Solving Combinatorial Optimization Problems Using Augmented Lagrange Chaotic Simulated Annealing

Chaotic simulated annealing (CSA) proposed by Chen and Aihara has been successfully used to solve a variety of combinatorial optimization problems. CSA uses a penalty term to enforce solution validity as in the original Hopfield–Tank approach. There exists a conflict between solution quality and solution validity in the penalty approach. It is often difficult to adjust the relative magnitude of the penalty term, so as to achieve a high quality solution which is at the same time valid. To overcome this, we incorporate augmented Lagrange multipliers into CSA, obtaining a method that we call augmented Lagrange chaotic simulated annealing (AL-CSA). Simulation results on two constrained optimization benchmarks derived from the Hopfield–Tank formulation of the traveling salesman problem show that AL-CSA can maintain CSA’s good solution quality while avoiding the potential difficulties associated with penalty terms. Furthermore, AL-CSA’s convergence time is shorter and choice of system parameters is easier compared to CSA.

[1]  Ehl Emile Aarts,et al.  Simulated annealing and Boltzmann machines , 2003 .

[2]  Gerhard Reinelt,et al.  TSPLIB - A Traveling Salesman Problem Library , 1991, INFORMS J. Comput..

[3]  Hiroshi Nozawa,et al.  A neural network model as a globally coupled map and applications based on chaos. , 1992, Chaos.

[4]  Kazuyuki Aihara,et al.  A modified algorithm for the quadratic assignment problem using chaotic-neuro-dynamics for VLSI implementation , 2001, IJCNN'01. International Joint Conference on Neural Networks. Proceedings (Cat. No.01CH37222).

[5]  Emilio Del-Moral-Hernandez,et al.  Chaotic Neural Networks , 2009, Encyclopedia of Artificial Intelligence.

[6]  Kazuyuki Aihara,et al.  Chaotic simulated annealing by a neural network model with transient chaos , 1995, Neural Networks.

[7]  Mohamad H. Hassoun.  Guest Editorial Neural Networks Council Awards , 1996, IEEE Trans. Neural Networks.

[8]  Lipo Wang,et al.  Image restoration using chaotic simulated annealing , 2003, Proceedings of the International Joint Conference on Neural Networks, 2003..

[9]  H. Nozawa,et al.  Solution of the optimization problem using the neural network model as a globally coupled map , 1994 .

[10]  Lipo Wang,et al.  Chaotic simulated annealing with augmented Lagrange for solving combinatorial optimization problems , 2000, 2000 26th Annual Conference of the IEEE Industrial Electronics Society. IECON 2000. 2000 IEEE International Conference on Industrial Electronics, Control and Instrumentation. 21st Century Technologies.

[11]  Chunru Wan,et al.  Minimizing Interference in Cellular Mobile Communications by Optimal Channel Assignment Using Chaotic Simulated Annealing , 2004 .

[12]  Lipo Wang,et al.  Artificial neural networks - oscillations, chaos, and sequence processing , 1993 .

[13]  Sanjit K. Mitra,et al.  Alternative networks for solving the traveling salesman problem and the list-matching problem , 1988, IEEE 1988 International Conference on Neural Networks.

[14]  Kazuyuki Aihara,et al.  Chaos engineering and its application to parallel distributed processing with chaotic neural networks , 2002, Proc. IEEE.

[15]  Kazuyuki Aihara,et al.  Global searching ability of chaotic neural networks , 1999 .

[16]  Kazuyuki Aihara,et al.  Combination of Chaotic Neurodynamics with the 2-opt Algorithm to Solve Traveling Salesman Problems , 1997 .

[17]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[18]  Shengwei Zhang,et al.  Lagrange programming neural networks , 1992 .

[19]  G. Pawley,et al.  On the stability of the Travelling Salesman Problem algorithm of Hopfield and Tank , 2004, Biological Cybernetics.

[20]  Kate Smith-Miles,et al.  On chaotic simulated annealing , 1998, IEEE Trans. Neural Networks.

[21]  Kate A. Smith,et al.  Solving Combinatorial Optimization Problems by Chaotic Neural Networks , 1998 .

[22]  Zhenya He,et al.  A transiently chaotic neural-network implementation of the CDMA multiuser detector , 1999, IEEE Trans. Neural Networks.

[23]  Kate A. Smith,et al.  Chaos in the discretized analog Hopfield neural network and potential applications to optimization , 1998, 1998 IEEE International Joint Conference on Neural Networks Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98CH36227).

[24]  Lipo Wang,et al.  Solving Optimization Problems Using Transiently Chaotic Neural Networks : Choices of Network Parameters , 2002 .

[25]  W. Zhong,et al.  Multiuser detection using time-varying scaling-parameter transiently chaotic neural networks , 1999 .

[26]  Lipo Wang,et al.  Oscillatory and chaotic dynamics in neural networks under varying operating conditions , 1996, IEEE Trans. Neural Networks.

[27]  S. Abe.  Theories on the Hopfield neural networks , 1989, International 1989 Joint Conference on Neural Networks.

[28]  Stan Z. Li,et al.  Improving convergence and solution quality of Hopfield-type neural networks with augmented Lagrange multipliers , 1996, IEEE Trans. Neural Networks.

[29]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.