Lecture15 - Advanced Operations Research Techniques IE316...

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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 15 Dr. Ted Ralphs IE316 Lecture 15 1 Reading for This Lecture Bertsimas Chapter 5 IE316 Lecture 15 2 Global Dependence on the Right-hand Side Vector Consider a family of polyhedra parameterized by the vector b P ( b ) = { x R n : Ax = b,x } Note that S = { b : P ( b ) is nonempty } = { Ax : x } is a convex set. We now consider the function F ( b ) = min x P ( b ) c T x . In what follows, we will assume feasibility of the dual and hence that F ( b ) is finite for all b S . We will try to characterize the function F ( b ) . IE316 Lecture 15 3 Characterizing F(b) For a particular vector b , suppose there is a nondenegenerate optimal basic feasible solution given by basis B . As before, nondegeneracy implies that we can perturb b without changing the optimal basis. Therefore, we have F ( b ) = c T B B- 1 b = p T b, for b close to b. This means that in the vicinity of b , F ( b ) is a linear function of b . Consider the extreme points p 1 ,...,p N of the dual polyhedron. There must be an extremal optimum to the dual and so we can rewrite F ( b ) as F ( b ) = max i =1 ,...,N ( p i ) T b Hence, F ( b ) is a piecewise linear convex function . IE316 Lecture 15 4 Another Parameterization Now consider the function f ( ) = F ( b + d ) for a particular vector b and direction d . Using the same approach, we obtain f ( ) = max i =1 ,...,N ( p i ) T ( b + d ) , b + d S. Again, this is a piecewise linear convex function . IE316 Lecture 15 5 The Set of all Dual Optimal Solutions Consider once more the function F ( b ) . Definition 1. A vector p R m is a subgradient of F at b if F ( b )+ p T ( b- b ) F ( b ) ....
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This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture15 - Advanced Operations Research Techniques IE316...

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