Lecture23 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 23 Dr. Ted Ralphs IE417 Lecture 23 1 Reading for This Lecture • Chapter 10 IE417 Lecture 23 2 Linear Programming • We are given A ∈ R m × n , b ∈ R m . Assume S = { x ∈ R n s.t. Ax = b,x ≥ is bounded. • We want to solve the LP min c > x s.t. Ax = b x ≥ • We need only consider the extreme points. IE417 Lecture 23 3 Characterization of Extreme Points • Arrange the columns of A such that A = [ B,N ] , where B is a non- singular n × n matrix. • Then x is an extreme point of S if and only if x = [ x B , 0] where x B = B- 1 b for some arrangement such that B- 1 b ≥ • This implies that the number of extreme points is finite (but still potentially very large). IE417 Lecture 23 4 The Simplex Algorithm • Note that x B = B- 1 b- B- 1 Nx N • Hence, c > x = c > B x B + c- N > x N = c > B B- 1 b + ( c > N- c > B B- 1 N ) x N • So if c > N- c > B B- 1 N ≥ , we have found the optimal solution (why?) • Otherwise, suppose some component of c > N- c > B B- 1 N is negative.is negative....
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Lecture23 - Advanced Mathematical Programming IE417 Lecture...

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