HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Full Text Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Atencia, M. Articles by Sandoval, F. Search for Related Content PubMed PubMed Citation Articles by Atencia, M. Articles by Sandoval, F.

Dynamical Analysis of Continuous Higher-Order Hopfield Networks for Combinatorial Optimization

matencia{at}ctima.uma.es, Departamento de Matemática Aplicada, ETSI Telecomunicación, Universidad de Málaga, 29071 Málaga, Spain

Gonzalo Joya

joya{at}dte.uma.es, Departamento de Tecnología Electrónica, ETSI Telecomunicación, Universidad de Málaga, 29071 Málaga, Spain

Francisco Sandoval

sandoval{at}dte.uma.es, Departamento de Tecnología Electrónica, ETSI Telecomunicación, Universidad de Málaga, 29071 Málaga, Spain

In this letter, the ability of higher-order Hopfield networks to solve combinatorial optimization problems is assessed by means of a rigorous analysis of their properties. The stability of the continuous network is almost completely clarified: (1) hyperbolic interior equilibria, which are unfeasible, are unstable; (2) the state cannot escape from the unitary hypercube; and (3) a Lyapunov function exists. Numerical methods used to implement the continuous equation on a computer should be designed with the aim of preserving these favorable properties. The case of nonhyperbolic fixed points, which occur when the Hessian of the target function is the null matrix, requires further study. We prove that these nonhyperbolic interior fixed points are unstable in networks with three neurons and order two. The conjecture that interior equilibria are unstable in the general case is left open.