lecture8 - CS 435 : Linear Optimization Fall 2008 Lecture...

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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 8: Extreme points Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay Definition 1 Given points p 1 ; p 2 ; p 3 ; : : : ; p n , the convex hull is the smallest convex set containing these points. An equivalent de nition (Why?) is convex hull ( p 1 ; : : : ; p n ) = f i p i ; i i = 1; 0 i 1 g . To simplify the discussion, we make the following assumptions on the nature of Ax b which hold for the rest of the course unless otherwise stated. 1 Assumptions on the nature of the convex set Ax b Assumption 1: Ax b is bounded. There is a real number B such that for every x satisfying Ax b ; x i B . Assumption 2: Ax b has no degeneracies. This means that not more than n hyperplanes pass through a point in an n-dimensional space. Assumption 3: Ax b should be full dimensional . This means that one should be able to place a n-dimensional sphere, however small, in the region de ned by Ax b . For two dimensions, the convex set should have area, and for three dimensions, the convex set should have volume. In n dimensions, the convex set should have an n-dimensional volume. Can you think of a body in three dimensions where four planes pass through a point on the boundary?...
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lecture8 - CS 435 : Linear Optimization Fall 2008 Lecture...

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