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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 roundoﬀ 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 roundoﬀ 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.
 Spring '06
 UNKNOWN

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