Simplex Algorithm.pdf - IE 534 Linear Programming Lecture 13 The Simplex Algorithm(3 Lizhi Wang October 2 2015 Lizhi Wang([email protected] IE 534

Simplex Algorithm.pdf - IE 534 Linear Programming Lecture...

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IE 534 Linear Programming Lecture 13: The Simplex Algorithm (3) Lizhi Wang October 2, 2015 Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 1 / 12
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The nuts and bolts 1 What exactly is a “corner point”? 2 Does the optimal solution always occur at a corner point? 3 How to find a corner point to start from? 4 How to check the optimality of a corner point? 5 How to find a better corner point if the current one is not optimal? 6 How to identify an infeasible or unbounded LP? 7 Is the Simplex algorithm guaranteed to terminate finitely? 8 Is the Simplex algorithm guaranteed to give the correct answer? 9 How efficient is the Simplex algorithm? Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 2 / 12
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No, but... No Some LPs may not even have a basic solution. An optimal solution may not be a basic solution. But A standard form LP always has a basic solution. If the optimal solution is unique, then it must be a basic solution. If a standard form LP has infinitely many optimal solutions, then one of them must be a basic solution. Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 3 / 12
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The Simplex focus Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 4 / 12
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The nuts and bolts 1 What exactly is a “corner point”? 2 Does the optimal solution always occur at a corner point? 3 How to find a corner point to start from? 4 How to check the optimality of a corner point? 5 How to find a better corner point if the current one is not optimal? 6 How to identify an infeasible or unbounded LP? 7 Is the Simplex algorithm guaranteed to terminate finitely? 8 Is the Simplex algorithm guaranteed to give the correct answer? 9 How efficient is the Simplex algorithm? Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 5 / 12
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Starting basic partition Define LP0 as max x,w { ζ = c > x : Ax + w = b, x 0 , w 0 } . LP0 is equivalent to max x { ζ = c > x : A x = b, x 0 } with c = c 0 m × 1 , A = A I m × m , and x = x w R n + m . ( x = 0 , w = b ) is a basic solution to LP0 . If b 0 , then the following basic partition ( B 0 , N 0 ) is a fbp B 0 = { n + 1 , ..., n + m } , N 0 = { 1 , ..., n } , A B 0 = I m × m , A N 0 = A, x B 0 = A - 1 B 0 b = b, x N 0 = 0 n × 1 . Otherwise we proceed using a technique called the two-phase method, in which we consider an auxiliary problem LP1 . Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 6 / 12
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The Simplex diagram Lizhi Wang ([email protected]) IE 534 Linear Programming October 2, 2015 7 / 12
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Example 1 Consider the following LP0 instance max x ζ = 13 x 1 + 7 x 2 - 12 x 3 s . t . 2 x 1 + 3 x 2 - x 3 5 - 4 x 1 - 7 x 2 + 2 x 3 - 11 3 x 1 - 4 x 2 - 2 x 3 - 8 x 1 , x 2 , x 3 0 . max x,w ζ = 13 x 1 + 7 x 2 - 12 x 3 s . t . 2 x 1 + 3 x 2 - x 3 + w 4 = 5 - 4 x 1 - 7 x 2 + 2 x 3 + w 5 = - 11 3 x 1 - 4 x 2 - 2 x 3 + w 6 = - 8 x 1 , x 2 , x 3 , w 4 , w 5 , w 6 0 . Consider the following auxiliary problem LP1 min x,w,t t 7 + t 8 s . t . 2 x 1 + 3 x 2 - x 3 + w 4 = 5 - 4 x 1 - 7 x 2 + 2 x 3 + w 5 - t 7 = - 11 3 x 1 - 4 x 2 - 2 x 3 + w 6 - t 8 = - 8 x 1 , x 2 , x 3 , w 4 , w 5 , w 6 , t 7 , t 8 0 .
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