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Unformatted text preview: C&O 355: Mathematical Programming Fall 2010 Lecture 11 Notes Nicholas Harvey http://www.math.uwaterloo.ca/~harvey/ 1 Faces of Polyhedra Recall from Lecture 10 the following definition. Definition 1.1. Let P R n be a polyhedron. A face of P is any set F of the form F = P n x R n : a T x = b o , (1.1) where a T x b is a valid inequality for P . We remark that unbounded polyhedra might not have any vertices, or edges, or even facets. For example, the polyhedron could be an affine space. On the other hand, a nonempty polytope always has at least one vertex. This follows from our theorem that feasible and bounded LPs always have an optimal solution at an extreme point. Fact 1.2. Let P be a polyhedron with dim P = d . Let F be a face of P . Then F : F is a face of F = F : F is a face of P , and F F . Furthermore, each facet of F can be obtained by intersecting F with another facet of P . Fact 1.3. Let P = x : a T i x b i i be a polyhedron in R n . Let x and y be two distinct vertices. Recall our notation I x = i : a T i x = b i . Suppose rank { a i : i I x I y } = n 1. Then the line segment L x , y = { x + (1 ) y : [0 , 1] } (1.2) is an edge of P . Moreover, every bounded edge arises in this way. Definition 1.4. Let P = x : a T i x b i i be a polyhedron in R n . An inequality a T i x b i is called facetdefining if the face P n x R n : a T x = b o is a facet. Fact 1.5. Let P = x : a T i x b i i be a fulldimensional polyhedron in R n . Let I = n i : the inequality a T i x b i is facetdefining o . Then P = n x : a T i x b i i I o . 1 2 Polyhedra and Graphs Recall from Lecture 10 that every polyhedron has finitely many vertices. By Fact 1.3, every bounded edge of a polyhedron can be described as the line segment L x , y connecting two par ticular vertices x and y . Thus the vertices and bounded edges of polyhedra naturally form a graph ....
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This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.
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