a nonlinear projection neural network for solving interval...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2010, Article ID 403749, 13 pagesdoi:10.1155/2010/403749

Research ArticleA Nonlinear Projection Neural Network forSolving Interval Quadratic Programming Problemsand Its Stability Analysis

Huaiqin Wu, Rui Shi, Leijie Qin, Feng Tao, and Lijun He

College of Science, Yanshan University, Qinhuangdao 066001, China

Correspondence should be addressed to Huaiqin Wu, [email protected]

Received 26 February 2010; Accepted 19 July 2010

Academic Editor: Jitao Sun

Copyright q 2010 Huaiqin Wu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper presents a nonlinear projection neural network for solving interval quadratic programssubject to box-set constraints in engineering applications. Based on the Saddle point theorem,the equilibrium point of the proposed neural network is proved to be equivalent to the optimalsolution of the interval quadratic optimization problems. By employing Lyapunov functionapproach, the global exponential stability of the proposed neural network is analyzed. Twoillustrative examples are provided to show the feasibility and the efficiency of the proposedmethod in this paper.

1. Introduction

In lots of engineering applications including regression analysis, image and signalprogressing, parameter estimation, filter design and robust control, and so forth �1�, it isnecessary to solve the following quadratic programming problem:

min12xTQx � cTx,

subject to x ∈ Ω,�1.1�

where Q ∈ Rn×n, c ∈ Rn, and Ω is a convex set. When Q is a positive definite matrix, theproblem �1.1� is said to be the convex quadratic program. When Q is a semipositive definitematrix, the problem �1.1� is said to be the degenerate convex quadratic program. In general,

2 Mathematical Problems in Engineering

the matrixQ is not precisely known, but can only be enclosed in intervals, that is,Q ≤ Q ≤ Q.Such quadratic program with interval data is named as interval quadratic program usually.In the recent years, there have been some project neural network approaches for solving theproblem �1.1�; see, for example, �2–15�, and the references therein. In �2�, Kennedy and Chuapresented a primal network for solving the convex quadratic program. This network containsa finite penalty parameter, so it converges an approximate solution only. To overcome thepenalty parameter, in �3, 4�, Xia proposed several primal projection neural networks forsolving the convex quadratic program and it dual, and analyzed the global asymptoticstability of the proposed neural networks when the constraint set Ω is a box set. In �5, 6�,Xia et al. presented a recurrent projection neural network for solving the convex quadraticprogram and related linear piecewise equation, and gave some conditions of the exponentialconvergence. In �7, 8�, Yang and Cao presented a delayed projection neural network forsolving problem �1.1�, and analyzed the global asymptotic stability and exponential stabilityof the proposed neural networks when the constraint set Ω is a unbounded box set.In order to solve the degenerate convex quadratic program, Tao et al. �9� and Xue andBian �10, 11� proposed two projection neural networks, and proved that the equilibriumpoint of the proposed neural networks was equivalent to the KT point of the quadraticprogramming problem. Particularly, in �10�, the proposed neural network was shown tohave complete convergence and finite-time convergence, and the nonsingular part of theoutput trajectory with respect to Q has an exponentially convergent rate. In �12, 13�, Hu andWang designed a general projection neural network for solving monotone linear variationalinequalities and extended linear-quadratic programming problems, and proved that theproposed network was exponentially convergent when the constraint set Ω is a polyhedralset.

In order to solve the interval quadratic program, in �14�, Ding and Huang presenteda new class of interval projection neural networks, and proved the equilibrium point ofthis neural networks is equivalent to the KT point of a class of interval quadratic program.Furthermore, some sufficient conditions to ensure the existence and global exponentialstability for the unique equilibrium point of interval projection neural networks are given.To the best of the authors knowledge, the work in �14� is first to study solving theinterval quadratic program by a projection neural network. However, the interval quadraticprogram discussed in �14� is only a quadratic program without constraints, thus has manylimitations in practice. It is well known that the quadratic program with constraints is morepopular.

Motivated by the above discussion, in the present paper, a new projectionneural network for solving the interval quadratic programming problem with box-setconstraints is presented. Based on the Saddle theorem, the equilibrium point of theproposed neural network is proved to be equivalent to the KT point of the intervalquadratic program. By using the fixed point theorem, the existence and uniqueness ofan equilibrium point of the proposed neural network are analyzed. By constructing asuitable Lyapunov function, a sufficient condition to ensure the existence and globalexponential stability for the unique equilibrium point of interval projection neural network isobtained.

