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Unformatted text preview: 1 IOE 510 Linear Programming I Wednesday February 17, 2010 Professor Amy Cohn 2 Announcements Exam solutions posted to Ctools Graded exams to be handed back on Monday 3 Recap from Last Time Proved optimality of extreme points Motivating the simplex method for solving linear programs 4 Candidate Direction Vectors Let x B represent the vector of variables basic at point x, and x N represent the nonbasic variables Given a BFS, we have n m candidate direction vectors those corresponding to increasing one element of x N Choose some p N what does the corresponding direction vector d p look like? 5 Ensuring d Is Feasible Let A B be the columns of A corresponding with the variables in x B We know that Ax = b Therefore A B x B = b (why?) Therefore x B = A B1 b (why?) We want to have A(x + d p ) remain feasible for some &gt; 0 Therefore Ad p = 0 6 What Is d p ? Let d p i represent the i th element of the vector d p d p p is 1 by convention d p i is 0 if i is in N but not p we are only relaxing a single active constraint What is d p j if j is in B? 7 What Is d p ? Let d p B be the vector in R m containing the elements of vector d p corresponding to the elements of the basis B Ad p = 0 implies A B d p B + A p = 0 Therefore d p B = A B1 A p 8 What Is the Cost Impact of d p ? If we move one step in direction d p , our change in cost will be cd p = c p c B A B1 A p (why?) For a given nonbasic variable x p , the corresponding candidate vector d p is improving if its reduced cost c p = c p c B A B1 A p is strictly negative 9...
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