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handout22 - Recap Simplex method for LPs in standard form...

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Math 171A: Linear Programming Lecture 22 The Simplex Method for Standard Form Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Monday, February 28th, 2011 Recap: Simplex method for LPs in standard form minimize x R n c T x subject to Ax = b | {z } equality constraints , x 0 | {z } simple bounds We apply “mixed-constraint” simplex with full matrix A I Each vertex is a feasible basic solution of Ax = b Simplex method involves solving with an m × m nonsingular B UCSD Center for Computational Mathematics Slide 2/38, Monday, February 28th, 2011 k th iteration STEP 1: Check for optimality (Implicitly solve A T k λ k = c ) Solve B T π = c B and set z N = c N - N T π . If [ z N ] i 0 for i = 1, 2, . . . , n - m , then STOP . Otherwise, define [ z N ] s = min( z N ). The s -th nonbasic variable (i.e., x ν s ) will become basic. N = { ν 1 , ν 2 , . . . , ν s s th element of N , . . . , ν n - m } UCSD Center for Computational Mathematics Slide 3/38, Monday, February 28th, 2011 STEP 2: Compute the search direction (Implicitly solve A k p k = e m + s ) B N 0 I N ! p B p N ! = 0 e s ! A step along p k increases x ν s but keeps other nonbasics fixed at 0 UCSD Center for Computational Mathematics Slide 4/38, Monday, February 28th, 2011
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STEP 2: (continued) Compute the search direction Solve Bp B + Np N = 0 p N = e s Bp B = - Np N = - Ne s = - ( s th column of N ) = - (column ν s of A ) = - a ν s we solve Bp B = - a ν s UCSD Center for Computational Mathematics Slide 5/38, Monday, February 28th, 2011 STEP 3: Step to an adjacent vertex If we take a step α along the vector p B p N ! = p B e s ! Then the basic variables change to x B + α p B all the basic variables change UCSD Center for Computational Mathematics Slide 6/38, Monday, February 28th, 2011 STEP 3: (continued) Step to an adjacent vertex What happens to the nonbasic variables? x N + α p N = x N + α e s = x N + 0 . . . 0 α 0 . . . 0 = 0 . . . 0 α 0 . . . 0 row s all nonbasics remain at 0 except x ν s , which wants to increase . UCSD Center for Computational Mathematics Slide 7/38, Monday, February 28th, 2011 STEP 3: (continued) Step to an adjacent vertex The changed variables are x B and x ν s (which increases from 0 to α ) We must ensure that x B + α p B 0 σ i = [ x B ] i - [ p B ] i if [ p B ] i < 0 + if [ p B ] i 0 This is known as the min ratio test . Define α = min { σ i } If α = + , the LP is unbounded, STOP . UCSD Center for Computational Mathematics Slide 8/38, Monday, February 28th, 2011
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STEP 3: (continued) Step to an adjacent vertex If α = σ t then [ x B ] t + σ t [ p B ] t = 0 the t -th basic becomes nonbasic t points to index β t of B s points to index ν s of N [ x B ] t 0 x β t goes from basic to nonbasic [ x N ] s α x ν s goes from nonbasic to basic UCSD Center for Computational Mathematics
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