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24vls_neighborhoodsearch - Very Large Scale Neighborhood...

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Very Large Scale Neighborhood Search Collaborators include: Ravi Ahuja, Ozlem Ergun, Abraham Punnen, Dushyant Sharma
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Neighborhood Search Combinatorial Optimization : minimize (f(S) : S ¡# F) ¡ f is typically linear, ¡ F is finite Neighborhood Function : ¡ For each S ¡# F, there is a neighborhood N(S); ¡ We say that S is a local optimum if f(S) d# f(T) for all T ¡# N(S); 2
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Neighborhood Search Neighborhood Search (local improvement algorithm) begin initialize with some S ¡# F; while S is not a local optimum do replace S by some T ¡# N(S) such that f(T) < f(S); end 3
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4 TSP and 2-exchanges 1 2 3 4 5 6 7 8 9 10 The original tour 1 2 3 9 10 4 5 6 7 8 A 2-neighbor of the original tour We say that a tour T’ is a 2- neighbor of a tour T if it is possible to obtain T’ from T by adding two edges and deleting two edges. The operation is called a 2-exchange. T’ = T + (3,8) + (4,9) - (3,4) - (8,9). Obtained by the operation Flip[4,8] When we say city i, we really mean the city that is in position i of the current tour. (Or you may assume that the current tour is 1, 2, 3, ..., n)
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5 A non-optimal tour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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6 1 13 14 15 16 17 18 19 20 Improving using a two exchange 8 9 10 11 12 7 6 5 4 3 2 A 2-exchange is obtained by deleting two edges of the tour and adding two edges
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Neighborhood Search We say that a tour T’ is a 2-neighbor of a tour T if it is possible to obtain T’ from T by adding two edges and deleting two edges. This can be generalized to 3-neighbors , and k-neighbors . Neighborhood Search: begin start with some tour T. while there is a neighbor T’ of T with a lower objective value, replace T by T’ end 7
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More on Neighborhood Search A solution is 2-optimum if there is no 2-neighbor that has a lower length. A solution is locally optimum if there is no neighbor with an improved objective. Neighborhood search in its “vanilla” form stops when a locally optimum solution is found. Most neighborhood search relies on neighborhoods that can be searched exhaustively. 8
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Some simple extensions Extension 1. For k = 1 to 1000, select a random tour T. Find a locally optimal tour starting with T. Then choose the best of these tours. Extension 2. 1. Let T be a tour 2. Find a locally optimal tour T’ starting at T 3. perturb T’ (perhaps using a random 5 exchange) 4. replace T by T’ and return to step 1. Other extensions: simulated annealing and tabu search 9
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Very Large Scale Nbhd (VLSN) search This talk: Focuses on VERY LARGE neighborhoods that can be searched very efficiently (preferably in polynomial time) or are searched heuristically. often exponentially large neighborhoods Rule of Thumb for Larger Neighborhoods: improved local optima greater search time 10
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relation to other search techniques min cost flows, cycle canceling Multicommodity flows: column generation But both these problems are special case of linear programming. We will use this technique for hard problems as well.
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