Constrained Optimization with a Continuous Hoppeld-lagrange Model

In this paper, a generalized Hoppeld model with continuous neurons using Lagrange multipliers, originally introduced in 12], is thoroughly analysed. We have termed the model the Hoppeld-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 Hoppeld 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 Hoppeld-Lagrange model to a penalty method. Furthermore, a diiculty is described which may arise if the method is applied to problems with`hard constraints'. The theoretical results suggest a method of using the Hoppeld-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.