Matrix analysis
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AIP Conference Proceedings 151 on Neural Networks for Computing
On the stability of the travelling salesman problem algorithm of Hopfield and Tank
Biological Cybernetics
Using dynamical systems methods to solve minimization problems
NUMDIFF-7 Selected papers of the seventh conference on Numerical treatment of differential equations
Handbook of mathematics (3rd ed.)
Handbook of mathematics (3rd ed.)
Fixed-point attractor analysis for a class of neurodynamics
Neural Computation
New theorems on global convergence of some dynamical systems
Neural Networks
Discretization of implicit ODEs for singular root-finding problems
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Neural Networks for Combinatorial Optimization: a Review of More Than a Decade of Research
INFORMS Journal on Computing
Spurious minima and basins of attraction in higher-order Hopfield networks
IWANN'03 Proceedings of the Artificial and natural neural networks 7th international conference on Computational methods in neural modeling - Volume 1
Convergence acceleration of the Hopfield neural network by optimizing integration step sizes
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
“Optimal” Hopfield network for combinatorial optimization with linear cost function
IEEE Transactions on Neural Networks
Microcode optimization with neural networks
IEEE Transactions on Neural Networks
Stability analysis of Hopfield-type neural networks
IEEE Transactions on Neural Networks
Hopfield Network as Static Optimizer: Learning the Weights and Eliminating the Guesswork
Neural Processing Letters
System Identification of Dengue Fever Epidemics in Cuba
IWANN '09 Proceedings of the 10th International Work-Conference on Artificial Neural Networks: Part I: Bio-Inspired Systems: Computational and Ambient Intelligence
Estimation of the rate of detection of infected individuals in an epidemiological model
IWANN'07 Proceedings of the 9th international work conference on Artificial neural networks
Fixed points of the Abe formulation of stochastic Hopfield networks
ICANN'07 Proceedings of the 17th international conference on Artificial neural networks
Robustness of the "hopfield estimator" for identification of dynamical systems
IWANN'11 Proceedings of the 11th international conference on Artificial neural networks conference on Advances in computational intelligence - Volume Part II
FPGA implementation of hopfield networks for systems identification
IWANN'05 Proceedings of the 8th international conference on Artificial Neural Networks: computational Intelligence and Bioinspired Systems
Numerical implementation of gradient algorithms
IWANN'13 Proceedings of the 12th international conference on Artificial Neural Networks: advences in computational intelligence - Volume Part II
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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.