Lecture21 - Integer Programming IE418 Lecture 21 Dr Ted...

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Unformatted text preview: Integer Programming IE418 Lecture 21 Dr. Ted Ralphs IE418 Lecture 21 1 Reading for This Lecture • Wolsey Section 9.6 • Nemhauser and Wolsey Section II.6 • Martin “Computational Issues for Branch-and-Cut Algorithms” (2001) • Linderoth and Ralphs “Noncommercial Software for Mixed-Integer Linear Programming” IE418 Lecture 21 2 Branch and Cut • Branch and cut is an LP-based branch and bound scheme in which the linear programming relaxations are augmented by valid inequalities. • The valid inequalities are generated dynamically using separation procedures. • We iteratively try to improve the current bound by adding valid inequalities. • In practrice, branch and cut is the method typically used for solving difficult mixed-integer linear programs. • Computational component of branch and cut – Preprocessing – Cut generation – Managing the LP relaxation – Search strategy – Branching strategy – Primal heuristics IE418 Lecture 21 3 Preprocessing and Probing • Often, it is possible to simplify a model using logical arguments. • Most commercial IP solvers have a built-in preprocessor. • Effective preprocessing can pay large dividends. • Let the upper and lower bounds on x j be u j and l j . • The most basic type of preprocessing is calculating implied bounds . • Let ( π, π ) be a valid inequality. • If π 1 > , then x 1 ≤ ( π- X j : π j > π j l j- X j : π j < π j u j ) /π 1 • If π 1 < , then x 1 ≥ ( π- X j : π j > π j l j- X j : π j < π j u j ) /π 1 IE418 Lecture 21 4 Basic Preprocessing • Again, let ( π, π ) be any valid inequality for S . • The constraint πx ≤ π is redundant if X j : π j > π j u j + X j : π j < π j l j ≤ π . • S is empty (IP is infeasible ) if X j : π j > π j l j + X j : π j < π j u j > π . • For any IP of the form max { c T x | Ax ≤ b, l ≤ x ≤ b } , x ∈ Z n , – If a ij ≥ ∀ i ∈ [1 ..m ] and c j < , then x j = l j in any optimal solution. – If a ij ≤ ∀ i ∈ [1 ..m ] and c j > , then x j = u j in any optimal solution. IE418 Lecture 21 5 Probing for Integer Programs • It is also possible in many cases to fix variables or generate new valid inequalities based on logical implications....
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Lecture21 - Integer Programming IE418 Lecture 21 Dr Ted...

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