This paper is organized as follows. Section 2 describes the system model and givessome necessary preliminaries; Section 3 gives the proof of the existence of equilibriumpoint of the proposed neural network, and discusses the global exponential stability of theproposed neural network; Section 4 provides two numerical examples to demonstrate thevalidity of the obtained results. Some conclusions are drawn in Section 5.

Mathematical Problems in Engineering 3

2. A Projection Neural Network Model

Consider the following interval quadratic programming problem:

min12xTQx � cTx

subject to g ≤ Dx ≤ h,

Q ≤ Q ≤ Q,

�2.1�

where Q �qij�n×n, Q �qij�n×n, Q �qij�n×n ∈ Rn×n; c, g, h ∈ Rn, and D diag�d1, d2, . . . , dn�is a positive definite diagonal matrix. Q ≤ Q ≤ Q means qij ≤ qij ≤ qij , i, j 1, . . . , n. TheLagrangian function of the problem �2.1� is

L(x, u, η

)

12xTQx � cTx − uT

(Dx − η

), Q ≤ Q ≤ Q, �2.2�

where u ∈ Rn is referred to as the Lagrange multiplier and η ∈ X {u ∈ Rn | g ≤ u ≤ h}.Based on the well-known Saddle point theorem �1�, x∗ is an optimal solution of �2.1� if andonly if there exist u∗ and η∗, satisfying L�x∗, u, η∗� ≤ L�x∗, u∗, η∗� ≤ L�x, u∗, η�, that is,

12x∗TQx∗ � cTx∗ − uT

(Dx∗ − η∗

)

≤ 12x∗TQx∗ � cTx∗ − u∗T

(Dx∗ − η∗

)

≤ 12xTQx � cTx − u∗T

(Dx − η

).

�2.3�

By the first inequality in �2.3�, �u − u∗�T �Dx∗ − η∗� ≥ 0, for all u ∈ Rn, hence Dx∗ η∗. Letf�x� �1/2�xTQx�cTx−u∗TDx. By the second inequality in �2.3�, f�x∗�−f�x� ≤ u∗T �η−η∗�,for all x ∈ Rn, η ∈ X. If there exists x ∈ Rn such that f�x∗� − f�x� > 0, then 0 < u∗T �η − η∗�,for all η ∈ X, which is contradictive when η η∗. Thus, for any x ∈ Rn, it follows thatf�x∗� − f�x� ≤ 0 and u∗T �η − η∗� ≥ 0, for all η ∈ X.

By using the project formulation �16�, the above inequality can be equivalentlyrepresented as η∗ PX�η∗ − u∗�, where PX�u� �PX�u1�, PX�u2�, . . . , PX�un��T is a projectfunction, and, for i 1, 2, . . . , n,

PX�ui�

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

gi, ui < gi,

ui, gi ≤ ui ≤ hi,

hi, ui > hi.

�2.4�

On the other hand, f�x∗� ≤ f�x�, for all x ∈ Rn. This implies that

∇f�x∗� Qx∗ � c −Du∗ 0. �2.5�


Thus, x∗ is an optimal solution of �2.1� if and only if there exist u∗ and η∗, such that �x∗, u∗, η∗�satisfies

Dx η,

Qx � c −Du 0,

η PX(η − u

).

�2.6�

From �2.6�, it follows that Dx PX�Dx − u�. Hence, x∗ is an optimal solution of �2.1� if andonly if there exists u∗ such that �x∗, u∗� satisfies

Qx � c −Du 0,

Dx PX�Dx − u�.�2.7�

Substituting u D−1�Qx � c� into the equation Dx PX�Dx − u�, we have

Dx PX(Dx −D−1Qx −D−1c

), �2.8�

where

D−1

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1d1

0 · · · 0

01d2

· · · 0...

.... . .

...

0 0 · · · 1dn

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, D−1Q

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1d1

q111d1

q12 · · ·1d1

q1n

1d2

q211d2

q22 · · ·1d2

q2n

......

. . ....

1dn

qn11dn

qn2 · · ·1dn

qnn

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. �2.9�

By the above discussion, we can obtain the following proposition.

Proposition 2.1. Let x∗ be a solution of the project equation

Dx PX(Dx −D−1Qx −D−1c

), �2.10�

then, x∗ is an optimal solution of the problem �2.1�.


d11

d1n

dn1

dnn

m11

m1n

mnn

mn1

...

