A New Relaxation Procedure in the Hopfield Network for Solving Optimization Problems
Neural Processing Letters
An Efficient Multivalued Hopfield Network for the Traveling Salesman Problem
Neural Processing Letters
Comparing Structures Using a Hopfield-Style Neural Network
Applied Intelligence
Modelling competitive Hopfield networks for the maximum clique problem
Computers and Operations Research
12.3 A Novel Routing Algorithm for MCM Substrate Verification Using Single-Ended Probe
VTS '98 Proceedings of the 16th IEEE VLSI Test Symposium
A Probe Scheduling Algorithm for MCM Substrates
ITC '99 Proceedings of the 1999 IEEE International Test Conference
Mathematics and Computers in Simulation
A review on evolution of production scheduling with neural networks
Computers and Industrial Engineering
MATH'06 Proceedings of the 10th WSEAS International Conference on APPLIED MATHEMATICS
Letters: A TCNN filter algorithm to maximum clique problem
Neurocomputing
Resolve redundancy with constraints for obstacle and singularity avoidance subgoals
International Journal of Robotics and Automation
Parallel ACS for weighted MAX-SAT
IWANN'03 Proceedings of the Artificial and natural neural networks 7th international conference on Computational methods in neural modeling - Volume 1
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Hopfield-type networks convert a combinatorial optimization to a constrained real optimization and solve the latter using the penalty method. There is a dilemma with such networks: when tuned to produce good-quality solutions, they can fail to converge to valid solutions; and when tuned to converge, they tend to give low-quality solutions. This paper proposes a new method, called the augmented Lagrange-Hopfield (ALH) method, to improve Hopfield-type neural networks in both the convergence and the solution quality in solving combinatorial optimization. It uses the augmented Lagrange method, which combines both the Lagrange and the penalty methods, to effectively solve the dilemma. Experimental results on the travelling salesman problem (TSP) show superiority of the ALH method over the existing Hopfield-type neural networks in the convergence and solution quality. For the ten-city TSPs, ALH finds the known optimal tour with 100% success rate, as the result of 1000 runs with different random initializations. For larger size problems, it also finds remarkably better solutions than the compared methods while always converging to valid tours