dual simplex - Math 407A Linear Optimization Lecture 11 The...

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Math 407A: Linear Optimization Lecture 11: The Dual Simplex Algorithm Math Dept, University of Washington
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The Dual Simplex Algorithm
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P maximize - 4 x 1 - 2 x 2 - x 3 subject to - x 1 - x 2 + 2 x 3 ≤ - 3 - 4 x 1 - 2 x 2 + x 3 ≤ - 4 x 1 + x 2 - 4 x 3 2 0 x 1 , x 2 , x 3
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P maximize - 4 x 1 - 2 x 2 - x 3 subject to - x 1 - x 2 + 2 x 3 ≤ - 3 - 4 x 1 - 2 x 2 + x 3 ≤ - 4 x 1 + x 2 - 4 x 3 2 0 x 1 , x 2 , x 3 D minimize - 3 y 1 - 4 y 2 + 2 y 3 subject to - y 1 - 4 y 2 + y 3 ≥ - 4 - y 1 - 2 y 2 + y 3 ≥ - 2 2 y 1 + y 2 - 4 y 3 ≥ - 1 0 y 1 , y 2 , y 3
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P maximize - 4 x 1 - 2 x 2 - x 3 subject to - x 1 - x 2 + 2 x 3 ≤ - 3 - 4 x 1 - 2 x 2 + x 3 ≤ - 4 x 1 + x 2 - 4 x 3 2 0 x 1 , x 2 , x 3 D minimize - 3 y 1 - 4 y 2 + 2 y 3 subject to - y 1 - 4 y 2 + y 3 ≥ - 4 - y 1 - 2 y 2 + y 3 ≥ - 2 2 y 1 + y 2 - 4 y 3 ≥ - 1 0 y 1 , y 2 , y 3 -1 -1 2 1 0 0 -3 -4 -2 1 0 1 0 -4 1 1 -4 0 0 1 2 -4 -2 -1 0 0 0 0
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P maximize - 4 x 1 - 2 x 2 - x 3 subject to - x 1 - x 2 + 2 x 3 ≤ - 3 - 4 x 1 - 2 x 2 + x 3 ≤ - 4 x 1 + x 2 - 4 x 3 2 0 x 1 , x 2 , x 3 D minimize - 3 y 1 - 4 y 2 + 2 y 3 subject to - y 1 - 4 y 2 + y 3 ≥ - 4 - y 1 - 2 y 2 + y 3 ≥ - 2 2 y 1 + y 2 - 4 y 3 ≥ - 1 0 y 1 , y 2 , y 3 -1 -1 2 1 0 0 -3 -4 -2 1 0 1 0 -4 1 1 -4 0 0 1 2 -4 -2 -1 0 0 0 0 Not primal feasible. Dual feasible!
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The tableau below is said to be dual feasible because the objective row coefficients are all non-positive, but it is not primal feasible . -1 -1 2 1 0 0 -3 -4 -2 1 0 1 0 -4 1 1 -4 0 0 1 2 -4 -2 -1 0 0 0 0
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The tableau below is said to be dual feasible because the objective row coefficients are all non-positive, but it is not primal feasible . -1 -1 2 1 0 0 -3 -4 -2 1 0 1 0 -4 1 1 -4 0 0 1 2 -4 -2 -1 0 0 0 0 A tableau is optimal if and only if it is both primal feasible and dual feasible.
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The tableau below is said to be dual feasible because the objective row coefficients are all non-positive, but it is not primal feasible . -1 -1 2 1 0 0 -3 -4 -2 1 0 1 0 -4 1 1 -4 0 0 1 2 -4 -2 -1 0 0 0 0 A tableau is optimal if and only if it is both primal feasible and dual feasible. Can we design a pivot for this tableau that tries to move it toward primal feasibility while retaining dual feasibility?
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D minimize - 3 y 1 - 4 y 2 + 2 y 3 subject to - y 1 - 4 y 2 + y 3 ≥ - 4 - y 1 - 2 y 2 + y 3 ≥ - 2 2 y 1 + y 2 - 4 y 3 ≥ - 1 0 y 1 , y 2 , y 3
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D minimize - 3 y 1 - 4 y 2 + 2 y 3 subject to - y 1 - 4 y 2 + y 3 ≥ - 4 - y 1 - 2 y 2 + y 3 ≥ - 2 2 y 1 + y 2 - 4 y 3 ≥ - 1 0 y 1 , y 2 , y 3 - 1 - 1 2 1 0 0 - 3 - 4 - 2 1 0 1 0 - 4 1 1 - 4 0 0 1 2 - 4 - 2 - 1 0 0 0 0
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dual simplex - Math 407A Linear Optimization Lecture 11 The...

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