Lecture6 - MS&E111 Introduction to Optimization Prof. Amin...

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Lecture 6 Introduction to Optimization April 24, 2006 Prof. Amin Saberi Scribed by: Yusuf Ozuysal 1 Simplex Algorithm and Duality Today we will begin by discussing some computational issues involving the implementation of the Simplex Algorithm. Then we will move to a new topic which will help us understand some of the most fundamental concepts related to defining, solving and interpreting a linear problem. Let us start with some key points about implementing the Simplex Algorithm. 1.1 Computational Challenges of the Simplex Algorithm Solving an LP by running the Simplex Algorithm offers some computational challenges. To start with, each step requires a lot of division operations and every division operation causes a certain amount of round-off error. So these errors add up if we have a considerable amount of division operations during a single run. Thus while implementing the Simplex Algorithm, it is good practice to pay attention and keep track of round-off errors. Also another major issue is the number of pivoting steps in a single run of the algorithm. We already know that it is possible to solve any LP or prove its infeasibility or unboundedness. Thus the challenge lies in solving any LP as fast as possible by minimizing the number of basic feasible solutions we pivot on during a single run. This significantly affects the amount of time a single run takes. To solve this latter problem, there have been several pivoting rules that have been proposed; however, no specific pivoting strategy has been found to minimize the number of pivots in a single run. For each given pivoting rule, there is at least one example LP in which the pivoting rule fails and results in larger number of pivots some other rule. One of the most well known conjectures about the minimum number of pivoting points or
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This note was uploaded on 09/23/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

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Lecture6 - MS&E111 Introduction to Optimization Prof. Amin...

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