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Unformatted text preview: Integer Programming IE418 Lecture 13 Dr. Ted Ralphs 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. 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 ): LR ( ) : z LR ( ) = b 1 + max x S LR { ( c A 1 ) x } . IE418 Lecture 13 4 The Lagrangian Dual The next step is to obtain a dual problem formed by allowing to vary. We are looking for the value of that yield the lowest upper bound . The Lagrangian dual problem, LD , is z LD = min z LR ( ) The Lagrangian dual can be rewritten as the following LP z LD = min , { + b 1  ( c A 1 ) x i , i 1 , . . . , T, } 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 . IE418 Lecture 13 5 Solving the Lagrangian Dual with Subgradient Optimization Note that ( c A 1 ) x is an affine function of for a fixed x ....
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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 .
 Spring '08
 Ralphs

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