Efficient neural networks for solving variational inequalities

In this paper, we propose efficient neural network models for solving a class of variational inequality problems. Our first model can be viewed as a generalization of the basic projection neural network proposed by Friesz et al. [3]. As the basic projection neural network, it only needs some function evaluations and projections onto the constraint set, which makes the model very easy to implement, especially when the constraint set has some special structure such as a box, or a ball. Under the condition that the underlying mapping F is pseudo-monotone with respect to a solution, a condition that is much weaker than those required by the basic projection neural network, we prove the global convergence of the proposed neural network. If F is strongly pseudo-monotone, we prove its globally exponential stability. Then to improve the efficient of the neural network, we modify it by choosing a new direction that is bounded away from zero. Under the condition that the underlying mapping F is co-coercive, a condition that is a little stronger than pseudo-monotone but is still weaker than those required by the basic projection neural network, we prove the exponential stability and global convergence of the improved model. We also reported some computational results, which illustrated that the new method is more efficient than that of Friesz et al. [3].

[1]  P. Marcotte,et al.  A note on a globally convergent Newton method for solving monotone variational inequalities , 1986 .

[2]  Jun Wang,et al.  A general methodology for designing globally convergent optimization neural networks , 1998, IEEE Trans. Neural Networks.

[3]  Zidong Wang,et al.  Stability and Synchronization of Discrete-Time Markovian Jumping Neural Networks With Mixed Mode-Dependent Time Delays , 2009, IEEE Transactions on Neural Networks.

[4]  Bingsheng He,et al.  A neural network model for monotone linear asymmetric variational inequalities , 2000, IEEE Trans. Neural Networks Learn. Syst..

[5]  John J. Hopfield,et al.  Simple 'neural' optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit , 1986 .

[6]  Wenyu Sun,et al.  A new modified Goldstein-Levitin-Polyakprojection method for variational inequality problems , 2004 .

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

[8]  Li-Zhi Liao,et al.  A Novel Neural Network for a Class of Convex Quadratic Minimax Problems , 2006, Neural Computation.

[9]  Xiaolin Hu,et al.  Solving Pseudomonotone Variational Inequalities and Pseudoconvex Optimization Problems Using the Projection Neural Network , 2006, IEEE Transactions on Neural Networks.

[10]  Xiaolin Hu,et al.  Applications of the general projection neural network in solving extended linear-quadratic programming problems with linear constraints , 2009, Neurocomputing.

[11]  Bart Kosko,et al.  Neural networks for signal processing , 1992 .

[12]  Zidong Wang,et al.  Global Synchronization for Discrete-Time Stochastic Complex Networks With Randomly Occurred Nonlinearities and Mixed Time Delays , 2010, IEEE Transactions on Neural Networks.

[13]  Yurong Liu,et al.  State estimation for jumping recurrent neural networks with discrete and distributed delays , 2009, Neural Networks.

[14]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[15]  P. Tseng,et al.  Modified Projection-Type Methods for Monotone Variational Inequalities , 1996 .

[16]  Defeng Sun,et al.  A class of iterative methods for solving nonlinear projection equations , 1996 .

[17]  Terry L. Friesz,et al.  Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems , 1994, Oper. Res..

[18]  B. He.  A class of projection and contraction methods for monotone variational inequalities , 1997 .

[19]  Hai Yang,et al.  Modified Goldstein–Levitin–Polyak Projection Method for Asymmetric Strongly Monotone Variational Inequalities , 2002 .

[20]  Jun Wang,et al.  On the Stability of Globally Projected Dynamical Systems , 2000 .

[21]  Masao Fukushima,et al.  A globally convergent Newton method for solving strongly monotone variational inequalities , 1993, Math. Program..

[22]  Xing-Bao Gao,et al.  A novel neural network for nonlinear convex programming , 2004, IEEE Trans. Neural Networks.

[23]  Tsuneo Yoshikawa,et al.  Foundations of Robotics: Analysis and Control , 1990 .

[24]  H. Lo,et al.  Two new self-adaptive projection methods for variational inequality problems , 2002 .

[25]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[26]  L. Liao,et al.  Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities , 2002 .

[27]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[28]  Liqun Qi,et al.  A novel neural network for variational inequalities with linear and nonlinear constraints , 2005, IEEE Transactions on Neural Networks.

[29]  Yongqing Yang,et al.  Global exponential system of projection neural networks for system of generalized variational inequalities and related nonlinear minimax problems , 2010, Neurocomputing.

[30]  Francesco Gianfelici.  Review of Random signals and systems by B. Pincibono Englewood Cliffs, NJ, Prentice-Hall, 2003 , 2009 .

[31]  Xing-Bao Gao,et al.  Exponential stability of globally projected dynamic systems , 2003, IEEE Trans. Neural Networks.

[32]  Boris Polyak,et al.  Constrained minimization methods , 1966 .

[33]  Zidong Wang,et al.  State Estimation for Coupled Uncertain Stochastic Networks With Missing Measurements and Time-Varying Delays: The Discrete-Time Case , 2009, IEEE Transactions on Neural Networks.

[34]  Jinde Cao,et al.  A Recurrent Neural Network Based on Projection Operator for Extended General Variational Inequalities , 2010, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[35]  A. Goldstein.  Convex programming in Hilbert space , 1964 .