## Simulated annealing

This decreasing step system gave birth to the Metropolis algorithm of 1953 and then to the annealing simulated by IBM in 1983.

## Presentation of the algorithm

The simulated annealing algorithm, or simulated annealing for English speakers, is therefore the algorithmic adaptation of the annealing process.

The simulated annealing algorithm successively generates configurations from an initial solution R_{0} and an initial temperature T_{0} decreasing with each iteration, until a stable configuration is obtained. The probability of accepting a new configuration is (Metropolis rule):

- 1 if the configuration improves the objective function;
- exp (diff (E) / T) with diff (E) the energy difference and T the temperature; the energy difference is an internal function, depending on the objective function.

## Sequence of the algorithm

The annealing algorithm is as follows:

- Take an initial solution R = R
_{0}and an initial temperature T = T_{0}. The initial state is as for the exact methods, obtained by a heuristic (descent or greedy). - Generate a random solution R
_{(i + 1)}in the neighborhood of the current solution:- compare R
_{(i + 1)}with R_{i}according to the Metropolis rule - repeat until a stable solution is found (or after a certain number of iterations)

- compare R
- Decrease the temperature T to a threshold temperature T
_{min}, or have a stable solution.

R_{0}T_{0}as long asT> T_{min}andE> e_{max}R_{(i + 1)}= neighbor (R_{i})ifE (R_{(i + 1)}) <E (R_{i})Whererandom () <P (diff (E) / T)soaccept R_{(i + 1)}update TreturnR_{not}

Since T is large at the start, many solutions degrading the current solution can be chosen. This makes it possible to escape from a local optimum. The following figure shows the value of the objective function as a function of the parameter vector X. The temperature therefore makes it possible to “skip” less “good” areas to leave a “valley”. The balls and signals in the following figure show up to what height the balls can "jump", that is to say the tolerance of negative variation of the objective function in obtaining a new solution.

If the temperature is low, it is more difficult to escape from a local optimum: