Constrained optimization with a continuous Hopfield-Lagrange model

Abstract

In this paper, a generalized Hopfield model with continuous neurons using Lagrange multipliers, originally introduced in [12], is thoroughly analysed. We have termed the model the Hopfield-Lagrange model. It can be used to resolve constrained optimization problems. In the theoretical part, we present a simple explanation of a fundamental energy term of the continuous Hopfield model. This term has caused some confusion as reported in [11]. It led to some misinterpretations which will be corrected. Next, a new Lyapunov function is derived which, under some dynamical conditions, guarantees stability of the the system. We explain why a certain type of frequently used quadratic constraints can degenerate the Hopfield-Lagrange model to a penalty method. Furthermore, a difficulty is described which may arise if the method is applied to problems with `hard constraints'. The theoretical results suggest a method of using the Hopfield-Lagrange model. This method is described and applied to severa...