The generic MFA algorithm appears in Figure 1.
Let N denote the total number of the utilized neurons.
At high T,
the mean-field solutions will tend to be states near to the
(symmetrical) maximum entropy state . Conversely at low
T, finding a mean-field solution will be equivalent to using a local
optimization method on the internal energy - a procedure highly
sensitive to the initial conditions and known to be ineffective [15].
These characteristics are similar to those of simulated annealing, which is no surprise since both it and the mean-field method compute thermal averages over Gibbs distributions of discrete states, the former stochastically and the later through a deterministic approximation. It is therefore natural to couple the mean-field method with the concept of annealing from high to low temperatures.
In addition to the structure of the energy function, there are three major interdependent issues which arise in completely specifying a mean-field annealing algorithm for a timetabling problem:
A quantity introduced by Peterson et al. [13] is called
saturation, , defined as
to characterize the degree of clustering of events in time and/or
space; clearly and
.
Figure 1: The Generic Mean-Field Annealing Algorithm
In our implementation, at each the algorithm
(Figure 1)
performs one update per neural variable
(defined as one sweep)
with sequential updating using equations 1 and 2.
After reaching
we check whether the
obtained solutions are legal, i.e.
. If this is not the
case the network is initialized with a different seed and is allowed to
resettle. We repeat this procedure a number of times until the best
solution is found.
A similar procedure was carried out on high-school
scheduling by Peterson et al. [13].
The MFA implementation was a little more complicated
than the implementation of simulated annealing and the expert system,
since it had many more parameters to handle, and it was
often more difficult to find optimal values for these parameters.
For example, one complication is the computation of the critical
temperature , which involved an iterative procedure of a
linearized dynamic system. On the other hand, we observed that the
convergence time was indeed much less than any of the convergence times of
the simulated annealing using the three annealing
schedules studied.
For more details on our MFA implementation, see Ref. [10].