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A NonEquidistant Elastic Net Algorithm
, 1997
"... 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 spacepoi ..."
Abstract

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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#,##
Generalized Hopfield Networks for Constrained Optimization
, 1998
"... A twofold generalization of the classical continuous Hop#eld neural network for modelling constrained optimization problems is proposed. On the one hand, nonquadratic cost functions are admitted corresponding to nonlinear output summation functions in the neurons. On the other hand it is shown und ..."
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A twofold generalization of the classical continuous Hop#eld neural network for modelling constrained optimization problems is proposed. On the one hand, nonquadratic cost functions are admitted corresponding to nonlinear 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#eldLagrange 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...