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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 10: Ax b as a convex combination of its extreme points Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay In this lecture, we complete the proof of a theorem stating that all points in the set Ax b can be expressed as a convex combination of its extreme points. We then prove that a linear function on such a set is maximized at an extreme point. This leads us to certain algorithms for linear programming. Theorem 1 Let p 1 ; p 2 ; p 3 ; : : : ; p t be the extreme points of the convex set S = f x : Ax b g Then every point in S can be represented as t X i =1 i p i , where t X i =1 i = 1 and i 1 Proof: The proof is by induction on the dimension of the object f x : Ax b g . The base step is when the dimension is zero and is trivial. For the inductive step consider an object f x : Ax b g in n-dimensions. Consider p 2 S . For simplicity of presentation, assume that every inequality is strict for p . That is Ap < b . Join p 1 to p and extend this line to meet a point q on the boundary of f x : Ax b g . Note that the segment joining p 1 and p must lie inside the set by convexity. Also, such a point q must exist since the object is bounded. What does it mean that q is a boundary point? As we said before, it means that if we draw a small sphereis a boundary point?...
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- Summer '09
- Computer Science, extreme points, Sundar Vishwanathan, fx X Ax