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hw4 - IE 418 Integer Programming Problem Set#4 AND Problem...

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IE 418 – Integer Programming Problem Set #4 AND Problem Set #5 Due Date: April 27 Do the following problems. You may work with one other classmate. If you work alone, you will receive a 10% bonus. You are allowed to examine outside sources, but you must cite any references that you use. Please don’t discuss the problems with other members of the class (other than your partner, if you are working with one). 1 Uncapacitated Facility Location 1.1 Problem Nemhauser and Wolsey II.2.6.3 2 Lifting Consider the following polyhedron: K 2 = conv( { x B 10 | 35 x 1 + 27 x 2 + 23 x 3 + 19 x 4 + 15 x 5 + 15 x 6 + 12 x 7 + 8 x 8 + 6 x 9 + 3 x 10 39 } ) The inequality x 4 + x 7 + x 9 2 (1) is a valid and facet-defining inequality for the polytope K 2 ( L, U ): K 2 ( L, U ) = K 2 ∩ { x 1 = x 2 = x 3 = x 5 = x 6 = x 8 = 0 , x 10 = 1 } . 2.1 Problem What is the lifting function Θ( α ) : ∪ {∞} for the inequality (1)? Prove your answer. 2.2 Problem Is Θ( α ) from Problem 2.1 superadditive? 2.3 Problem If the answer to Problem 2.2 is “no”, then give a superadditive lower bound ( φ ( α )) for Θ( α ) in Problem 2.1.
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