title: Dynamical analysis of continuous higher-order hopfield networks for combinatorial optimization.
abstract: 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.
authors.affiliation: Departamento de Matemática Aplicada, ETSI Telecomunicación, Universidad de Málaga, 29071 Málaga, Spain. matencia@ctima.uma.es
authors.forename: Miguel
authors.initials: M
authors.lastname: Atencia
authors.forename: Gonzalo
authors.initials: G
authors.lastname: Joya
authors.forename: Francisco
authors.initials: F
authors.lastname: Sandoval
mesh_terms: D000818:Animals; D006801:Humans; D008959:Models, Neurological; D016571:Neural Networks, Computer; D017711:Nonlinear Dynamics