consider the following linear programs. max( c T x : Ax = b,x 0) (P1) and max( c T x : Ax = d,x 0) . (P2) Prove that, if (P1) is unbounded and (P2) is feasible, then (P2) is unbounded. Hint: Consider the duals of (P1) and (P2). You may use the Duality Theorem for Linear Programming. Exercise 3. Suppose we have numbers a 1 ,...,a n and b > 0 consider the linear program (P) maximize x 1 + x 2 + ... + x n subject to a 1 x 1 + a 2 x 2 + ... + a n x n b x 1 ,...,x n
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Unformatted text preview: (1) Write the dual (D) of (P). How many variables does (D) have? (2) Show that if any a i then (D) is infeasible. (3) Show that if any a i then (P) is unbounded. Exercise 4. Consider the following linear program (P): max c T x s.t. Ax = b Fx d Note, the variables x are free. (1) Rewrite (P) so that it is in standard equality form. Compute then the dual (D) of the LP you obtain using the formula for problems in standard equality form. (2) Give a self contained proof that (D) is indeed the dual of (P) along the same line as the proof of Theorem 4.1. 1...
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