In 1986 Tank and Hopfield first proposed a neural network for linear Variational Inequality that was mapped onto a closed loop circuit. Then the quadratic programming problems arise in a wide variety of scientific and engineering applications including regression analysis, image and signal processing, parameter estimation, filter design, robot control, etc. Recently In [1], a linear projection neural network for solving quadratic programming problems was developed. It was shown that the linear projection neural network in [1] was globally convergent to an equilibrium point if M is symmetric and positive semi-definite and the neural network in [1] was globally asymptotically and exponentially stable to a unique equilibrium point if M is positive definite. In [2] [3] and [4] the delayed projection neural network for solving quadratic programming problems were developed, they were shown that the linear projection neural network in [2] [3] and [4] were globally exponentially stable. In [5] and [6] two neutral-type delayed Projection Neural Network for Solving Programming Problems were proposed, and they were shown the projection neural network were globally exponentially stable.
In my letter, I present a neutral-type delayed projection neural network for solving quadratic programming problems. This neutral-type delayed projection neural network has no penalty parameter and no Lagrange multipliers. And the network has a simple structure. In the letter the network is showed to be globally exponentially stable. So it can be used to solve quadratic programming problems.
My paper is organized as follows, in the second part a neutral-type delayed projection neural network was described. In the third section, we show the proposed neural network is globally exponentially stable under certain conditions. In the forth section, we present a illustrative example to show the effectiveness of the proposed network. In the last section, we give the conclusions of this paper.
1.1 The proposed neural network model
Considering the following convex quadratic programming problems:
(1)
where Q is a positive-definite(positive semi-definite) matrix and A is a row full rank matrix.
Assume that feasible domain is not empty and there exists the optimal solution for (1). (1) can be solved by the two following projection neural network. (2)
(3)
Where , is a constant and denote the transmission delay.
, denotes the projection operator defined by [3].
(4)
In the letter, we proposed a neutral-type delayed projection neural network for solving (1):
(5)
Where, is a constant and denote the transmission delay. , is defined by (4).
Let be an equilibrium point set of (5) and be an optimal solution set of (1). It is easy to see that is an equilibrium point of (5) if and only if is an equilibrium point of (2) and (3) if and only if is an equilibrium point of (1). Hence we have .
For the convenience of later discussions, we first introduce some related definitions and lemmas.
Definition 1 [6]: A system is said to be globally exponentially stable at the equilibrium point if every trajectory starting at any initial point satisfies the following:
Where k and are positive constants independent of the initial point and
.
Lemma 1 [6]: (Gronwall) Let , be real-valued nonnegative continuous function with domain , and
For all , then
Where with being monotone increasing function.
Proof: See [7].
1.2 Main Results
In this section, we will study global exponential stability of (5).
Theorem 1: Let , if there exists a constant , such that and ,then the equilibrium point of the neutral-type delayed projection neural network defined by (5) is globally exponentially stable.
Proof: Since is the equilibrium point of (5), then it can be obtained that
By the variation-of-constants formula, the above functional differential equation can be solved as follows:
It is noted that, for , there exists an integer such that . And by above equation,
it follows that, for any integer
We let , and we can see , then, it can be obtained that
So we have:
The following two cases will then be discussed.
Case 1: if , by noting that , it follows that
Therefore, when , it is easy to see that the proposed neural network defined by (5) is globally exponentially stable.
Case 2: if
Here we use the assumption. By the above inequality, we can get the following inequality:
Where
。
From lemma 1 we have:
Then
Therefore, if then the proposed neural network defined by (3) is globally exponentially stable at its equilibrium point . And it is easy to see that it is equivalent to: The proof is over.
Remark 1: By letting , the proposed neural network reduces to the model used in [3]. So the proposed neutral-type delayed projection neural network contains the neural network in [3]. Therefore, the proposed neural network improves the existing results and is more useful.
2 Conclusions
In this paper we propose a neutral-type delayed projection neural network for solving quadratic programming problems. A globally exponential stability criterion is derived by the theory of functional differential equation. Which improves the existing asymptotic stability criteria for the neutral-type neural network, Moreover the proposed delayed neural network can be guaranteed to solve a class of quadratic programming problems and linear variational inequalities.
References
[1] Xia You-shen and Wang Jun. ‘A recurrent neural network for solving linear projection equations’ [J], Neural Netw, vol. 13, pp. 337-350, 2000.
[2] Liu Qing-shan, Cao Jin-de and Xia You-shen, ‘A Delayed Neural Network for Solving Linear Projection Equations and its Analysis’ [J],IEEE Trans on neural networks, vol. 16, NO. 4, pp. 834-843, July 2005.
[3] Yang Yong-qing and Cao Jin-de, ‘Solving Quadratic Programming Problems by Delayed Projection Neural Network’ [J], IEEE Trans on neural networks, vol. 17, NO. 6, pp. 1630-1634, Nov 2006.
[4] Liu Zi-xin, Lu Shu, and Zhong Shou-ming, ‘A new delayed projection neural network for solving linear variational inequalities and quadratic optimization problems’ [C] , 2008 International Symposium on Computational Intelligence and Design, pp. 1630-1634, 2008.
[5] Cheng C.-J., Liao T.-L., Yan J.-J., and Hwang C.-C., ‘Globally asymptotic stability of a class of neutral-type neural networks with delays ’ [J], IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 36, no. 5, pp. 1191-1195, Oct. 2006
[6] Cheng Long, Hou Zeng-Guang, and Tan Min, ‘A Neutral-Type Delayed Projection Neural Network for Solving Nonlinear Variational Inequalities’ [J], IEEE Trans on circuits andsystems-II, VOL. 55, NO. 8, pp. 806-810, Augu 2008.
[7] J. K. Hale and S. M. V. Lunel, ‘Introduction to Functional Differential Equations’ [J], New York: Springer, 1993
作者简介:
闫淑丽:电子科技大学应用数学学院硕士
王毅:电子科技大学骨干教师, 应用数学学院教授,硕士生导师
钟守铭: 电子科技大学骨干教师, 应用数学学院教授,硕士生导师,博士生导师.中
国生物数学学会常务理事,四川省数学会理事,四川省生物数学专业委员会
主任,成都市应用数学学会副理事长,期刊“生物数学学报”编委。在国内
外重要刊物发表科研论文80余篇,参与完成国家自然科学基金项目3项,在研
国家自然科学基金重点项目1项.
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