ip_lecture_6

ip_lecture_6 - Industrial Engineering 634 Fall 2011 Integer...

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Unformatted text preview: Industrial Engineering 634 Fall 2011 Integer Programming Lecturer: Nelson Uhan Scribe: Richard Mouawad 02 September 2011 Lecture 6: Cones, convex hull and polytopes 1 Previous results First we will list some previous results that are needed throughout this lecture. Corollary 1.1. Let P = { x R n : Ax b } . All minimal faces of P have dimension n- rank ( A ) . The minimal faces of polytopes are vertices (from Lecture 3). Theorem 1.1. There exists x such that Ax b if and only if yb for all y such that yA = 0 (from Lecture 5). 2 Cones A subset C of R n is called a cone if it is closed under taking conic combinations: that is, if x,y C and , 0, we have that x + y C . Given a set C R n , we define the cone generated by C to be all nonnegative linear combinations of the vectors in C ; namely if x 1 ,...,x k C and for all x C , there exists 1 ,..., k 0 with k i =1 i x i . A cone is finitely generated if some finite set of vectors generates it. A polyhedral cone is a polyhedron of type { x : Ax } ....
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ip_lecture_6 - Industrial Engineering 634 Fall 2011 Integer...

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