03 - Industrial Engineering 634 Fall 2011 Integer...

Unformatted text preview: Industrial Engineering 634 Fall 2011 Integer Programming Lecturer: Nelson Uhan Scribe: Nathan Hartje August 26, 2011 Lecture 3. Faces of Polyhedra 1 Review Up to now, we have been investigating mathematical programs. When we inspected the generic linear program in the previous lecture, we began to conceptualize these programs in terms their feasible sets. Diving deeper into this concept, we defined a polyhedron in R n as a set of type P = { x ∈ R n : Ax ≤ b } for some matrix A ∈ R m × n and vector b ∈ R m . We further described the structure of a polyhedron P = { x ∈ R n : Ax ≤ b } as follows. Definition 1.1. If c ∈ R n is a nonzero vector such that δ = max { cx : x ∈ P } is finite, then { x ∈ R n : cx = δ } is a supporting hyperplane of P . Definition 1.2. A face of P is P itself or the intersection of P with a supporting hyperplane of P . Definition 1.3. A point x for which { x } is a face is called a vertex of P , or a basic solution of Ax ≤ b . In the last lecture, we began to discuss faces of P by establishing a proposition, and, today, we continue our proof of this proposition. 2 Faces of polyhedra Proposition 2.1. Let P = { x ∈ R : Ax ≤ b } be a polyhedron and F ⊆ P . Then the following statements are equivalent: (a) F is a face of P . (b) There exists c such that δ = max { cx : x ∈ P } is finite and F = { x ∈ P : cx = δ } . (c) F = { x ∈ P : A x = b } 6 = ∅ for some subsystem A x ≤ b of Ax ≤ b . Proof. From the previous lecture, we concluded the following two statements. F is a face of P if and only if F = { x ∈ P : A x = b } 6 = ∅ for some subsystem A x ≤ b of Ax ≤ b (i.e. the equivalence of ( a ) and ( c )). If F = { x ∈ P : A x = b } 6 = ∅ for some subsystem A x ≤ b of Ax ≤ b , then there exists c such that δ = max { cx : x ∈ P } is finite and F = { x ∈ P : cx = δ } (i.e. ( c ) implies ( b ))....
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