convex - ORIE 320 Professor Bland Convexity Polyhedra...

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Unformatted text preview: ORIE 320 9/10/07 Professor Bland 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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convex - ORIE 320 Professor Bland Convexity Polyhedra...

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