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Temperature Scheduling in Simulated Annealing (view source)
Revision as of 08:12, 21 December 2007
, 08:12, 21 December 2007→Convergence and Prospective strategies
The primary objective for potential cooling strategies lies in the determination of the α and ''K'' factors given in the ParSA section on improving solution quality in a lesser amount of time. By tracking both the chain length ''n'' and speed of convergence P(X<sub>n</sub> ∉ Cost<sub>min</sub>), one can find a linear plot relating all four of the quantities through the following relationship:
The primary objective for potential cooling strategies lies in the determination of the α and ''K'' factors given in the ParSA section on improving solution quality in a lesser amount of time. By tracking both the chain length ''n'' and speed of convergence P(X<sub>n</sub> ∉ Cost<sub>min</sub>), one can find a linear plot relating all four of the quantities through the following relationship:
<math>\ln{P(X_n \not\in Cost_{min})} = \alpha \ln{K} - \alpha \ln{n}</math>
<math>\ln{P(X_n \not\in Cost_{min})} = \alpha \ln{K} - \alpha \ln{n}</math>
Once a sufficient number of runs have been completed, the α and ''K'' factors will be known and can thereby be exploited to find the most effective chain length to run multiple independent Markov chains. Given the potential size of the search space, one can muse that the α factor will most likely be closer to one rather than to zero, because with the multiple run strategy, faster cooling will result in a particular chain settling very quickly to a minimum (which may be a local minimum). After settling, the cluster can then move on to a new chain to settle to another minima. If this is repeated, the chances of finding the global minima among one of the solutions is much greater than if only one chain were used.
Once a sufficient number of runs have been completed, the α and ''K'' factors will be known and can thereby be exploited to find the most effective chain length to run multiple independent Markov chains. Given the potential size of the search space, one can muse that the α factor will most likely be closer to one rather than to zero, because with the multiple run strategy, faster cooling will result in a particular chain settling very quickly to a minimum (which may be a local minimum). After settling, the cluster can then move on to a new chain to settle to another minima. If this is repeated, the chances of finding the global minima among one of the solutions is much greater than if only one chain were used.
The figures below represent convergence graphs of different optimization algorithms. In figure 2 the lines with equilateral triangles dispersed throughout represent simulated annealing algorithms similar to ParSA. The graph with the base of the triangle up represents adaptive simulated annealing, while the graph with the base of the triangles down represents non-adaptive simulated annealing. In figures 3 and 4, the dotted line denoted by SA represents simulated annealing.
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