Lecture11 - Integer Programming IE418 Lecture 11 Dr Ted...

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Integer Programming IE418 Lecture 11 Dr. Ted Ralphs
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IE418 Lecture 11 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4
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IE418 Lecture 11 2 Relaxation For simplicity, we now consider a pure integer program IP defined by z IP = max { cx | x S } , S = { x Z n + | Ax b } . Definition 1. A relaxation of IP is a maximization problem defined as z R = max { z R ( x ) | x S R } with the following two properties: S S R (1) cx z R ( x ) , x S. (2)
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IE418 Lecture 11 3 Importance of Relaxations The main purpose of a relaxation is to obtain an upper bound on z IP . Relaxation is used as a method of bounding in branch and bound. The idea is to choose a relaxation that is much easier to solve than the original problem. Note that the relaxation must be solved to optimality to yield a valid bound. We will consider three basic types of relaxations. LP relaxation Combinatorial relaxation Lagrangian relaxation Relaxations are also used in some other bounding schemes we will look at.
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