Lecture21

# Lecture21 - Introduction to Mathematical Programming IE406...

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Introduction to Mathematical Programming IE406 Lecture 21 Dr. Ted Ralphs

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IE406 Lecture 21 1 Reading for This Lecture Bertsimas Sections 10.2, 10.3, 11.1, 11.2
IE406 Lecture 21 2 Branch and Bound Branch and bound is the most commonly-used algorithm for solving MILPs. It is a divide and conquer approach. Suppose F is the feasible region for some MILP and we wish to solve min x F c ± x . Consider a partition of F into subsets F 1 , . . . F k . Then min x F c ± x = min { 1 i k } { min x F i c ± x } In other words, we can optimize over each subset separately. Idea : If we can’t solve the original problem directly, we might be able to solve the smaller subproblems recursively. Dividing the original problem into subproblems is called branching . Taken to the extreme, this scheme is equivalent to complete enumeration.

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IE406 Lecture 21 3 Branch and Bound Next, we discuss the role of bounding . For the rest of the lecture, assume all variables have ﬁnite upper and lower bounds. Any feasible solution to the problem provides an upper bound u ( F ) on the optimal solution value. We can use approximate methods to obtain an upper bound. Idea : After branching, try to obtain a lower bound b ( F i ) on the optimal solution value for each of the subproblems. If b ( F i ) u ( F ) , then we don’t need to consider subproblem i . One easy way to obtain a lower bound is by solving the LP relaxation obtained by dropping the integrality constraints.
IE406 Lecture 21 4 LP-based Branch and Bound In LP-based branch and bound , we ﬁrst solve the LP relaxation of the original problem. The result is one of the following: 1. The LP is infeasible MILP is infeasible . 2. We obtain a feasible solution for the MILP optimal solution . 3. We obtain an optimal solution to the LP that is not feasible for the MILP lower bound . In the ﬁrst two cases, we are ﬁnished . In the third case, we must branch and recursively solve the resulting subproblems.

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IE406 Lecture 21 5 Branching in LP-based Branch and Bound The most common way to branch is as follows: Select a variable i whose value ˆ x i is fractional in the LP solution. Create two subproblems. * In one subproblem, impose the constraint x i ≤ ± ˆ x i ² . * In the other subproblem, impose the constraint x i ≥ ³ ˆ x i ´ . Such a method of branching is called a branching rule . Why is this a valid branching rule ? What does it mean in a 0-1 integer program?
IE406 Lecture 21 6 Continuing the Algorithm After Branching After branching, we solve each of the subproblems recursively . Now we have an additional factor to consider. If the optimal solution value to the LP relaxation is greater than the

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Lecture21 - Introduction to Mathematical Programming IE406...

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