Lecture17

# Lecture17 - Integer Programming IE418 Lecture 17 Dr. Ted...

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Integer Programming IE418 Lecture 17 Dr. Ted Ralphs

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IE418 Lecture 17 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 17 2 Describing conv( S ) We now switch to the case of a pure integer program with explicit nonnegativity constraints (the reason will become clear). z IP = max { cx | x S } , S = { x Z n + | Ax b } . We have just seen that in theory, it would be possible to generate a complete description of conv( S ) . So why aren’t IPs easy to solve? The number of inequalities is generally HUGE ! The number of facets of the TSP polytope for an instance with 120 nodes is more than 10 100 times the number of atoms in the universe . It is physically impossible to write down a description of this polytope. Not only that, but it is very diﬃcult in general to generate these facets (this problem is not in P in general).

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IE418 Lecture 17 3 For Example For a TSP of size 15 The number of subtour elimination constraints is 16,368 . The number of comb inequalities is 1 , 993 , 711 , 339 , 620 . These are only two of the know classes of facets for the TSP. For a TSP of size 120 The number of subtour elimination constraints is 0 . 6 × 10 36 ! The number of comb inequalities is approximately 2 × 10 179 !
IE418 Lecture 17 4 Basic Bounding Methods Our discussions of branch and bound have so far focused on the use of three basic bounding methods. LP relaxation Lagrangian relaxation Dantzig-Wolfe decomposition Recall from Lecture 13 that the bound produced by Lagrangian relaxation and Dantzig-Wolfe decomposition is z D = max { cx | A 1 x b 1 , x conv( S LR ) } , which is an improvement over that produced by solving the LP relaxation. Producing the bound z D depends on our ability to eﬃciently optimize over conv( S LR ) . Can we improve the LP relaxation in some way?

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IE418 Lecture 17 5 Cutting Planes Because of the equivalence of optimization and separation, we could also produce the bound z D by dynamic generation of valid inequalities. Recall that the inequality denoted by ( π, π 0 ) is valid for a polyhedron P if πx π 0 x ∈ P . The term cutting plane usually refers to an inequality valid for conv( S ) , but which is violated by the solution obtained by solving the (current) LP relaxation. Note that this is not a very precise deﬁnition and the term is a bit colloquial, but we will use it anyway. Cutting plane methods attempt to improve the bound produced by the LP
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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 .

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Lecture17 - Integer Programming IE418 Lecture 17 Dr. Ted...

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