Neural Networks without Training for Optimization - Neural Networks: Methodology and Applications - page 473

The above two constraints are not su cient to guarantee the validity of
a solution. In fact, without any new constraint, the minimization process
would naturally drive the network in a state where all the neuron outputs
are 0. It does not make sense. The validity of a solution requires that exactly
N neurons have an output equal to 1 after convergence. Therefore, a third
constraint function is defined,

⎡ ⎤

2

N N

F 3 = 1
2

⎣ ⎦

y i,j − N .

i =1 j =1

8.6.4.7 Energy of the Neural Network

The total energy of the Hopfield network is the weighted sum of the above
functions,

E = F c + a 1 F 1 + a 2 F 2 + a 3 F 3 .
Constants a 1 ,a 2 and a 3 must be adjusted according to the relative weights
of the various constraints.

Step 3: Finding the Equations of the Neurons

The energy function of the problem can be expressed as a quadratic function,
which is the energy of a Hopfield neural network,

N N N N N N

1
2

E ( y )= − w ij,kl y i,j y k,l − I i,j y i,j .

i =1 j =1 i =1 j =1

k =1 l =1

In that equation, the weights w ij,kl are determined from the analytical
form of the cost and of the constraints. Considering the first constraint F 1 ,
its contribution to the synaptic coe cients is given by

− δ i,k (1 − δ j,l )wh e δ x,y = 1 if and only if x = y.

Similarly, the contribution of the constraint F 2 to the synaptic coe cients is

− δ j,l (1 − δ i,k ) .

The contribution of the constraint F 3 is 1.

Finally, the contribution of the cost function is given by

−

− d i,k ( δ l,j +1 + δ l,j− 1 ) .

Therefore, the final expression of the weights is

w ij,kl = − 2 d i,k δ 1 ,j⊕l − a 1 δ i,k (1 − δ j,l ) − a 2 δ j,l (1 − δ i,k ) − a 3 .

The external input on the neuron ( i,j )is: I i,j = a 3 N .