One of the potential drawbacks of using simulated annealing for hard optimization problems is that finding a good solution can often take an unacceptably long time. Mean-field annealing (MFA) attempts to avoid this problem by using a deterministic approximation to simulated annealing, by attempting to average over the statistics of the annealing process. The result is improved execution speed at the expense of solution quality. Although not strictly a continuous descent technique, MFA is closely related to the Hopfield neural network [15].
Mean-field annealing has been successfully applied to high-school class
scheduling [13].
For scheduling, it is advantageous to use a
Potts neural encoding to specify discrete neural variables (or neurons)
for the problem.
This is defined in its simplest form as a mapping
of events onto space-time slots,
for example events defined by a professor-class pair (p,q) are mapped onto
space-time slots (x,t).
Potts neurons are defined to be 1 if event (p,q) takes place in
the space-time slot (x,t), and 0 otherwise.
In this way, the constraints involved can be embedded in the neural net in
terms of a Potts normalization condition such as
.
For a full derivation of the mean-field
annealing algorithm from its roots in statistical physics, see Peterson
et al. [27].
Here we will give a brief overview,
taking as our starting point the saddle point equations
and
which generate a set of self-consistency
equations in the mean-field (MF) variables (E is the energy function):
where is the dominant energy function of the system. The
MFA algorithm involves solving equations 1 and 2 at a
series of progressively lower temperature T: this process is known
as temperature annealing. The iterations of the MF dynamics over
equations 1 and 2 can either be synchronous or serial
updating.
The solutions to 1 and 2 correspond to stable
states of the Hopfield network [15] with transfer functions of
the form of equation 3 and
, so both
techniques appear to be seeking the same solution points.
Observe from equation 2 that at a MF solution the variables
respect a continuous version of the Potts condition (over all states a):