Optimization problems are typically implemented on a feedback network. These networks interconnect the neurons with a feedback path. A typical feedback neural network is the Hopfield neural network [Hop85]. Figure 4 shows the circuit structure of the neuron and its functional structure. This differential equation describes the neuron:
where (j=1,2..., n), and g( ) is the sigmoid activation function. It is shown in [Hop85] how to choose the values of synapse so that (1) represents the dynamics corresponding to a given energy function. If the energy function corresponds to an optimization objective, the initialization of the 's to an initial configuration will result in an equilibration which settles to a local minimum of the objective function. One famous example using the neural networks is the Traveling Salesman Problem (TSP) [Wil88], in which a salesman is supposed to tour a number of cities (visiting each exactly once, then returning to where he started) and desires to minimize the total distance of the tour. The intercity distances are given as the input, and the desired output is the shortest (or near-shortest tour). The operation of the feedback network implies a descent on energy surface. In order to implement a solution to the TSP, or any other optimization problem on a feedback network, the energy function is used as a medium. By designing the network so that the minimum of the energy function coincides with a minimum-length tour, the network becomes a computer that searches for the minimum tour. This example stirred great excitement since it suggested that the TSP could be solved approximately, using an analog circuit in microseconds.
The highly interconnected architecture of switching networks suggests similarities to neural networks. The papers [Tro91, Mar89, Ali88] considered how to schedule packet transmissions through a complete crossbar switch. They constructed an traffic matrix. The optimization problem is to select a set of 1's in the traffic matrix to form a permutation matrix, whose overlap with the traffic matrix is maximized - this minimizes the blocking for that time slot. The neural network is constructed with one neuron for each switch crosspoint, the network is initialized with the traffic values and the network is allowed to settle to form the permutation matrix.
A more complex approach by using the neural network is to assign an optimal path arrangement in a three-stage Clos' packet switch. The optimization problem is not only to select one permutation matrix in a traffic matrix, but several permutation matrices, the sum of whose overlap is maximized. This approach has been proposed in publications 7, 8 and 9.
Figure 4 Hopfield model of a dynamic basic neuron cell.