Mathematical improvement of the Hopfield model for TSP feasible solutions by synapse dynamical systems

Abstract It is well known that the Hopfield Model (HM) for neural networks to solve the TSP suffers from three major drawbacks: (D1) it can converge to non-optimal local minimum solutions; (D2) it can also converge to infeasible solutions; (D3) results are very sensitive to the careful tuning of its parameters. A number of methods have been proposed to overcome (D1) well. In contrast, work on (D2) and (D3) has not been sufficient; techniques has not been generalized to more general optimization problems. This paper mathematically resolves (D2) and (D3) of the HM to such an extent that the resolution can be applied to solving with networks continuous optimization problems including the Hopfield version of the TSP. It first constructs Extended HMs (E-HMs) that overcome both (D2) and (D3). The extension of the E-HM lies in the addition of a synapse dynamical system cooperated with the current HM unit dynamical system. It is this synapse dynamical system that makes the TSP constraint hold at any final states for whatever choices of the HM parameters and an initial state. The paper then generalizes the E-HM further into a network that can solve a class of continuous optimization problems with a constraint equation where both of the objective function and the constraint function are non-negative and continuously differentiable.

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