Editing Simulated annealing (section)

===Efficient candidate generation===
When choosing the candidate generator {{code|neighbour()}}, one must consider that after a few iterations of the simulated annealing algorithm, the current state is expected to have much lower energy than a random state. Therefore, as a general rule, one should skew the generator towards candidate moves where the energy of the destination state <math>s'</math> is likely to be similar to that of the current state. This [[heuristic]] (which is the main principle of the [[Metropolis–Hastings algorithm]]) tends to exclude ''very good'' candidate moves as well as ''very bad'' ones; however, the former are usually much less common than the latter, so the heuristic is generally quite effective.

In the traveling salesman problem above, for example, swapping two ''consecutive'' cities in a low-energy tour is expected to have a modest effect on its energy (length); whereas swapping two ''arbitrary'' cities is far more likely to increase its length than to decrease it. Thus, the consecutive-swap neighbor generator is expected to perform better than the arbitrary-swap one, even though the latter could provide a somewhat shorter path to the optimum (with <math>n-1</math> swaps, instead of <math>n(n-1)/2</math>).

A more precise statement of the heuristic is that one should try the first candidate states <math>s'</math> for which <math>P(E(s), E(s'), T)</math> is large. For the "standard" acceptance function <math>P</math> above, it means that <math>E(s') - E(s)</math> is on the order of <math>T</math> or less. Thus, in the traveling salesman example above, one could use a {{code|neighbour()}} function that swaps two random cities, where the probability of choosing a city-pair vanishes as their distance increases beyond <math>T</math>.