Information Sciences, vol. 180, pp. 697-711, 2010

**A neural network based on the generalized Fischer-Burmeister** **function for nonlinear complementarity problems**

Jein-Shan Chen ^{1}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan

Chun-Hsu Ko ^{2}

Department of Electrical Engineering I-Shou University

Kaohsiung 840, Taiwan

Shaohua Pan^{3}

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China

February 18, 2008

(revised on June 11, 2008, February 25, 2009, March 16, 2009, August 18, 2009)
**Abstract. In this paper, we consider a neural network model for solving the nonlinear**
complementarity problem (NCP). The neural network is derived from an equivalent un-
constrained minimization reformulation of the NCP, which is based on the generalized
*Fischer-Burmeister function ϕ*_{p}*(a, b) =∥(a, b)∥**p**− (a + b). We establish the existence and*
the convergence of the trajectory of the neural network, and study its Lyapunov stability,
*asymptotic stability as well as exponential stability. It was found that a larger p leads*
to a better convergence rate of the trajectory. Numerical simulations verify the obtained
theoretical results.

**Key words: The NCP, neural network, exponentially convergent, generalized Fischer-**
Burmeister function.

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Oﬃce.

The author’s work is partially supported by National Science Council of Taiwan. E-mail:

jschen@math.ntnu.edu.tw

2E-mail: chko@isu.edu.tw

3E-mail:shhpan@scut.edu.cn

**1** **Introduction**

For decades, the nonlinear complementarity problem (NCP) has attracted a lot of atten-
tion because of its wide applications in operations research, economics, and engineering
*[9, 12]. Given a mapping F : IR*^{n}*→ IR*^{n}*, the NCP is to ﬁnd a point x∈ IR** ^{n}* such that

*x≥ 0, F (x) ≥ 0, ⟨x, F (x)⟩ = 0,* (1)

where *⟨·, ·⟩ is the Euclidean inner product. Throughout this paper, we assume that F is*
*continuously diﬀerentiable, and let F = (F*1*, . . . , F**n*)^{T}*with F**i* : IR^{n}*→ IR for i = 1, . . . , n.*

There have been many methods proposed for solving the NCP [9, 12]. One of the most popular approaches is to reformulate the NCP as an unconstrained minimization problem via a merit function; see [14, 19, 20, 21]. A merit function is a function whose global minimizers coincide with the solutions of the NCP. The class of NCP-functions deﬁned below is used to construct a merit function.

**Definition 1.1 A function ϕ : IR**× IR → IR is called an NCP-function if it satisﬁes

*ϕ(a, b) = 0* *⇐⇒ a ≥ 0, b ≥ 0, ab = 0.* (2)

A popular NCP-function is the Fischer-Burmeister (FB) function [10, 11], which is deﬁned as

*ϕ*_{FB}*(a, b) =√*

*a*^{2}*+ b*^{2}*− (a + b).* (3)

*The FB merit function ψ*_{FB}: IR*× IR → IR*+ *can be obtained by taking the square of ϕ*_{FB},
i.e.,

*ψ*_{FB}*(a, b) :=* 1

2*|ϕ*FB*(a, b)|*^{2}*.* (4)

*In [1, 3, 4], we studied a family of NCP-functions that subsumes the FB function ϕ*_{FB}
*as a special case. More speciﬁcally, we deﬁne ϕ** _{p}* : IR

*× IR → IR by*

*ϕ*_{p}*(a, b) :=∥(a, b)∥**p**− (a + b),* (5)
*where p is any ﬁxed real number from (1, +∞) and ∥(a, b)∥**p* *denotes the p-norm of (a, b),*
i.e., *∥(a, b)∥**p* = √^{p}

*|a|** ^{p}*+

*|b|*

^{p}*. In other words, in the function ϕ*

*, we replace the 2-norm*

_{p}*of (a, b) in the FB function ϕ*

_{FB}

*by a more general p-norm of (a, b). The function ϕ*

*is still an NCP-function, as noted in Tseng’s paper [30]. There has been no further study*

_{p}*on this NCP-function, even for p = 3, until recently [1, 3, 4]. Similar to ϕ*

_{FB}, the square

*of ϕ*

_{p}*induces a nonnegative NCP-function ψ*

*: IR*

_{p}*× IR → IR*+:

*ψ*_{p}*(a, b) :=* 1

2*|ϕ**p**(a, b)|*^{2}*.* (6)

*The function ψ** _{p}* is continuously diﬀerentiable and it has some favorable properties; see
[1, 3, 4]. Moreover, if we deﬁne the function Ψ

