Constrained optimization with a continuous Hopfield-Lagrange model
In this paper, a generalized Hopfield model with continuous neurons using Lagrange multipliers, originally introduced Wacholder, Han &Mann , 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 Takefuji . 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 several problems like Weighted Matching, Crossbar Switch Scheduling and the Travelling Salesman Problem. The relevant theoretical results are applied and compared to the computational ones. Various formulations of the constraints are tried, of which one is a new approach, where a multiplier is used for every single constraint.