In this article, we present a solution to the maximum clique problem using a gradient-ascent learning algorithm of the Hopfield neural network. This method provides a near-optimum parallel algorithm for finding a maximum clique. To do this, we use the Hopfield neural network to generate a near-maximum clique and then modify weights in a gradient-ascent direction to allow the network to escape from the state of near-maximum clique to maximum clique or better. The proposed parallel algorithm is tested on two types of random graphs and some benchmark graphs from the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS). The simulation results show that the proposed learning algorithm can find good solutions in reasonable computation time.

A method of supervised learning for multilayer artificial neural networks to escape local minima is proposed. The learning model has two phases: a backpropagation phase and a gradient ascent phase. The backpropagation phase performs steepest descent on a surface in weight space whose height at any point in weight space is equal to an error measure, and it finds a set of weights minimizing this error measure. When the backpropagation gets stuck in local minima, the gradient ascent phase attempts to fill up the valley by modifying gain parameters in a gradient ascent direction of the error measure. The two phases are repeated until the network gets out of local minima. The algorithm has been tested on benchmark problems, such as exclusive-or (XOR), parity, alphabetic characters learning, Arabic numerals with a noise recognition problem, and a realistic real-world problem: classification of radar returns from the ionosphere. For all of these problems, the systems are shown to be capable of escaping from the backpropagation local minima and converge faster when using the new proposed method than using the simulated annealing techniques.