...

...

...

...

...

++

++

++

+++

+

−

−

−

−PX

PX

∑

∑

∑ ∑

∑

∑

∫

∫

C1

Cn

x1

xn

Figure 1: Architecture of the proposed neural network in �2.11�.

In the following, we propose a neural network, which is said to be the intervalprojection neural network, for solving �2.1� and �2.10�, whose dynamical equation is definedas follows:

dx�t�dt

PX(Dx −D−1Qx −D−1c

)−Dx, t > t0,

x�t0� x0,

Q ≤ Q ≤ Q.

�2.11�

The neural networks �2.11� can be equivalently written as

dxi�t�dt

PX

⎛

⎝dixi −1di

n∑

j1

qijxj −cidi

⎞

⎠ − dixi, t > t0,

xi�t0� xi0,

qij ≤ qij ≤ qij , i 1, . . . , n, j 1, . . . , n.

�2.12�

Figure 1 shows the architecture of the neural network �2.11�, whereM �mij�n×n D−D−1Q,C D−1c, and D �dij�n×n.


Definition 2.2. The point x∗ is said to be an equilibrium point of interval projection neuralnetwork �2.11�, if x∗ satisfies

0 PX(Dx∗ −D−1Qx∗ −D−1c

)−Dx∗. �2.13�

By Proposition 2.1 and Definition 2.2, we have the following theorem.

Theorem 2.3. The point x∗ is an equilibrium point of the interval projection neural network �2.11� ifand only if it is an optimal solution of the interval quadratic program �2.1�.

Definition 2.4. The equilibrium point x∗ of the neural network �2.11� is said to be globallyexponentially stable, if the trajectory x�t� of the neural network �2.11� with the initial valuex0 satisfies

‖x�t� − x∗‖ ≤ c0 exp(−β�t − t0�

), ∀t ≥ t0, �2.14�

where β > 0 is a constant independent of the initial value x0 and c0 > 0 is a constant dependenton the initial value x0. ‖ · ‖ denotes the 1-norm of Rn, that is, ‖x‖

∑ni1 |xi|.

Lemma 2.5 �see �17��. Let Ω ⊂ Rn be a closed convex set. Then,

�v − PΩ�v��T �PΩ�v� − u� ≥ 0, ∀u ∈ Ω, v ∈ Rn,

‖PΩ�u� − PΩ�v�‖ ≤ ‖u − v‖, ∀u, v ∈ Rn,�2.15�

where PΩ�u� is a project function on Ω, given by PΩ�u� arg miny∈Ω‖u − y‖.

3. Stability Analysis

In order to obtain the results in this paper, we make the following assumption for the neuralnetwork �2.11�:

H1: qii ≤ d2i , d2i −

did∗

< qii −n∑

j1,j / i

q∗ji < d2i , i 1, 2, . . . , n, �3.1�

where d∗ max1≤i≤n1/di, q∗ji max{|qji|, |qji|}.

Theorem 3.1. If the assumption H1 is satisfied, then there exists a unique equilibrium point for theneural network �2.11�.

Proof. Let T�x� D−1PX�Dx − D−1Qx − D−1c�, x ∈ Rn. By Definition 2.2, it is obvious thatthe neural network �2.11� has a unique equilibrium point if and only if T has a unique fixedpoint in Rn. In the following, by using fixed point theorem, we prove that T has a unique


fixed point in Rn. For any x, y ∈ Rn, by Lemma 2.5 and the assumption H1, we can obtainthat

