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19_Lagrangian_Relaxation_1

# 19_Lagrangian_Relaxation_1 - 15.082 and 6.855J Lagrangian...

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1 15.082 and 6.855J Lagrangian Relaxation I never missed the opportunity to remove obstacles in the way of unity. —Mohandas Gandhi

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2 On bounding in optimization In solving network flow problems, we not only solve the problem, but we provide a guarantee that we solved the problem. Guarantees are one of the major contributions of an optimization approach. But what can we do if a minimization problem is too hard to solve to optimality? Sometimes, the best we can do is to offer a lower bound on the best objective value. If the bound is close to the best solution found, it is almost as good as optimizing.
3 Overview Decomposition based approach. Start with Easy constraints Complicating Constraints. Put the complicating constraints into the objective. We will obtain a lower bound on the optimal solution for minimization problems. In many situations, this bound is close to the optimal solution value.

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4 An Example: Constrained Shortest Paths Given: a network G = (N,A) c ij cost for arc (i,j) t ij traversal time for arc (i,j) ( , ) ij ij i j A c x j j Z* = Min s. t. 1 if i = s 1 if i = t 0 otherwise ij ji j j x x j j - = - 0 or 1 for all ( , ) ij x i j A = ( , ) ij ij i j A t x T j j j Complicating constraint
5 Example (1, 1 ) (12, 3 ) (1, 2 ) (10, 1 ) 1 2 4 5 3 6 Find the shortest path from node 1 to node 6 with a transit time at most 10 (c ij , t ij ) i j

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6 Shortest Paths with Transit Time Restrictions Shortest path problems are easy. Shortest path problems with transit time restrictions are NP-hard. We say that constrained optimization problem Y is a relaxation of problem X if Y is obtained from X by eliminating one or more constraints. We will “relax” the complicating constraint, and then use a “heuristic” of penalizing too much transit time. We will then connect it to the theory of Lagrangian relaxations.
7 Shortest Paths with Transit Time Restrictions Step 1. (A relaxation approach). Solve the problem without the complicating constraint. If the solution satisfies the complicating constraint, then it is optimal for the original problem. ( , ) ij ij i j A c x j j 1 if i = s 1 if i = t 0 otherwise ij ji j j x x j j - = - 0 or 1 for all ( , ) ij x i j A =

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8 What is the shortest path from node 1 to node 6? (1, 1 ) (12, 3 ) ) (1, 2 ) (10, 1 ) 1 2 4 5 3 6
9 The shortest path is 1-2-4-6 (1, (1, 1 ) (12, 3 ) (1, 2 ) (10, 1 ) 1 2 4 5 3 6 The transit time of 1-2-4-6 is 18.

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19_Lagrangian_Relaxation_1 - 15.082 and 6.855J Lagrangian...

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