lecture9 - CS 435 : Linear Optimization Fall 2008 Lecture...

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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 9: Convex Hull of Extreme Points Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay In this lecture, we complete the proof of the theorem on extreme points mentioned in the previous lecture and begin the last part of understanding the object f x : Ax b g . Proof: (Continuing Part 2.) Here we prove that every extreme point of f x : Ax b g can be expressed as an intersection of n linearly independent hyperplanes. We show that if a point in the set cannot be expressed as an intersection of n linearly independent hyperplanes then we can express it as a convex combination of two other points in the set. Let x be such a point. We split Ax b into two parts A x = b (1) A 00 x < b 00 (2) By the assumption on x , A has rank strictly less than n . We wish to express x as the convex combination of two other points y and z . Where do we nd this y and z ? At this point, we recommend visualising examples in two and three dimensions. It will not take long to conjecture thatvisualising examples in two and three dimensions....
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lecture9 - CS 435 : Linear Optimization Fall 2008 Lecture...

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