IE534_Lecture_15 - IE 534 Linear Programming Lecture 15 The Simplex Algorithm(5 Lizhi Wang Lizhi Wang([email protected] IE 534 Linear Programming 1 20

IE534_Lecture_15 - IE 534 Linear Programming Lecture 15 The...

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IE 534 Linear Programming Lecture 15: The Simplex Algorithm (5) Lizhi Wang September 26, 2012 Lizhi Wang ([email protected]) IE 534 Linear Programming September 26, 2012 1 / 20
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Outline 1 Review 2 How to check the unboundedness of an LP? 3 How to find a better fbp? Lizhi Wang ([email protected]) IE 534 Linear Programming September 26, 2012 2 / 20
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The nuts and bolts 1 What exactly is a “corner point”? 2 Is the optimal solution always basic? 3 How to find a basic solution? 4 How to find a fbp to start from? What if LP is infeasible? 5 How to check the optimality of a fbp? 6 How to check the unboundedness of an LP? 7 How to find a better fbp? 8 Is the Simplex algorithm guaranteed to terminate finitely? 9 Is the Simplex algorithm guaranteed to give the correct answer? 10 How efficient is the Simplex algorithm? Lizhi Wang ([email protected]) IE 534 Linear Programming September 26, 2012 3 / 20
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The Simplex diagram Lizhi Wang ([email protected]) IE 534 Linear Programming September 26, 2012 4 / 20
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The Simplex dictionary For any fbp ( B , N ) , the Simplex standard form can be written as max ζ = c > x s . t . A x = b x 0 max ζ = c > B x B + c > N x N s . t . A B x B + A N x N = b x B 0 , x N 0 A Simplex dictionary is defined as ζ = c > B A - 1 B b + ( c > N - c > B A - 1 B A N ) x N x B = A - 1 B b - A - 1 B A N x N . The term ( c > N - c > B A - 1 B A N ) is called the reduced cost . Lizhi Wang ([email protected]) IE 534 Linear Programming September 26, 2012 5 / 20
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Optimality check Proposition For a fbp ( B , N ) , if ( c > N - c > B A - 1 B A N ) 0 , then the fbs ( x B = A - 1 B b, x N = 0) is optimal.
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