ENG  Vol.7 No.9 , September 2015
A Smoothing Neural Network Algorithm for Absolute Value Equations
Abstract: In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.
Cite this paper: Wang, F. , Yu, Z. and Gao, C. (2015) A Smoothing Neural Network Algorithm for Absolute Value Equations. Engineering, 7, 567-576. doi: 10.4236/eng.2015.79052.

[1]   Mangasarian, O.L. and Meyer, R.R. (2006) Absolute Value Equations. Linear Algebra and Its Applications, 419, 359-367.

[2]   Rohn, J. (2004) A Theorem of the Alternatives for the Equation Ax + B|x| = b. Linear and Multilinear Algebra, 52, 421-426.

[3]   Han, J.Y., Xiu, N.H. and Qi, H.D. (2006) Nonlinear Complementary Theory and Algorithm. Shanghai Science and Technology Press, Shanghai.

[4]   Rohn, J.R. (1989) Systems of Linear Interval Equations. Linear Algebra and Its Applications, 126, 39-78.

[5]   Mangasarian, O.L. (2009) Knapsack Feasibility as an Absolute Value Equation Solvable by Successive Linear Programming. Optimization Letter, 3, 161-170.

[6]   Yamashita, S. and Fukushima, M. (2001) A Branch-and-Bound Method for Absolute Value Program and Its Application to Facility Location Problems. Kyoto University, Kyoto.

[7]   Mangasarian, O.L. (2013) Absolute Value Equation Solution via Dual Complementarity. Optimization Letters, 7, 625-630.

[8]   Mangasarian, O.L. (2009) A Generalized Newton Method for Absolute Value Equations. Optimization Letters, 3, 101-108.

[9]   Caccetta, L., Qu, B. and Zhou, G.L. (2011) A Globally and Quadratically Convergent Method for Absolute Value Equations. Computational Optimization and Applications, 48, 45-58.

[10]   Rohn, J. (2009) An Algorithm for Solving the Absolute Value Equation. Electronic Journal of Linear Algebra, 18, 589-599.

[11]   Yong, L.Q. and Tuo, S.H. (2012) Quasi-Newton Method to Absolute Value Equations Based on Aggregate Function. Journal of System science and Mathematics, 32, 1427-1436.

[12]   Yong, L.Q., Liu, S.Y. and Tuo, S.H. (2014) Transformation of the Linear Complementarity Problem and the Absolute Value Equation. Journal of Jilin University (Science Edition), 4, 638-686

[13]   Li, X.S. (1991) The Aggregation Function Method for Solving the Problem of Nonlinear Mini-Max Problem. Computational Structural Mechanics and Its Application, 8, 85-92.

[14]   Chen, J.-S., Ko, C.-H. and Pan, S.H. (2010) A Neural Network Based on the Generalized Fischer-Burmeister Function for Nonlinear Complementarity Problems. Information Sciences, 180, 697-711.

[15]   Ma, H.Q. and Huang, N.-J. (20145) Neural Network Smoothing Approximation Method for Stochastic Variational Inequality Problems. Journal of Industrial and Management Optimization, 7, 645-660.

[16]   Liao, L.Z., Qi, H.D. and Qi, L.Q. (2001) Solving Nonlinear Complementarity Problems with Neural Networks: A Reformulation Method Approach. Journal of Computational and Applied Mathematics, 131, 343-359.

[17]   Miller, R.K. and Michel, A.N. (1982) Ordinary Differential Equations. Academic Press, Waltham.