rem 1 In mean #eld approximation, the free energy F c of constrained stochastic binary Hop#eld networks, submitted to the constraint P i S i =1equals F c #V#=, 1 2 X ij w ij V i V j , 1 # ln# X i exp#,## X j w ij V j + I i ###: #3# The stationary points of F c are found at state spacepoints where V i =P#S i =1 8j 6= i : S j =0#= exp#,## P j w ij V j + I i ## P l exp#,##

A twofold generalization of the classical continuous Hop#eld neural network for modelling constrained optimization problems is proposed. On the one hand, non-quadratic cost functions are admitted corresponding to non-linear output summation functions in the neurons. On the other hand it is shown under which conditions various #new# types of constraints can be incorporated directly. The stability properties of several relaxation schemes are shown. If a direct incorporation of the constraints appears to be impossible, the Hop#eld-Lagrange model can be applied, the stability properties of which are analyzed as well. Another good way to deal with constraints is by means of dynamic penalty terms, using mean #eld annealing in order to end up in a feasible solution. A famous example in this context is the elastic net, although it seems impossible - contrary to what is suggested in the literature - to derive the architecture of this network from a constrained Hop- #eld model. Furthermore, a no...