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Lecture4

Lecture4 - Integer Programming IE418 Lecture 4 Dr Ted...

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Integer Programming IE418 Lecture 4 Dr. Ted Ralphs

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IE418 Lecture 4 1 Reading for This Lecture Wolsey, Chapters 8 and 9 N&W Sections I.4.1-I.4.3
IE418 Lecture 4 2 Dimension of Polyhedra A polyhedron P is of dimension k , denoted dim ( P ) = k , if the maximum number of affinely independent points in P is k + 1 . A polyhedron P ⊆ R n is full-dimensional if dim ( P ) = n . Let M = { 1 , . . . , m } , M = = { i M | a i x = b i x ∈ P} (the equality set ), M = M \ M = (the inequality set ). Let ( A = , b = ) , ( A , b ) be the corresponding rows of ( A, b ) . Proposition 1. If P ⊆ R n , then dim ( P ) + rank ( A = , b = ) = n

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IE418 Lecture 4 3 Dimension and Rank x ∈ P is called an inner point of P if a i x < b i i M . x ∈ P is called an interior point of P if a i x < b i i M . Every nonempty polyhedron has an inner point . The previous proposition showed that a polyhedron has an interior point if and only if it is full-dimensional .
IE418 Lecture 4 4 Computing the Dimension of a Polyhedron To compute the dimension of a polyhedron, we generally use these two equations dim ( P ) = n - rank ( A = , b = ) , and dim ( P ) = max {| S | : S ⊆ P and the points in S are aff. indep. } - 1 . In general, it is difficult to determine dim ( P ) using either one of these formulas alone, so we use them together .

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Lecture4 - Integer Programming IE418 Lecture 4 Dr Ted...

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