L24 - CO350 Linear Programming Chapter 8 Degeneracy and...

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Unformatted text preview: CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 Recap The perturbation method ( P ) max c T x s.t. Ax = b x ≥ Assumption: B is a feasible basis with A B = I . ( P ) : perturb b to b = b + [ ε,ε 2 ,...,ε m ] T . Theorem 8.3 (pg 111) (a) ( P ) is nondegenerate. (b) Feasible bases of ( P ) are feasible bases of ( P ) . (c) Optimal bases of ( P ) are optimal bases of ( P ) . (d) Valid pivots for ( P ) are valid for ( P ) . (e) Bases detecting ( P ) unbounded also detects ( P ) un- bounded. Corollary 8.3 (pg 112) The simplex method applied to ( P ) starting from B will terminate after a finite number of iterations. Moreover, B optimal for ( P ) = ⇒ B optimal for ( P ) , and ( P ) unbounded = ⇒ ( P ) unbounded. Chapter 8: Finite Termination 2 Recap The lexicographical simplex method Simplex method on ( P ) with a special choice rule for leav- ing variables that mimics the choice of leaving variables for ( P ) . This special rule is called the lexicographical rule . The lexicographical rule: pick the lexicographical minimum of ( ¯ b i ,β i 1 ,β i 2 ,...,β im ) ¯ a ik over { i ∈ B : ¯ a ik > } where [ β i 1 ,β i 2 ,...,β im ] is the h-th row of the matrix A- 1 B and i is the h-th index in the basis B . A shortcut: A- 1 B appears in the tableau corresponding to B as columns indexed by the initial basis B . Chapter 8: Finite Termination 3 Smallest-subscript rules ( § 8.5) Recall: smallest subscript rule for entering variable “Pick the nonbasic variable with the least subscript among those with positive reduced cost”. Similar rule for leaving variable: “Pick the basic variable with the least subscript among those that tie for minimum ratio”....
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L24 - CO350 Linear Programming Chapter 8 Degeneracy and...

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