The.Algorithm.Design.Manual.Springer-Verlag.1998

Simulated Annealing
Return S
then
until (no change in C(S))
If then
else if
Reduce temperature t
There are three components to any simulated annealing algorithm for combinatorial search:
● Concise problem representation - The problem representation includes both a representation of the solution space and
an appropriate and easily computable cost function C() measuring the quality of a given solution.
● Transition mechanism between solutions - To move from one state to the next, we need a collection of simple
transition mechanisms that slightly modify the current solution. Typical transition mechanisms include swapping the
position of a pair of items or inserting/deleting a single item. Ideally, the effect that these incremental changes have on
measuring the quality of the solution can be computed incrementally, so cost function evaluation takes time
proportional to the size of the change (typically constant) instead of linear in the size of the solution.
● Cooling schedule - These parameters govern how likely we are to accept a bad transition as a function of time. At the
beginning of the search, we are eager to use randomness to explore the search space widely, so the probability of
accepting a negative transition is high. As the search progresses, we seek to limit transitions to local improvements
and optimizations. The cooling schedule can be regulated by the following parameters:
❍ Initial system temperature - Typically .
❍ Temperature decrement function - Typically , where . This implies an exponential
decay in the temperature, as opposed to a linear decay.
❍ Number of iterations between temperature change - Typically, 100 to 1,000 iterations might be permitted
before lowering the temperature.
❍ Acceptance criteria - A typical criterion is to accept any transition from to when and to
accept a negative transition whenever
where r is a random number . The constant c normalizes this cost function, so that almost all
transitions are accepted at the starting temperature.
❍ Stop criteria - Typically, when the value of the current solution has not changed or improved within the last
iteration or so, the search is terminated and the current solution reported.
Creating the proper cooling schedule is somewhat of a trial and error process. It might pay to start from an existing
implementation of simulated annealing, pointers to which are provided in Section .
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