∥∥T�x� − T

(y)∥∥

∥∥∥D−1PX

(Dx −D−1Qx −D−1c

)−D−1PX

(Dy −D−1Qy −D−1c

)∥∥∥

≤∥∥∥D−1

∥∥∥∥∥∥PX


)− PX

(Dy −D−1Qy −D−1c

)∥∥∥

≤∥∥∥D−1

∥∥∥∥∥∥(Dx −D−1Qx −D−1c

)−(Dy −D−1Qy −D−1c

)∥∥∥

∥∥∥D−1

∥∥∥∥∥∥(D −D−1Q

)(x − y

)∥∥∥

≤∥∥∥D−1

∥∥∥∥∥∥D −D−1Q

∥∥∥∥∥x − y

∥∥

max1≤i≤n

1di

·max1≤i≤n

⎛

⎝di −1diqii �

1di

n∑

j1,j / i

∣∣qji∣∣

⎞

⎠ ·∥∥x − y

∥∥

≤ d∗ max1≤i≤n

⎛

⎝di −1diqii �

1di

n∑

j1,j / i

q∗ji

⎞

⎠∥∥x − y

∥∥

max1≤i≤n

�i∥∥x − y

∥∥,

�3.2�

where �i d∗�di − �1/di�qii � �1/di�∑n

j1,j / i q∗ji�. By the assumptionH1, 0 < d

∗�di − �1/di�qii ��1/di�

∑nj1,j / i q

∗ji� < 1, i 1, 2, . . . , n. This implies that 0 < max1≤i≤n�i < 1. Equation �3.2�

shows that T is a contractive mapping, and hence T has a unique fixed point. This completesthe proof.

Proposition 3.2. If the assumption H1 holds, then for any x0 ∈ Rn, there exists a solution with theinitial value x�0� x0 for the neural network �2.11�.

Proof. Let F D�T − I�, where I is an identity mapping, then F�x� PX�Dx − D−1Qx −D−1c� −Dx. By �3.2�, we have

∥∥F�x� − F(y)∥∥

∥∥D�T − I��x� −D�T − I�(y)∥∥

≤ max1≤i≤n

di

(1 �max

1≤i≤n�i

)∥∥x − y∥∥, ∀x, y ∈ Rn.

�3.3�

Equation �3.3� means that the mapping F is globally Lipschitz. Hence, for any x0 ∈ Rn, thereexists a solution with the initial value x�0� x0 for the neural network �2.11�. This completesthe proof.

Proposition 3.2 shows the existence of the solution for the neural network �2.11�.


Theorem 3.3. If the assumption H1 is satisfied, then the equilibrium point of the neural network�2.11� is globally exponentially stable.

Proof. By Theorem 3.1, the neural network �2.11� has a unique equilibrium point. We denotethe equilibrium point of the neural network �2.11� by x∗.

Consider Lyapunov function V�t� ‖x�t� − x∗‖ ∑n

i1 |xi�t� − x∗i |. Calculate thederivative of V�t� along the solution x�t� of the neural network �2.11�. When t > t0, we have

dV�t�dt

n∑

i1

xi�t� − x∗i∣∣xi�t� − x∗i

∣∣d(xi�t� − x∗i

)

dt

n∑

i1

xi�t� − x∗i∣∣xi�t� − x∗i

∣∣

⎛

⎝PX

⎛

⎝dixi −1di

n∑

j1

qijxj −cidi

⎞

⎠ − dixi

⎞

⎠

n∑

i1

xi�t� − x∗i∣∣xi�t� − x∗i∣∣

⎛

⎝PX

⎛

⎝dixi −1di

n∑

j1

qijxj −cidi

⎞

⎠ − dix∗i � dix∗i − dixi

⎞

⎠

−n∑

i1

di∣∣xi�t� − x∗i

∣∣ �n∑

i1

xi�t� − x∗i∣∣xi�t� − x∗i∣∣

⎛

⎝PX

⎛

⎝dixi −1di

n∑

j1

qijxj −cidi

⎞

⎠ − dix∗i

⎞

⎠

≤(−min1≤i≤n

di

) n∑

i1

∣∣xi�t� − x∗i∣∣ �

n∑

i1

∣∣∣∣∣∣PX

⎛

⎝dixi −1di

n∑

j1

qijxj −cidi

⎞

⎠ − dix∗i

∣∣∣∣∣∣

(−min1≤i≤n

di

)‖x − x∗‖ �

∥∥∥PX(Dx −D−1Qx −D−1c

)−Dx∗

∥∥∥.

�3.4�

Noting Dx∗ PX�Dx∗ −D−1Qx∗ −D−1c�, by Lemma 2.5, we have

∥∥∥PX(Dx −D−1Qx −D−1c

)−Dx∗

∥∥∥

∥∥∥PX


)− PX

(Dx∗ −D−1Qx∗ −D−1c

)∥∥∥

≤∥∥∥(Dx −D−1Qx −D−1c

)−(Dx∗ −D−1Qx∗ −D−1c

)∥∥∥

≤∥∥∥(D −D−1Q

)�x − x∗�

∥∥∥

≤∥∥∥D −D−1Q

∥∥∥‖x − x∗‖.

