quiz4solutions

quiz4solutions - 640:354:01, 03/07/2007 Quiz #4, with...

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640:354:01, 03/07/2007 Quiz #4, with solutions 1. Write the dual of the following LP problem. Maximize z = 3 x 1 + 2 x 2 - 5 x 3 subject to 2 x 1 - x 2 + 3 x 3 6 x 1 + x 2 - 4 x 3 = 10 x 1 ,x 2 0 Solution: The dual problem is Minimize z 0 = 6 y 1 + 10 y 2 subject to 2 y 1 + y 2 3 - y 1 + y 2 2 3 y 1 - 4 y 2 = - 5 y 1 0 2. Let ( P ) denote a certain linear programming problem, and ( D ) its dual. For each of the following cases, determine whether it can occur or not. (A) ( P ) and ( D ) are both infeasible. YES, this is possible. For example, there are infeasible problems which are self-dual. (A problem is self-dual if its dual is equivalent to the problem.) More speciﬁcally, consider the following problem in standard form: Maximize z = x 1 - x 2 subject to x 2 ≤ - 1 - x 1 1 x 1 , x 2 0 The problem is clearly infeasible. Its dual is

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Minimize z 0 = - y 1 + y 2 subject to - y 2 1 y 1 ≥ - 1 y 1 , y 2 0 which is equivalent to the primal. (B) ( P ) and ( D ) are both unbounded. NO. If a problem is unbounded, then its dual must be infeasible (by WEAK DUALITY which says that c T x b T y whenever x is a feasible solution to the primal,
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This note was uploaded on 09/12/2008 for the course MATH 354 taught by Professor Vogelius during the Spring '08 term at Rutgers.

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quiz4solutions - 640:354:01, 03/07/2007 Quiz #4, with...

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