08 - Industrial Engineering 634 Fall 2011 Integer...

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Unformatted text preview: Industrial Engineering 634 Fall 2011 Integer Programming Lecturer: Nelson Uhan Scribe: Jinhak Kim September 9, 2011 Lecture 8. Strong Formulations, cont. What is an ideal formulation? Let F = ¨ x 1 ; ¡ ¡ ¡ ; x k © be the set of feasible solutions to some IP. Assume the feasible set F is bounded and hence F is nite. Lemma. min f cx : x 2 Fg = min f cx : x 2 conv( F ) g . Proof. Since F conv( F ), it is clear that min f cx : x 2 conv( F ) g min f cx : x 2 Fg : For the reverse inequality, let x £ be an optimal solution to min f cx : x 2 conv( F ) g . That is, cx £ = min f cx : x 2 conv( F ) g . Then there exist 1 ; ¡ ¡ ¡ ; k with i 0 for all i = 1 ; ¡ ¡ ¡ ; k and P k i =1 i = 1 such that x £ = k X i =1 i x i . By multiplying c on both sides, we have cx £ = k X i =1 i ( cx i ). Since cx £ is a weighted average of cx 1 ; ¡ ¡ ¡ ; cx k , cx £ cx i for some x i 2 F . Thus min f cx : x 2 conv( F ) g = cx £ min f cx : x 2 Fg : Recall conv( F ) is a polytope. Hence if we can represent conv() is a polytope....
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This note was uploaded on 02/14/2012 for the course IE 530 taught by Professor Ravindran during the Spring '97 term at Purdue.

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08 - Industrial Engineering 634 Fall 2011 Integer...

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