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