*: IR*

_{p}

^{n}*→ IR*+ by

Ψ_{p}*(x) :=*

∑*n*
*i=1*

*ψ*_{p}*(x*_{i}*, F*_{i}*(x)) =* 1

2*∥Φ**p**(x)∥*^{2} (7)

where Φ* _{p}* : IR

^{n}*→ IR*

*is a mapping given as*

^{n}Φ_{p}*(x) =*

*ϕ*_{p}*(x*_{1}*, F*_{1}*(x))*
...
*ϕ*_{p}*(x*_{n}*, F*_{n}*(x))*

* ,* (8)

then the NCP can be reformulated into the following smooth minimization problem:

*x*min*∈IR** ^{n}*Ψ

_{p}*(x).*(9)

Thus, Ψ_{p}*(x) in (7) is a smooth merit function for the NCP.*

Eﬀective gradient-type methods can be applied to the unconstrained smooth min- imization problem (9). However, in many scientiﬁc and engineering applications, it is desirable to have a real-time solution of the NCP. Thus, traditional unconstrained optimization algorithms may not be suitable for real-time implementation because the computing time required for a solution greatly depends on the dimension and structure of the problem. One promising way to overcome this problem is to apply neural networks.

Neural networks for optimization were ﬁrst introduced in the 1980s by Hopﬁeld and Tank [16, 29]. Since then, neural networks have been applied to various optimization problems, including linear programming, nonlinear programming, variational inequali- ties, and linear and nonlinear complementarity problems; see [6, 8, 7, 15, 17, 18, 23, 25, 32, 33, 34, 35, 36]. There have been many studies on neural-network approaches to real- world problems in some other ﬁelds, such as [27, 28, 37]. The main idea of the neural network approach for optimization is to construct a nonnegative energy function and establish a dynamic system that represents an artiﬁcial neural network. The dynamic system is usually in the form of ﬁrst order ordinary diﬀerential equations. Furthermore, it is expected that the dynamic system will approach its static state (or an equilibrium point), which corresponds to the solution for the underlying optimization problem, start- ing from an initial point. In addition, neural networks for solving optimization problems are hardware-implementable; that is, the neural networks can be implemented using in- tegrated circuits.

In this paper, we focus on a neural network approach to the NCP. We utilize Ψ_{p}*(x)*
as the traditional energy function. As mentioned above, the NCP is equivalent to the

unconstrained smooth minimization problem (9). Therefore, it is natural to adopt the following steepest descent-based neural network model for NCP:

*dx(t)*

*dt* =*−ρ∇Ψ**p**(x(t)),* *x(0) = x*0*,* (10)

*where ρ > 0 is a scaling factor. Most neural networks in the existing literature are*
projection-type ones based on other kinds of NCP-functions, such as natural residual
function (e.g. [18, 34]) and the regularized gap function (e.g. [6]). Recently, neural
networks based on the FB function have been designed for linear and quadratic pro-
gramming, and for nonlinear complementarity problems [8, 25]. Our model is based on
the generalized FB function, which is a generalization of the functions used in [8, 25].

We show that the neural network (10) is Lyapunov stable, asymptotically stable, and
*exponentially stable. We observed in [2] that p has a great inﬂuence on the numerical*
*performance of certain descent-type methods; a larger p yields a better convergence rate,*
*whereas a smaller p often gives a better global convergence. Thus, whether such phe-*
nomena occur in our neural network model is also investigated.

Throughout this paper, IR^{n}*denotes the space of n-dimensional real column vectors*
and ^{T}*denotes the transpose. For any diﬀerentiable function f : IR*^{n}*→ IR, ∇f(x) means*
*the gradient of f at x. For any diﬀerentiable mapping F = (F*_{1}*, . . . , F** _{m}*)

*: IR*

^{T}

^{n}*→ IR*

*,*

^{m}*∇F (x) = [∇F*1*(x)* *· · · ∇F**m**(x)]∈ IR*^{n}^{×m}*denotes the transposed Jacobian of F at x. The*
*p-norm of x is denoted by∥x∥**p**and the Euclidean norm of x is denoted by∥x∥. Besides, e**i*

*is the n-dimensional vector whose i-th component is 1 and 0 elsewhere. Unless otherwise*
*stated, we assume that p in the sequel is any ﬁxed real number in (1, +∞) if not speciﬁed.*

**2** **Preliminaries**

*In this section, we review some properties of ϕ*_{p}*and ψ** _{p}*, as well as materials of ordinary
diﬀerential equations that will play an important role in the subsequent analysis. We
start with some concepts for a nonlinear mapping.