�3.5�


Hence,

dV�t�dt

≤

⎛

⎝−min1≤i≤n

di �max1≤i≤n

⎛

⎝di −1di

∣∣qii

∣∣ �

1di

n∑

j1,j / i

∣∣qji

∣∣

⎞

⎠

⎞

⎠ · ‖x − x∗‖

≤

⎛

⎝− 1d∗

�max1≤i≤n

⎛

⎝di −1diqii �

1di

n∑

j1,j / i

q∗ji

⎞

⎠

⎞

⎠ · ‖x − x∗‖

max1≤i≤n

�′i‖x − x∗‖,

�3.6�

where �′i di − �1/di�qii � �1/di�∑n

j1,j / i q∗ji − �1/d∗�. By the assumption H1, �

′i < 0. Hence,

max1≤i≤n�′i < 0. Let �∗ min1≤i≤n|�′i|, then �∗ > 0. Equation �3.6� can be rewritten as dV�t�/dt ≤

−�∗‖x − x∗‖. It follows easily that ‖x�t� − x∗‖ ≤ ‖x0 − x∗‖ exp�−�∗�t − t0��, for all t > t0. Thisshows that the equilibrium point x∗ of the neural network �2.11� is globally exponentiallystable. This completes the proof.

4. Illustrative Examples

Example 4.1. Consider the interval quadratic program defined by D diag�2, 1�, g �2, 2�T ,h �3, 2�T , cT �−1,−1�, Q

(3 0.10.1 0.7

), Q

(3.1 0.20.2 0.8

), and Q∗ �q∗ji�

(3.1 0.20.2 0.8

).

The optimal solution of this quadratic program is �1, 2� under Q Q or Q Q. It iseasy to check that

3.0 < q11 < 3.1, q11 < d21 4,

0.7 < q22 < 0.8, q22 < d22 1,

d21 −d1d∗

2 < q11 − q∗21 3.0 − 0.2 < d21 4,

d22 −d2d∗

0 < q22 − q∗12 0.7 − 0.2 < d22 1.

�4.1�

The assumption H1 holds. By Theorems 3.1 and 3.3, the neural network �2.11� has a uniqueequilibrium point which is globally exponentially stable, and the unique equilibrium point�1, 2� is the optimal solution of this quadratic programming problem.

In the case of Q Q, Figure 2 reveals that the projection neural network �2.11�with random initial value �2.5,−0.5� has a unique equilibrium point �1, 2� which is globallyexponentially stable. In the case ofQ Q, Figure 3 reveals that the projection neural network�2.11� with random initial value �−2.5, 3� has the same unique equilibrium point �1, 2� whichis globally exponentially stable. These are in accordance with the conclusion of Theorems 3.1and 3.3.


0 5 10 15 20−0.5

0

0.5

1

1.5

2

2.5

t

x1x2

x(t)

Figure 2: Convergence of the state trajectory of the neural network with random initial value �2.5,−0.5�;Q Q in this example.

0 5 10 15 20−3

−2

−1

0

1

2

3

t

x1x2

x(t)

Figure 3: Convergence of the state trajectory of the neural network with random initial value �−2.5,3�;Q Q in this example.

Example 4.2. Consider the interval quadratic program defined by D diag�1, 2, 2�, g

�1, 2, 3�T , h �2, 3, 4�T , cT �1, 1, 1�, Q ( 0.8 0.2 0.3

0.2 3 0.10.3 0.1 3.5

), Q

( 0.9 0.3 0.40.3 3.1 0.20.4 0.2 3.6

), and Q∗ �q∗ji� ( 0.9 0.3 0.4

0.3 3.1 0.20.4 0.2 3.6

).


0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

t

x1x2x3

x(t)

Figure 4: Convergence of the state trajectory of the neural network with random initial value�−0.5,0.6,−0.8�; Q Q in this example.

The optimal solution of this quadratic program is �1, 1, 1.5� under Q Q or Q Q. Itis easy to check that

0.8 < q11 < 0.9, q11 < d21 1,

3.0 < q22 < 3.1, q22 < d22 4,

3.5 < q33 < 3.6, q33 < d23 4,

d21 −d1d∗

0 < q11 −(q∗21 � q

∗31

) 0.8 − �0.3 � 0.4� < d21 1,

d22 −d2d∗

2 < q22 −(q∗12 � q

∗32) 3.0 − �0.3 � 0.2� < d22 4,

d23 −d3d∗

0 < q33 −(q∗13 � q

∗23) 3.5 − �0.4 � 0.2� < d23 4.

