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L23 - CO350 Linear Programming Chapter 8 Degeneracy and...

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CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 24th June 2005
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Chapter 8: Finite Termination 1 Recap The perturbation method ( P ) max c T x s.t. Ax = b x 0 Assumption: B is a feasible basis with A B = I . Perturb the right hand side to b = b + [ ε, ε 2 , . . . , ε m ] T to get ( P ) max c T x s.t. Ax = b x 0 We showed that B is also a feasible basis of ( P ) . Tableaux for ( P ) and ( P ) differ in right hand side only = choices of leaving variables are affected. Lemma 8.2 If ε is positive and sufficiently small, then α 0 + α 1 ε + α 2 ε 2 + · · · + α m ε m < β 0 + β 1 ε + β 2 ε 2 + · · · + β m ε m ⇐⇒ ( α 0 , α 1 , α 2 , . . . , α m ) L < ( β 0 , β 1 , β 2 , . . . , β m )
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Chapter 8: Finite Termination 2 Example (cycling example on pg 107) Initial tableau: z - 2 x 1 - 3 x 2 + x 3 + 12 x 4 = 0 - 2 x 1 - 9 x 2 + x 3 + 9 x 4 + x 5 = 0 1 3 x 1 + x 2 - 1 3 x 3 - 2 x 4 + x 6 = 0 Tableau for perturbed problem: z - 2 x 1 - 3 x 2 + x 3 + 12 x 4 = 0 - 2 x 1 - 9 x 2 + x 3 + 9 x 4 + x 5 = ε 1 3 x 1 + x 2 - 1 3 x 3 - 2 x 4 + x 6 = ε 2 ¯ c 2 is largest positive reduced cost, so x 2 enters. min {- , ε 2 / 1 } = ε 2 , so x 6 leaves. Pivot on (6 , 2) : z - x 1 + 6 x 4 + 3 x 6 = 3 ε 2 x 1 - 2 x 3 - 9 x 4 + x 5 + 9 x 6 = ε + 9 ε 2 1 3 x 1 + x 2 - 1 3 x 3 - 2 x 4 + x 6 = ε 2 ¯ c 1 is only positive reduced costs, so x 1 enters. min { ( ε + 9 ε 2 ) / 1 , ε 2 / 1 3 } = 3 ε 2 , so x 2 leaves. Pivot on (2 , 1) : z + 3 x 2 - x 3 + 6 x 6 = 6 ε 2 - 3 x 2 - x 3 - 3 x 4 + x 5 + 6 x 6 = ε + 6 ε 2 x 1 + 3 x 2 - x 3 - 6 x 4 + 3 x 6 = 3 ε 2 The perturbed problem is unbounded. Same pivots on original problem gives same conclusion.
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Chapter 8: Finite Termination 3 Theorem 8.3 (pg 111) (a) ( P ) is nondegenerate. (b) B is a feasible basis of ( P ) = B is a feasible basis of ( P ) . (c) B is an optimal basis of ( P ) = B is an optimal basis of ( P ) . (d) x k can enter and x r can leave in tableau for ( P ) cor- responding to B = same for tableau for ( P ) corresponding to B .
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