**Definition 2.1 Let F = (F**_{1}*, . . . , F** _{n}*)

*: IR*

^{T}

^{n}*→ IR*

^{n}*. Then, the mapping F is said to be*

**(a) monotone if***⟨x − y, F (x) − F (y)⟩ ≥ 0 for all x, y ∈ IR*

^{n}*.*

**(b) strongly monotone with modulus µ > 0 if***⟨x − y, F (x) − F (y)⟩ ≥ µ∥x − y∥*^{2} *for all*
*x, y* *∈ IR*^{n}*.*

* (c) an P*0

*-function if max*

1*≤i≤n*
*xi̸=yi*

*(x**i**− y**i**)(F**i**(x)− F**i**(y))≥ 0 for all x, y ∈ IR*^{n}*and x̸= y.*

**(d) a uniform P -function with modulus κ > 0 if max**

1≤i≤n*(x*_{i}*−y**i**)(F*_{i}*(x)−F**i**(y))≥ κ∥x−y∥*^{2}*,*
*for all x, y∈ IR*^{n}*.*

**(e) Lipschitz continuous if there exists a constant L > 0 such that***∥F (x) − F (y)∥ ≤*
*L∥x − y∥ for all x, y ∈ IR*^{n}*.*

From Deﬁnition 2.1, the following one-sided implications can be obtained:

*F is strongly monotone =⇒ F is a uniform P -function =⇒ F is an P*0 function;

*∇F is positive semideﬁnite =⇒ F is monotone =⇒ F is an P*0 function.

*Nevertheless, we point out that F being a uniform P -function does not necessarily imply*
*that F is monotone. The following two lemmas summarize some favorable properties of*
*ϕ*_{p}*and ψ** _{p}*, respectively. Since their proofs can be found in [2, 3, 4], we here omit them.

**Lemma 2.1 Let ϕ*** _{p}* : IR

*× IR → IR be given by (5). Then, the following properties hold.*

**(a) ϕ**_{p}*is a positive homogeneous and sub-additive NCP-function.*

**(b) ϕ**_{p}*is Lipschitz continuous with L =√*

2 + 2^{(1/p}^{−1/2)}*for 1 < p < 2, and L =* *√*
2 + 1
*for p≥ 2.*

**(c) ϕ**_{p}*is strongly semismooth.*

**(d) If***{(a*^{k}*, b** ^{k}*)

*} ⊆ IR × IR with a*

^{k}*→ −∞, or b*

^{k}*→ −∞, or a*

^{k}*→ ∞, b*

^{k}*→ ∞, then*

*|ϕ**p**(a*^{k}*, b** ^{k}*)

*| → ∞ when k → ∞.*

**(e) Given a point (a, b)**∈ IR×IR, every element in the generalized gradient ∂ϕ*p**(a, b) has*
*the representation (ξ− 1, ζ − 1) with*

*ξ =* *sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}*p*^{−1}

*and ζ =* *sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}*p*^{−1}

*for (a, b)̸= (0, 0), where sgn(·) represents the sign function; otherwise, ξ and ζ are*
*real numbers that satisfy* *|ξ|*^{p}^{−1}* ^{p}* +

*|ζ|*

^{p}

^{−1}

^{p}*≤ 1.*

**Lemma 2.2 Let ϕ**_{p}*and ψ*_{p}*be deﬁned as in (5) and (6), respectively. Then,*
**(a) ψ**_{p}*(a, b)≥ 0 for all a, b ∈ IR and ψ**p* *is an NCP-function, i.e., it satisﬁes (2).*

**(b) ψ**_{p}*is continuously diﬀerentiable everywhere. Moreover,* *∇**a**ψ*_{p}*(a, b) =∇**b**ψ*_{p}*(a, b) = 0*
*if (a, b) = (0, 0); otherwise,*

*∇**a**ψ*_{p}*(a, b) =*

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}*p*^{−1}

*− 1*
)

*ϕ*_{p}*(a, b),*

*∇**b**ψ*_{p}*(a, b) =*

(*sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}*p*^{−1}

*− 1*
)

*ϕ*_{p}*(a, b).* (11)

**(c)** *∇**a**ψ*_{p}*(a, b)· ∇**b**ψ*_{p}*(a, b)* *≥ 0 for all a, b ∈ IR. The inequality becomes an equality if*
*and only if ϕ*_{p}*(a, b) = 0.*

**(d)** *∇**a**ψ*_{p}*(a, b) = 0* *⇐⇒ ∇**b**ψ*_{p}*(a, b) = 0* *⇐⇒ ϕ**p**(a, b) = 0* *⇐⇒ ψ**p**(a, b) = 0.*

**(e) The gradient of ψ**_{p}*is Lipschitz continuous for p* *≥ 2, i.e., there exists L > 0 such*
*that*

*∥∇ψ**p**(a, b)− ∇ψ**p**(c, d)∥ ≤ L∥(a, b) − (c, d)∥ for all (a, b), (c, d) ∈ IR*^{2} *and p≥ 2.*