�4.2�

The assumption H1 holds. By Theorems 3.1 and 3.3, the neural network �2.11� has a uniqueequilibrium point which is globally exponentially stable, and the unique equilibrium point�1, 1, 1.5� is the optimal solution of this quadratic programming problem.

In the case of Q Q, Figure 4 reveals that the projection neural network �2.11�with random initial value �−0.5, 0.6,−0.8� has a unique equilibrium point �1, 1, 1.5� which isglobally exponentially stable. In the case ofQ Q, Figure 5 reveals that the projection neuralnetwork �2.11� with random initial value �0.8,−0.6, 0.3� has the same unique equilibriumpoint �1, 1, 1.5� which is globally exponentially stable. These are in accordance with theconclusion of Theorems 3.1 and 3.3.


0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

t

x1x2x3

x(t)

Figure 5: Convergence of the state trajectory of the neural network with random initial value �0.8,−0.6,0.3�;Q Q in this example.

5. Conclusion

In this paper, we have developed a new projection neural network for solving intervalquadratic programs, the equilibrium point of the proposed neural network is equivalentto the solution of interval quadratic programs. A condition is derived which ensures theexistence, uniqueness, and global exponential stability of the equilibrium point. The resultsobtained are highly valuable in both theory and practice for solving interval quadraticprograms in engineering.

Acknowledgment

This paper was supported by the Hebei Province Education Foundation of China �2009157�.

References

�1� M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithm, JohnWiley & Sons, Hoboken, NJ, USA, 2nd edition, 1993.

�2� M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Transactions onCircuits and Systems, vol. 35, no. 5, pp. 554–562, 1988.

�3� Y. Xia, “A new neural network for solving linear and quadratic programming problems,” IEEETransactions on Neural Networks, vol. 7, no. 6, pp. 1544–1547, 1996.

�4� Y. Xia and J. Wang, “A general projection neural network for solving monotone variationalinequalities and related optimization problems,” IEEE Transactions on Neural Networks, vol. 15, no.2, pp. 318–328, 2004.

�5� Y. Xia, G. Feng, and J. Wang, “A recurrent neural network with exponential convergence for solvingconvex quadratic program and related linear piecewise equations,”Neural Networks, vol. 17, no. 7, pp.1003–1015, 2004.


�6� Y. Xia and G. Feng, “An improved neural network for convex quadratic optimization with applicationto real-time beamforming,” Neurocomputing, vol. 64, no. 1–4, pp. 359–374, 2005.

�7� Y. Yang and J. Cao, “Solving quadratic programming problems by delayed projection neuralnetwork,” IEEE Transactions on Neural Networks, vol. 17, no. 6, pp. 1630–1634, 2006.

�8� Y. Yang and J. Cao, “A delayed neural network method for solving convex optimization problems,”International Journal of Neural Systems, vol. 16, no. 4, pp. 295–303, 2006.

�9� Q. Tao, J. Cao, and D. Sun, “A simple and high performance neural network for quadraticprogramming problems,” Applied Mathematics and Computation, vol. 124, no. 2, pp. 251–260, 2001.

�10� X. Xue and W. Bian, “A project neural network for solving degenerate convex quadratic program,”Neurocomputing, vol. 70, no. 13–15, pp. 2449–2459, 2007.

�11� X. Xue and W. Bian, “A project neural network for solving degenerate quadratic minimax problemwith linear constraints,” Neurocomputing, vol. 72, no. 7–9, pp. 1826–1838, 2009.

�12� X. Hu, “Applications of the general projection neural network in solving extended linear-quadraticprogramming problems with linear constraints,” Neurocomputing, vol. 72, no. 4–6, pp. 1131–1137,2009.

�13� X. Hu and J. Wang, “Design of general projection neural networks for solving monotone linearvariational inequalities and linear and quadratic optimization problems,” IEEE Transactions onSystems, Man, and Cybernetics, Part B, vol. 37, no. 5, pp. 1414–1421, 2007.

�14� K. Ding and N.-J. Huang, “A new class of interval projection neural networks for solving intervalquadratic program,” Chaos, Solitons and Fractals, vol. 35, no. 4, pp. 718–725, 2008.

�15� Y. Yang and J. Cao, “A feedback neural network for solving convex constraint optimizationproblems,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 340–350, 2008.

�16� D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.

�17� C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free BoundaryProblems, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1984.

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