Lecture14

# Lecture14 - Integer Programming IE418 Lecture 14 Ashutosh...

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Integer Programming IE418 Lecture 14 Ashutosh Mahajan Dr. Ted Ralphs

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IE418 Lecture 14 1 Reading for This Lecture Wolsey Sections 7.4-7.5 Nemhauser and Wolsey Section II.4.2 Linderoth and Savelsburgh, (1999) Martin (2001) Kopman (2001) Achterberg, Koch, Martin (2005) Karamanov and Cornuejols, Branching on General Disjunctions (2007) Achterberg, Conﬂict Analysis in Mixed Integer Programming (2007)
IE418 Lecture 14 2 Branch and Bound Recap Suppose F is the feasible region for some MILP and we wish to solve max x F c ± x . Consider a partition of F into subsets F 1 , . . . F k . Then max x F c ± x = max 1 i k max x F i c ± x. In other words, we can optimize over each subset separately. Idea : If we can’t solve the original problem directly, we might be able to solve the smaller subproblems recursively. Dividing the original problem into subproblems is called branching . Taken to the extreme, this scheme is equivalent to complete enumeration.

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IE418 Lecture 14 3 Branching We have now spent several lectures discussing methods for bounding . Obtaining tight bounds is the most important aspect of the branch and bound algorithm. Branching eﬀectively is a very close second. Choosing an eﬀective method of branching can make orders of magnitude diﬀerence in the size of the search tree and the solution time.
IE418 Lecture 14 4 Baby BIP Consider a small binary integer program: min 3 x 1 + x 2 2 x 1 + x 2 1 - 2 x 1 + 2 x 2 1 x 1 , x 2 ∈ { 0 , 1 } What is the optimal solution? What is the proof?

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IE418 Lecture 14 5 Branch and Bound example Consider the baby BIP again: Use the LP-based Branch and Bound method. Step 1: Solve the LP relaxation. x 1 = 1 6 , x 2 = 2 3 How do we branch?
IE418 Lecture 14 6 Branch and Bound example X X 1.5 3 4 2.5 ( 1 6 , 2 3 ) (1 , 1) (1 , 0) ( 1 2 , 1) ( 1 2 , 0) x 2 0 7 6 x 1 0 x 1 0 x 2 1

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IE418 Lecture 14 7 Branch and Bound example Let us try a diﬀerent branching order. 3 X ( 1 6 , 2 3 ) x 1 0 7 6 x 1 1 Observe that the size of tree has been halved. This type of variance can be seen frequently in practice. For a pure binary integer program, we can have up to 2 n nodes in the search tree. By branching judiciously, we can reduce this as much as possible.
IE418 Lecture 14 8 Some Deﬁnitions Let us consider again a pure IP with feasible set S Z n . A

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## This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .

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Lecture14 - Integer Programming IE418 Lecture 14 Ashutosh...

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