An Extended Projection Neural Network for Constrained Optimization

2004 ◽  
Vol 16 (4) ◽  
pp. 863-883 ◽  
Author(s):  
Youshen Xia
Keyword(s):  
Neural Network ◽  
Variational Inequality ◽  
Parallel Implementation ◽  
Variational Inequality Problems ◽  
Extended Projection ◽  
Linear And Nonlinear Constraints ◽  
Lower Complexity ◽  
Monotone Variational Inequality ◽  
Projection Neural Network ◽  
Special Case

Recently, a projection neural network has been shown to be a promising computational model for solving variational inequality problems with box constraints. This letter presents an extended projection neural network for solving monotone variational inequality problems with linear and nonlinear constraints. In particular, the proposed neural network can include the projection neural network as a special case. Compared with the modified projection-type methods for solving constrained monotone variational inequality problems, the proposed neural network has a lower complexity and is suitable for parallel implementation. Furthermore, the proposed neural network is theoretically proven to be exponentially convergent to an exact solution without a Lipschitz condition. Illustrative examples show that the extended projection neural network can be used to solve constrained monotone variational inequality problems.

On Exponential Convergence Conditions of an Extended Projection Neural Network

2008 ◽  
Vol 20 (9) ◽  
pp. 2227-2237 ◽  
Author(s):  
Youshen Xia ◽  
Dongyi Ye
Keyword(s):  
Neural Network ◽  
Variational Inequality ◽  
Jacobian Matrix ◽  
Exponential Convergence ◽  
Positive Semidefinite ◽  
Variational Inequality Problems ◽  
Nonlinear Mapping ◽  
Extended Projection ◽  
Monotone Variational Inequality ◽  
Projection Neural Network

Recently the extended projection neural network was proposed to solve constrained monotone variational inequality problems and a class of constrained nonmonotontic variational inequality problems. Its exponential convergence was developed under the positive definiteness condition of the Jacobian matrix of the nonlinear mapping. This note proposes new results on the exponential convergence of the output trajectory of the extended projection neural network under the weak conditions that the Jacobian matrix of the nonlinear mapping may be positive semidefinite or not. Therefore, new results further demonstrate that the extended projection neural network has a fast convergence rate when solving a class of constrained monotone variational inequality problems and nonmonotonic variational inequality problems. Illustrative examples show the significance of the obtained results.

scholarly journals On the $O(1/k)$ Convergence Rate of He's Alternating Directions Method for a Kind of Structured Variational Inequality Problem

2015 ◽  
Vol 7 (2) ◽  
pp. 69
Author(s):  
Haiwen Xu
Keyword(s):  
Variational Inequality ◽  
Convergence Rate ◽  
Variational Inequality Problem ◽  
Variational Inequality Problems ◽  
System Of Nonlinear Equations ◽  
Inequality Problem ◽  
Quadratic Minimization ◽  
Projection And Contraction Methods ◽  
Monotone Variational Inequality ◽  
Alternating Directions Method

The  alternating directions method for a kind of structured variational inequality problem (He, 2001) is an attractive method for structured monotone variational inequality problems. In each iteration, the subproblemsare  convex quadratic minimization problem with simple constraintsand a well-conditioned system of nonlinear equations that can be efficiently solvedusing classical methods. Researchers have recently described the convergence rateof projection and contraction methods for variational inequality problems andthe original ADM and its linearized variant. Motivated and inspired by researchinto the convergence rate of these methods, we provide a simple proof to show the $O(1/k)$ convergencerate of  alternating directions methods for structured monotone variational inequality problems (He, 2001).

Sign in / Sign up

Export Citation Format

Share Document