convex - ORIE 3300/5300 Professor Bland Fall 2011 Review of...

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ORIE 3300/5300 Fall 2011 Professor Bland Review of Convexity, Polyhedra, Extreme Points A set S I R n is convex if for all points x 1 ,x 2 S the point (1 - λ ) x 1 + λx 2 S , for all 0 λ 1. Can you give a geometric interpretation of this property? Note that the intersection of convex sets is also convex. Let α 1 ,...,α n and β be constants, with at least one of α 1 ,...,α n nonzero. The set of all vectors x = ( x 1 , ··· ,x n ) such that α 1 x 1 + ... + α n x n β is called a halfspace of I R n . It should be clear that every halfspace is convex. A polyhedron in I R n is the intersection of finitely many half-spaces. For example, the set of all feasible solutions ( x 1 ,x 2 ) T for the original version of Grandma’s Sausage Problem, or the set of all feasible solutions x = ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) T that satisfy Ax = b for the 3 × 5 matrix A and the 3 × 1 vector b from the version of Grandma’s Sausage Problem in which the slack variables x 3
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This note was uploaded on 03/08/2012 for the course ORIE 3300 at Cornell.

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convex - ORIE 3300/5300 Professor Bland Fall 2011 Review of...

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