TY - JOUR
T1 - A novel neural network for variational inequalities with linear and nonlinear constraints
AU - Gao, Xing Bao
AU - LIAO, Lizhi
AU - Qi, Liqun
N1 - Funding Information:
Manuscript received October 11, 2003; revised December 9, 2004. This work was supported in part by the Hong Kong Baptist University, the Research Grant Council of Hong Kong, and NSFC, China, under Grant 10471083. X.-B. Gao is with the College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, China (e-mail: xinbaog@ snnu.edu.cn). L.-Z. Liao is with the Department of Mathematics, Hong Kong Baptist University Kowloon Tong, Hong Kong, China (e-mail: [email protected]). L. Qi is with the Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TNN.2005.852974
PY - 2005/11
Y1 - 2005/11
N2 - Variational inequality is a uniform approach for many important optimization and equilibrium problems. Based on the sufficient and necessary conditions of the solution, this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints. Three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of Lyapunov and converges to an exact solution of the original problem. Meanwhile, the proposed network with a gradient mapping is also proved to be stable in the sense of Lyapunov and to have a finite-time convergence under some mild conditions by using a new energy function. Compared with the existing neural networks, the new model can be applied to solve some nonmonotone problems, has no adjustable parameter, and has lower complexity. Thus, the structure of the proposed network is very simple. Since the proposed network can be used to solve a broad class of optimization problems, it has great application potential. The validity and transient behavior of the proposed neural network are demonstrated by several numerical examples.
AB - Variational inequality is a uniform approach for many important optimization and equilibrium problems. Based on the sufficient and necessary conditions of the solution, this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints. Three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of Lyapunov and converges to an exact solution of the original problem. Meanwhile, the proposed network with a gradient mapping is also proved to be stable in the sense of Lyapunov and to have a finite-time convergence under some mild conditions by using a new energy function. Compared with the existing neural networks, the new model can be applied to solve some nonmonotone problems, has no adjustable parameter, and has lower complexity. Thus, the structure of the proposed network is very simple. Since the proposed network can be used to solve a broad class of optimization problems, it has great application potential. The validity and transient behavior of the proposed neural network are demonstrated by several numerical examples.
KW - Convergence
KW - Neural network
KW - Stability
KW - Variational inequality
UR - http://www.scopus.com/inward/record.url?scp=28244495859&partnerID=8YFLogxK
U2 - 10.1109/TNN.2005.852974
DO - 10.1109/TNN.2005.852974
M3 - Journal article
C2 - 16342476
AN - SCOPUS:28244495859
SN - 1045-9227
VL - 16
SP - 1305
EP - 1317
JO - IEEE Transactions on Neural Networks
JF - IEEE Transactions on Neural Networks
IS - 6
ER -