"Deceptive" problems for Hopfield neural networks

In this paper, we analyze Hopfield neural networks (HNN) using eigenvalues of the connection matrix. The result makes clear that a well-tuned HNN always converges to a certain solution. Although this solution is good enough for most problems, there are some "deceptive" problems whose optimal solutions are far from those obtained by ordinary HNN. Next, we analyze the dynamics of HNN, and show that there are two deceiving phases in these deceptive problems. After that, we propose a modified method to deal with constraint conditions. This method improves the solutions of ordinary problems, and even solves at least one of these deceiving phases.