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Lecture13

# Lecture13 - Integer Programming IE418 Lecture 13 Dr Ted...

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Integer Programming IE418 Lecture 13 Dr. Ted Ralphs

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IE418 Lecture 13 1 Reading for This Lecture Wolsey, Chapters 10 and 11 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.3.7, II.5.4 “Decoposition in Integer Programming,” Ralphs and Galati.
IE418 Lecture 13 2 The Decomposition Principle Again, we consider a pure integer program IP defined by z IP = max { cx | x S } , S = { x Z n + | Ax b } . We also assume all variables have finite upper and lower bounds. Recall the concept of Lagrangian relaxation: we relax some constraints and then penalize their violation. The principle of decomposition is to divide the inequalities describing S into two sets: the “ easy constraints ,” and the “ complicating constraints ,” and is such a way that removing the complicating constraints results in a integer program we can solve effectively.

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IE418 Lecture 13 3 The Lagrangian Relaxation Suppose as before that our IP is defined by max cx s.t. A 1 x b 1 (the “complicating” constraints) A 2 x b 2 (the “nice” constraints) x Z n where optimizing over S LR = { x Z n | A 2 x b 2 } is “ easy .” Lagrangian Relaxation (for λ 0 ): LR ( λ ) : z LR ( λ ) = λb 1 + max x S LR { ( c - λA 1 ) x } .
IE418 Lecture 13 4 The Lagrangian Dual The next step is to obtain a dual problem formed by allowing λ to vary. We are looking for the value of λ 0 that yield the lowest upper bound . The Lagrangian dual problem, LD , is z LD = min λ 0 z LR ( λ ) The Lagrangian dual can be rewritten as the following LP z LD = min η,λ { η + λb 1 | η ( c - λA 1 ) x i , i 1 , . . . , T, λ 0 } where { x i } T i =1 are the extreme points of conv( S LR ) . This can be solved using a cutting plane algorithm where the separation problem is an optimization problem over the set S LR .

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IE418 Lecture 13 5 Solving the Lagrangian Dual with Subgradient Optimization Note that ( c - λA 1 ) x is an affine function of λ for a fixed x . This tells us that z LR ( λ ) , when viewed as a function of λ , is the maximum of a finite number of affine functions.
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Lecture13 - Integer Programming IE418 Lecture 13 Dr Ted...

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