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handout18 - Recap: the simplex method Math 171A: Linear...

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Math 171A: Linear Programming Lecture 18 Finding a Feasible Point Philip E. Gill c ± 2011 http://ccom.ucsd.edu/~peg/math171a Wednesday, February 16th, 2011 Recap: the simplex method One step moves from a vertex to an adjacent vertex . Each step requires the solution of two sets of equations: A T k λ k = c and A k p k = e s where A k is the nonsingular working-set matrix. Given a nonsingular A 0 , all subsequent A k are nonsingular. If there is a tie in the choice of constraint to enter or leave the working set, the simplex method may stall at a vertex. The method cycles infinitely if there is a repeat of a sequence of constraint changes at a stalled vertex. UCSD Center for Computational Mathematics Slide 2/33, Wednesday, February 16th, 2011 Recap: Bland’s Least-index rule Apply the simplex method with the least-index rules: w s = min { w i : ( λ k ) i < 0 } t = min { j : σ j = α k } Bland’s rule is not useful computationally . it usually needs more iterations than the Dantzig rule. Better anti-cycling rules are based on constraint perturbation . Nevertheless, Bland’s rule is useful as a theoretical tool . UCSD Center for Computational Mathematics Slide 3/33, Wednesday, February 16th, 2011 Theorem A vertex x 0 with active-set matrix A a is a solution of an LP if and only if there is a nonnegative basic solution of A T a λ a = c. Proof: If x 0 is a nondegenerate vertex then A a is nonsingular, λ a is unique and the result follows. Assume that x 0 is a degenerate vertex, with A T a = UCSD Center for Computational Mathematics Slide 4/33, Wednesday, February 16th, 2011
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Consider the auxiliary problem: minimize x R n c T x subject to A a x A a x 0 x 0 is an initial vertex for this problem. Starting at x 0 , run the simplex method with Bland’s rule. The simplex method solves a sequence of square systems A T w λ w = c A w p = e s and must terminate or declare the problem unbounded . UCSD Center for Computational Mathematics Slide 5/33, Wednesday, February 16th, 2011 EITHER (A) x 0 is optimal OR (B) there is an unbounded direction p In other words: EITHER (A) x 0 is optimal c = A T w λ w for some λ w 0 OR (B) p such that c T p < 0, A w p = e s and A a p 0 UCSD Center for Computational Mathematics Slide 6/33, Wednesday, February 16th, 2011
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This note was uploaded on 03/19/2012 for the course MATH 171a taught by Professor Staff during the Spring '08 term at UCSD.

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handout18 - Recap: the simplex method Math 171A: Linear...

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