Cooperative annealing Hopfield network for unconstrained binary quadratic programming problem

Abstract

The updating rule of the original discrete Hopfield neural network (DHNN) is based on gradient descent dynamics, which always leads to the local minima problem. In this paper, by introducing the idea of the simulated annealing (SA) into the DHNN, we first propose an annealing HNN (AHNN) that permits temporary energy ascent to help the DHNN escape from local minima. Then, from a cooperative perspective, a population of the AHNN processes are simultaneously implemented and coupled by their acceptance probabilities, and thus a cooperative AHNN (CoAHNN) is proposed. The primary objective of the coupling in the CoAHNN is to create cooperative behavior via information exchange among neural networks. This objective helps in the decision of whether uphill moves will be accepted. In addition, coupling can provide information used online to guide the networks toward the global optimum. The CoAHNN is tested on 21 unconstrained binary quadratic programming problems (UBQP) with the size ranging from 3000 to 7000, and 48 maximum cut benchmark problems, a special case of the UBQP, with the size ranging from 512 to 3375. The UBQP consists in maximizing a quadratic 0–1 function. It is a well known NP-hard problem and is considered as a unified model for a variety of combinatorial optimization problems. Simulation results show that the CoAHNN is better than or competitive with other HNN based algorithms, metaheuristic algorithms and state-of-the-art algorithms.