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The Mean-Field Annealing Algorithm

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 tex2html_wrap_inline809 . 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, tex2html_wrap_inline819 , defined as

  equation432

to characterize the degree of clustering of events in time and/or space; clearly tex2html_wrap_inline821 and tex2html_wrap_inline823 .

   figure115
Figure 1: The Generic Mean-Field Annealing Algorithm

In our implementation, at each tex2html_wrap_inline853 the algorithm (Figure 1) performs one update per neural variable (defined as one sweep) with sequential updating using equations 1 and 2. After reaching tex2html_wrap_inline855 we check whether the obtained solutions are legal, i.e. tex2html_wrap_inline857 . 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 tex2html_wrap_inline859 , 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].


next up previous
Next: The Rule-Based System Up: Mean-Field Annealing Previous: Mean-Field Annealing

Saleh Elmohamed
Tue Apr 29 19:08:49 EDT 1997