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Unformatted text preview: Integer Programming IE418 Lecture 8 Dr. Ted Ralphs IE418 Lecture 8 1 Reading for This Lecture • Wolsey Chapter 3 • Nemhauser and Wolsey Sections III.1.1III.1.3 IE418 Lecture 8 2 Total Dual Integrality Definition 1. A system of linear inequalities Ax ≤ b is called totally dual integral (TDI) if, for all c ∈ Z n such z LP = max { cx  Ax ≤ b } is finite, the dual min { yb  yA = c, y ∈ R m + } has an integral optimal solution. • Note that this definition does not pertain to polyhedra, but to systems of inequalities. • The importance of this definition is that if Ax ≤ b is TDI and b is integral, then P = { x ∈ R n  Ax ≤ b } must be integral ( why ?). • Note that the property of being TDI is sensitive to scaling. • Every polyhedron has a representation that is TDI . • In fact, a polyhedron is integral if and only if it has a TDI representation where the righthand side is integral. IE418 Lecture 8 3 Total Unimodularity Definition 2. An m × n integral matrix A is totally unimodular (TU) if the determinant of every square submatrix is 0, 1, or 1....
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This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .
 Spring '08
 Ralphs

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