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Unformatted text preview: Review Modeling PPP IE426: Optimization Models and Applications: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University October 12, 2006 Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound Got MILP? Mixed Integer Linear Program minimize c T x subject to Ax ≥ b x ≥ x j ∈ Z ∀ j ∈ I ⊆ N. Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound It’s All About Relaxations Let z S = min f ( x ) : x ∈ S Let z T = min f ( x ) : x ∈ T S T Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound Obvious, but Important Stuff z T ≤ z S !!! If x * T is an optimal solution to min f ( x ) : x ∈ T And x * T ∈ S , then x * T is an optimal solution to min f ( x ) : x ∈ S If we were to replace min by max, then z T ≥ z S . Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound LP relaxation minimize z IP = c T x subject to Ax ≥ b x ≥ x j ∈ Z j ∈ P ⊆ N. minimize z LP = c T x subject to Ax ≥ b x ≥ z LP ≤ z IP If x * solves the LP relaxation and x * satisfies the integrality requirements, then x * solves IP Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound x * ∈ X ? Then we branch! Partition the problem into smaller pieces. x * ∈ X ⇒ ∃ k ∈ P such that x * k is fractional. Create two new problems... 1 In one problem, add the constraint x k ≤ x * k 2 In the other problem, add the constraint x k ≥ x * k Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound The Branch and Bound Algorithm All of the following assumes a maximization problem. It works equally well for minimization, but you need to replace “lower” by “upper” (and vice verse) everywhere Let z L be a lower bound on optimal objective value. Originally z L =∞ . The initial list of LP relaxations to solve consists of just the LP relaxation to the entire problem. Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound Branch and Bound (1 of 2) 1 Select an LP Relaxation 2 Solve the selected LP relaxation. If the LP relaxation is infeasible, Go to 1. 3 Otherwise, let x * be the optimal solution to the LP relaxation and let z * be its objective value. 4 If x * satsifies integer restrictions, then z * is the optimal value for the restricted IP problem. If z * > z L , then z L := z * . (Also keep track of x * as candidate solution). Go to 1. Jeff Linderoth IE426:Lecture 12 Review Modeling PPP MILP Relaxations Branch and Bound Branch and Bound (2 of 2) 5 Otherwise, x * does not satisfy the integer restrictions. z * is an upper bound on the optimal value of the restricted IP problem....
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This note was uploaded on 02/29/2008 for the course IE 426 taught by Professor Linderoth during the Spring '08 term at Lehigh University .
 Spring '08
 Linderoth
 Optimization, Systems Engineering

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