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8 Integral Polyhedron_2

# 8 Integral Polyhedron_2 - IEE 598 8(III.1 Integral...

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IEE 598 - 8 (III.1) Integral Polyhedra Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 25, 2010

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Integral Polyhedron Definition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point. Zhang IEE 376 Introduction to OR 2 / 12
Integral Polyhedron Definition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point. Proposition: A nonempty polyhedron P = { x R n : Ax b } with rank ( A ) = n is integral if and only if all of its extreme points are integral. Zhang IEE 376 Introduction to OR 2 / 12

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Integral Polyhedron Definition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point. Proposition: A nonempty polyhedron P = { x R n : Ax b } with rank ( A ) = n is integral if and only if all of its extreme points are integral. Consider the optimization problem z * = max { cx : x P } , given a polyhedron P . Zhang IEE 376 Introduction to OR 2 / 12
Integral Polyhedron Definition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point. Proposition: A nonempty polyhedron P = { x R n : Ax b } with rank ( A ) = n is integral if and only if all of its extreme points are integral. Consider the optimization problem z * = max { cx : x P } , given a polyhedron P . Proposition: The following statements are equivalent: 1 P is integral; 2 There is an integral optimal solution for all c R n for which an optimal solution exists; 3 There is an integral optimal solution for all c Z n for which an optimal solution exists; 4 z * is integral for all c Z n for which an optimal solution exists. Zhang IEE 376 Introduction to OR 2 / 12

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Totally Unimodularity Definition: An m × n matrix A is totally unimodular (TU) if the determinant of every square submatrix is 0, 1, or - 1. Zhang IEE 376 Introduction to OR 3 / 12
Totally Unimodularity Definition: An m × n matrix A is totally unimodular (TU) if the determinant of every square submatrix is 0, 1, or - 1. Proposition: If A is a totally unimodular matrix and b Z m , then P = { x R n : Ax b } is an integral polyhedron. Zhang IEE 376 Introduction to OR 3 / 12

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Totally Unimodularity Definition: An m × n matrix A is totally unimodular (TU) if the determinant of every square submatrix is 0, 1, or - 1. Proposition: If A is a totally unimodular matrix and b Z m , then P = { x R n : Ax b } is an integral polyhedron. Proposition: The following statements are equivalent: 1 A is TU; 2 The transpose of A is TU; 3 ( A , I ) is TU; 4 A matrix obtained by deleting a unit row/column from A is TU; 5 A matrix obtained by multiplying a row/column of A by - 1 is TU; 6 A matrix obtained by interchanging two rows/columns of A is TU; 7 A matrix obtained by duplicating a row/column of A is TU; 8 A matrix obtained by a pivot operation of A is TU.
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8 Integral Polyhedron_2 - IEE 598 8(III.1 Integral...

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