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Mean-Field Annealing

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 tex2html_wrap_inline783 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 tex2html_wrap_inline789 .

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 tex2html_wrap_inline791 and tex2html_wrap_inline793 which generate a set of self-consistency equations in the mean-field (MF) variables (E is the energy function):

  equation420

  equation423

  equation426

where tex2html_wrap_inline797 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 tex2html_wrap_inline801 , 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):

  equation429




next up previous
Next: The Mean-Field Annealing Algorithm Up: No Title Previous: The Timetabling Problem

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