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handout16 - Math 171A: Linear Programming The simplex...

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Math 171A: Linear Programming Lecture 16 The Simplex Method Philip E. Gill c ± 2011 http://ccom.ucsd.edu/~peg/math171a Friday, February 11th, 2011 The simplex method defines a sequence of vertices: x 0 , x 1 , x 2 , ..., x k vertex number , ..., such that c T x 0 c T x 1 c T x 2 ≥ ··· ≥ c T x k ≥ ··· For the moment, we assume that a starting vertex x 0 is known. UCSD Center for Computational Mathematics Slide 2/35, Friday, February 11th, 2011 Three ingredients of the k th iteration: S1. A test for optimality at vertex k using Lagrange multipliers: A T k λ k = c S2. If the vertex is not optimal, define direction p k that moves off of one active constraint associated with the vertex. A k p k = e s S3. Take the maximum possible feasible step along p k . x k +1 = x k + α k p k We choose A k so that we can easily solve for λ k and p k . UCSD Center for Computational Mathematics Slide 3/35, Friday, February 11th, 2011 The rows of A k are the normals of a working set of active constraints. These rows are chosen so that A k is a nonsingular matrix. UCSD Center for Computational Mathematics Slide 4/35, Friday, February 11th, 2011
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At x k we define an index set W k (the working set ) with three properties: W1. W k contains exactly n indices; W2. j ∈ W k implies that constraint j is active at x k ; W3. the normals of constraints with indices in W k are the rows of the nonsingular matrix A k (the working-set matrix ). UCSD Center for Computational Mathematics Slide 5/35, Friday, February 11th, 2011 When the vertex x k is nondegenerate : W k = A ( x k ) , (the active set ) , and A k = A a ( x k ) UCSD Center for Computational Mathematics Slide 6/35, Friday, February 11th, 2011 When the vertex x k is degenerate , W k ⊂ A ( x k ), with: A a m a × n A k n × n Active-set matrix Working-set matrix The j -th index in W k will be denoted by w j , so that W k = { w 1 , w 2 ,..., w n | {z } row indices of A } The working-set matrix and right-hand side is A k = a T w 1 a T w 2 . . . a T wn , b k = b w 1 b w 2 . . . b wn The vertex x k satisfies Ax k b and A k x k = b k UCSD Center for Computational Mathematics Slide 8/35, Friday, February 11th, 2011
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It is important to understand “what refers to what” W k = { w 1 , w 2 ,..., w s s th working-set index ,..., w n } Constraint a T ws x b ws is associated with row s of A k Constraint a T ws x b ws is associated with row w s of A Constraint a T t x b t is associated with row t of A
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handout16 - Math 171A: Linear Programming The simplex...

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