Lecture20Notes

# Lecture20Notes - C&O 355 Mathematical Programming Fall 2010...

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Fall 2010 Lecture 20 Notes Nicholas Harvey http://www.math.uwaterloo.ca/~harvey/ 1 Finding Neighbouring Vertices in the Simplex Method Let x be a basic feasible solution of the polyhedron P = { x : Ax b } . Let B be the subset of the constraints that are tight at x . Let A B denote the submatrix of A corresponding to these constraints. Similarly, let b B denote the portion of b corresponding to these constraints. So A B x = b B holds. Assume that we have perturbed the matrix A such that each vertex of P has exactly n tight constraints. Then | B | = n , so A B is square. Since x is a basic feasible solution, rank A B = n , and so A B is invertible. Since A B is invertible, we can express the objective function c as a linear combination of the tight constraints. That is, there exists a vector u such that c T = u T A B . Case 1: u 0 . In this case, we have expressed the objective function as a non-negative linear combination of the constraints that are tight at

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Lecture20Notes - C&O 355 Mathematical Programming Fall 2010...

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