# lect20 - provide details, as long as you clearly indicate...

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ISE 536–Fall03: Linear Programming and Extensions November 17, 2003 Lecture 20: Large Scale LP, Examples Lecturer: Fernando Ord´o˜nez 1 Bounds on Benders Decomposition Let ( x * , γ * ) be the optimal solution, and z * ( x * , γ * ) be the optimal objective function value to the master problem: min x c t x + k i =1 α i γ i s . t . Ax = b x 0 d t i z k - x t B i z k γ i for z k i BFS of D t z f d t i w k - x t B i w k 0 for w k i extreme ray of D t z f Let z * be the optimal objective function value of the complete problem min x,y 1 ,...,y k c t x + α 1 f t y 1 . . . α k f t y k s . t . Ax = b B 1 x Dy 1 = d 1 B 2 x Dy 2 = d 2 . . . . . . = . . . B k x Dy k = d k and φ i ( x ) = min y i f t y i s . t . Dy i = d i - B i x y i 0 x, y 1 , . . . , y k 0 Show that z * ( x * , γ * ) z * c t x * + k i =1 α i φ i ( x * ). 1

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2 Example Exercise 6.6 Consider a linear programming problem in standard form in which the matrix A has the following structure: A 00 A 01 ··· ··· A 0 n A 10 A 11 . . . A 22 . . . . . . A n 0 A nn Show how a decomposition method can be applied to a problem with this structure. Do not
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Unformatted text preview: provide details, as long as you clearly indicate the master problem and the subproblems. Hint: Decompose twice. Exercise 6.8 consider the Dantzig-Wolfe decomposition method and suppose that we are at a basicd feasible solutin to the master problem. 1. Show that at least one of the variables λ j 1 must be a basic variable. 2. Let r 1 be the current value of the simplex multiplier associated with the ﬁrst convexity constraint (6.12), and let z 1 be the optimal cost in the ﬁrst subproblem. Show that z 1 ≤ r 1 . 2...
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## This note was uploaded on 02/13/2012 for the course ISE 536 taught by Professor Yy during the Spring '05 term at South Carolina.

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lect20 - provide details, as long as you clearly indicate...

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