Differential Equations Solutions 93

# Differential Equations Solutions 93 - informal and...

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103 0 1 2 3 4 5 6 7 8 9 10 x 10 4 10.7 10.8 10.9 11 11.1 11.2 11.3 11.4 11.5 Trial number Score Very late stage score, 100 cities, T0=0.0046, 77 temperature changes Figure 17.7. TSP scores by simulated annealing, T =0 . 0046 , logarithmic cooling schedule. exp( 1 /T )isexp(1 /T ) on average, and so 1 /T should look something like log( k ). This is the famous logarithmic cooling schedule [1]. Figures 17.6, 17.7 and 17.8 illustrate application of simulated annealing with a logarithmic cooling schedule to a TSP with 100 random locations. The ±rst two graphs show how the score evolves over 100,000 trials at a low temperature. Note that not many proposed moves that increase the score are accepted and that the score does not improve very much. The last ±gure is a picture of the best tour obtained. Because it crosses itself, it’s not the optimal tour. Getting that requires more computation and/or more sophisticated cooling schedules. Solving TSP for 100 random locations is really quite diﬃcult! If you think this use of simulated annealing to attack the TSP seems quite
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Unformatted text preview: informal and heuristic rather than analytic, youre right. In fact, some have argued that simulated annealing is not really an optimization method but rather a collec-tion of heuristic techniques that help in some cases. However, there is an important, recently discovered connection between the central idea of simulated annealing and use of Monte Carlo to approximate solutions to NP-hard problems, including de-termining the volume of a bounded convex region K in R n . If n is large, nding V ol ( K ) can be a very hard problem. The most well-developed approach is to dene a sequence of convex sets: K K 1 K 2 . . . K m = K where V ol ( K ) is easy to evaluate. For each i , perform a random walk in K i and count how many walkers happen to be in K i 1 . This gives an estimate of V ol ( K i 1 ) /V ol ( K i ) and the product of these estimates for all i is an estimate for...
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