# Hopfield Neural Network With Negative Weights

Hopfield and Tank have shown that a neural network can find solutions for complex optimization problems although it can be trapped in a local minimum of the objective function returning a suboptimal solution. When the problem has constraints they can be added to the objective function as penalty terms using Lagrange multipliers. In this paper, we introduce an approach inspired by the work of Andrew, Chu, and Kate to implement a neural network to obtain solutions satisfying the linear equality constraints using the Moore- Penrose pseudo inverse matrix to construct a projection matrix to send any configuration to the subspace of configuration space that satisfies all the constraints. The objective function of the problem is modified to include Lagrange-multipliers terms for the equations of constraints. Furthermore, we have found that such condition makes the network converge to a set of stable states even if some diagonal elements of the weight matrix are negative. If after several steps the network does not converge to a stable state, we just solve the problem using simulated annealing. We use this technique to solve the NP-hard Light Up puzzle.

## History

## Email Address of Submitting Author

cursino@ita.br## ORCID of Submitting Author

0000-0002-7718-5897## Submitting Author's Institution

Instituto Tecnológico de Aeronáutica - ITA## Submitting Author's Country

- Brazil