lec19 - 15.053 z Branch Tuesday May 1 and Bound 1 Quotes of...

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1 15.053 Tuesday, May 1 z Branch and Bound
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Quotes of the Day The time to relax is when you don't have time for it. -- Attributed to Jim Goodwin and Sydney J. Harris There is more to life than increasing its speed. -- Mohandas K. Gandhi 2
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Overview z Branch and bound is a clever way of enumerating all possible solutions. z We first start by understanding “bounding.” In particular, how can solving LPs provide useful information for solving IPs? You will recall that when we are maximizing, any solution to the dual of an LP gives an upper bound on an LP. But here, we want upper bounds on an integer program when we are maximizing. It turns out that there is a straightforward way to get an upper bound on an integer program in which we are maximizing. We will see it soon. 3
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4 The 053 Chocolate Store Problem 1 3 4 5 6 7 8 9 10 12 14 15 16 2 11 13 Locate a minimum number of stores so that each district has one or is adjacent to one. z IP = Min = x 1 + … x 16 x 1 + x 2 + x 4 + x 5 1 x 1 + x 2 + x 3 + x 5 +x 6 1 x 13 + x 15 + x 16 1 x j {0,1} for all j The upper bound is obtained by dropping the integrality constraints, and thereby giving the decision maker many more options. That is, the feasible region will be enlarged, and so the optimum solution will be better (or at least it will be no worse). We call the problem obtained by dropping the integrality constraints the “linear programming relaxation.” When we relax binary constraints on x j , we replace them by the linear constraints 0 x j 1. It is important to remember to include these constraints as part of any linear programming relaxation.
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5 Using LPs solution to get bounds z LP = Min x 1 + … x 16 x 1 + x 2 + x 4 + x 5 1 x 1 + x 2 + x 3 + x 5 +x 6 1 x 13 + x 15 + x 16 1 0 x j 1 for all j The LP relaxation of an integer program is what you get if you remove the integrality constraints from an integer program. Note: we replace x j {0,1} with 0 x j 1. Theorem: For a minimization problem Z IP Z LP. (Why?) Since the covering problem is a minimization problem, the LP relaxation will give a lower bound on the optimum objective value. That is, by eliminating constraints and making the feasible region larger, the objective will improve or stay the same; that is, it will get lower or stay the same. We can write this as Z LP Z IP , as was done on this slide, where Z IP is the optimum solution value for the original integer program, and Z LP is the optimum solution value for the linear programming relaxation.
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6 Using the LP bounds to help solve the IP 1 3 4 5 6 7 8 9 10 12 14 15 16 2 11 13 The LP solution to this problem was integral. x 2 = 1, x 11 = 1, x 13 = 1 all other variables were 0. The solution value is 3. Theorem: If the optimal solution for an LP relaxation is feasible for the IP, then it is also optimal for the IP.
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