lecture12a - CS 435 : Linear Optimization Fall 2008 Lecture...

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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 12: The Simplex Algorithm: Proof of Correctness Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay In this lecture we will analyse the stopping condition of the Simplex algorithm and prove that the algorithm is correct. To recap, consider an given extreme point x ; given by A x = b and A 00 x b 00 . The directions of the neighbouring extreme points are the columns of the matrix   A   1 . Stopping Condition: The algorithm stops at an extreme point x and returns it as optimal when the cost at x is greater than or equal to the cost at the neighbouring extreme points. 1 Proof of Correctness We now prove that the simplex algorithm is correct, i.e. , when it terminates, we indeed have found the globally optimal point. Note that the stopping condition does not say that x is a local maximum. This is because we only know that the cost at x is maximum as compared to the values at its neighbours {not compared to the values at all points in a small enough neighbourhood around it....
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This note was uploaded on 01/04/2010 for the course CSE CS435 taught by Professor Profsundar during the Summer '09 term at IIT Bombay.

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lecture12a - CS 435 : Linear Optimization Fall 2008 Lecture...

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