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Lecture18

# Lecture18 - Integer Programming IE418 Lecture 18 Dr Ted...

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Integer Programming IE418 Lecture 18 Dr. Ted Ralphs

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IE418 Lecture 18 1 Reading for This Lecture Nemhauser and Wolsey Sections II.1.1-II.1.3, II.1.6 Wolsey Chapter 8 Valid Inequalities for Mixed Integer Linear Programs, G. Cornuejols (2006)
IE418 Lecture 18 2 Valid Inequalities from Disjunctions We continue to focus primarily on the case of a pure integer program z IP = max { cx | x S } , S = { x Z n + | Ax b } . Valid inequalities for conv( S ) can also be generated based on disjunctions. Let P i = { x R n + | A i x b i } for i = 1 , . . . , k be such that S ⊆ ∪ k i =1 P i . Then inequalities valid for k i =1 P i are also valid for conv( S ) .

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IE418 Lecture 18 3 The Union of Polyhedra Note that the convex hull of the union of polyhedra is not necessarily a polyhedron. Under mild conditions, we can characterize it, however. Let Y be the polyhedron described by the following constraints: A i x i b i y i i = 1 , . . . , k k X i =1 x i = x k X i =1 y i = 1 y 0 Furthermore, for polyhedron P i , let C i = { x : A i x 0 and let P i = Q i + C i where Q i is a polytope.
IE418 Lecture 18 4 The Convex Hull of the Union of Polyhedra Under the assumptions on the previous slide, we have the following result. Proposition 1. If either ∪P i = or C j cone i : P i = C i for all j such that P j = , then the following sets are identical: conv( k i =1 P i } conv( Q i ) + cone ( C i ) proj x Y . Note that the assumptions of the proposition are necessary, but are automatically satisfied if C i = { 0 } whenever P i = , or all the polyhedra have the same recession cone.

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IE418 Lecture 18 5 The Convex Hull of the Union of Polyhedra (cont.) Note also that if all the polyhedra have the same recession cones, then conv( k i =1 P i ) = conv( k i =1 P i ) and is the projection of A i x i b i y i i = 1 , . . . , k k X i =1 x i = x k X i =1 y i = 1 y { 0 , 1 } This is the case when the polyhedra only differ in their right-hand sides, as is the case when branching on variables.
IE418 Lecture 18 6 Valid Inequalities from Disjunctions Another viewpoint for constructing valid inequalities based on disjunctions comes from the following result: Proposition 2. If n j =1 π 1 j π 1 0

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Lecture18 - Integer Programming IE418 Lecture 18 Dr Ted...

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