Attractor Basin Analysis of the Hopfield Model: The Generalized Quadratic Knapsack Problem

The Continuous Hopfield Neural Network (CHN) is a neural network which can be used to solve some optimization problems. The weights of the network are selected based upon a set of parameters which are deduced by mapping the optimization problem to its associated CHN. When the optimization problem is the Traveling Salesman Problem, for instance, this mapping process leaves one free parameter; as this parameter decreases, better solutions are obtained. For the general case, a Generalized Quadratic Knapsack Problem (GQKP), there are some free parameters which can be related to the saddle point of the CHN. Whereas in simple instances of the GQKP, this result guarantees that the global optimum is always obtained, in more complex instances, this is far more complicated. However, it is shown how in the surroundings of the saddle point the attractor basins for the best solutions grow as the free parameter decreases, making saddle point neighbors excellent starting point candidates for the CHN. Some technical results and some computational experiences validate this behavior.