Lecture 9 Simplex and Duality.pdf - Comp 360 Algorithm Design(Fall 2017 Lecture 9 The Simplex Algorithm and Duality Instructor Yang Cai Based on Kevin

Lecture 9 Simplex and Duality.pdf - Comp 360 Algorithm...

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Comp 360: Algorithm Design (Fall 2017) Lecture 9: The Simplex Algorithm and Duality Instructor: Yang Cai Based on Kevin Wayne’s Slides.
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2 Basic Feasible Solution Theorem. Let P = { x : Ax = b , x 0 }. For x P , define B = { j : x j > 0 }. Then x is a vertex iff A B has linearly independent columns. Notation. Let B = set of column indices. Define A B to be the subset of columns of A indexed by B . Ex.
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3 Basic Feasible Solution Theorem. Let P = { x : Ax = b , x 0 }. For x P , define B = { j : x j > 0 }. Then x is a vertex iff A B has linearly independent columns. Pf. n Assume x is not a vertex. n There exist direction d 0 such that x ± d P . n A d = 0 because A ( x ± d ) = b and Ax=b . n Define B' = { j : d j 0 }. n ࠵? "# ࠵? = ࠵? ( ⋅ ࠵? ( (∈"# = 0 , where ࠵? ( is the ࠵? -th column of ࠵? . n A B' has linearly dependent columns since d 0 . n Moreover, d j = 0 whenever x j = 0 because x ± d 0. n Thus B' B , so A B' is a submatrix of A B . n Therefore, A B has linearly dependent columns.
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4 Basic Feasible Solution Theorem. Let P = { x : Ax = b , x 0 }. For x P , define B = { j : x j > 0 }. Then x is a vertex iff A B has linearly independent columns. Pf. n Assume A B has linearly dependent columns. n There exist d 0 such that A B d = 0. n Extend d to n by adding 0 components. n Now, A d = 0 and d j = 0 whenever x j = 0 . n For sufficiently small λ , x ± λ d P x is not a vertex.
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5 Basic Feasible Solution Theorem. Given P = { x : Ax = b , x 0 }, x is a vertex iff there exists B { 1, …, n } such | B | = m and: ° A B
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