**(f ) For all a, b**∈ IR, we have (2 − 2* ^{1/p}*) min

*{a, b} ≤ |ϕ*

*p*

*(a, b)| ≤ (2 + 2*

*) min*

^{1/p}*{a, b}.*

Next, we recall some materials about ﬁrst order diﬀerential equations (ODE):

*˙x(t) = H(x(t)),* *x(t*_{0}*) = x*_{0} *∈ IR** ^{n}* (12)

*where H : IR*

^{n}*→ IR*

*is a mapping. We also introduce three kinds of stability that will be discussed later. These materials can be found in ODE textbooks; see [26].*

^{n}**Definition 2.2 A point x**^{∗}*= x(t*^{∗}*) is called an equilibrium point or a steady state of the*
*dynamic system (12) if H(x*^{∗}*) = 0. If there is a neighborhood Ω*^{∗}*⊆ IR*^{n}*of x*^{∗}*such that*
*H(x*^{∗}*) = 0 and H(x)̸= 0 ∀x ∈ Ω*^{∗}*\{x*^{∗}*}, then x*^{∗}*is called an isolated equilibrium point.*

**Lemma 2.3 Assume that H : IR**^{n}*→ IR*^{n}*is a continuous mapping. Then, for any t*_{0} *≥ 0*
*and x*_{0} *∈ IR*^{n}*, there exists a local solution x(t) for (12) with t∈ [t*0*, τ ) for some τ > t*_{0}*.*
*If, in addition, H is locally Lipschitz continuous at x*0*, then the solution is unique; if H*
*is Lipschitz continuous in IR*^{n}*, then τ can be extended to* *∞.*

*If a local solution deﬁned on [t*_{0}*, τ ) cannot be extended to a local solution on a larger*
*interval [t*_{0}*, τ*_{1}*), τ*_{1} *> τ , then it is called a maximal solution, and the interval [t*_{0}*, τ ) is the*
maximal interval of existence. Clearly, any local solution has an extension to a maximal
*one. We denote [t*_{0}*, τ (x*_{0}*)) by the maximal interval of existence associated with x*_{0}.
**Lemma 2.4 Assume that H : IR**^{n}*→ IR*^{n}*is continuous. If x(t) with t* *∈ [t*0*, τ (x*_{0}*)) is a*
*maximal solution and τ (x*0*) <∞, then lim*

*t**↑τ(x*^{0})*∥x(t)∥ = ∞.*

**Definition 2.3 (Stability in the sense of Lyapunov) Let x(t) be a solution for (12). An***isolated equilibrium point x*^{∗}*is Lyapunov stable if for any x*_{0} *= x(t*_{0}*) and any ε > 0,*
*there exists a δ > 0 such that* *∥x(t) − x*^{∗}*∥ < ε for all t ≥ t*0 *and* *∥x(t*0)*− x*^{∗}*∥ < δ.*

**Definition 2.4 (Asymptotic stability) An isolated equilibrium point x**^{∗}*is said to be*
*asymptotically stable if in addition to being Lyapunov stable, it has the property that*
*x(t)→ x*^{∗}*as t→ ∞ for all ∥x(t*0)*− x*^{∗}*∥ < δ.*

**Definition 2.5 (Lyapunov function) Let Ω***⊆ IR*^{n}*be an open neighborhood of ¯x. A*
*continuously diﬀerentiable function W : IR*^{n}*→ IR is said to be a Lyapunov function at*
*the state ¯x over the set Ω for equation (12) if*

*W (¯x) = 0,* *W (x) > 0,* *∀x ∈ Ω\{¯x}.*

*dW (x(t))*

*dt* =*∇W (x(t))*^{T}*H(x(t))≤ 0, ∀x ∈ Ω.* (13)

**Lemma 2.5 (a) An isolated equilibrium point x**^{∗}*is Lyapunov stable if there exists a*
*Lyapunov function over some neighborhood Ω*^{∗}*of x*^{∗}*.*

**(b) An isolated equilibrium point x**^{∗}*is asymptotically stable if there is a Lyapunov func-*
*tion over some neighborhood Ω*^{∗}*of x*^{∗}*such that* *dW (x(t))*

*dt* *< 0 for all x∈ Ω*^{∗}*\{x*^{∗}*}.*

**Definition 2.6 (Exponential stability) An isolated equilibrium point x**^{∗}*is exponentially*
*stable if there exists a δ > 0 such that arbitrary point x(t) of (10) with the initial condition*
*x(t*_{0}*) = x*_{0} *and* *∥x(t*0)*− x*^{∗}*∥ < δ is well-deﬁned on [0, +∞) and satisﬁes*

*∥x(t) − x*^{∗}*∥*2 *≤ ce*^{−ωt}*∥x(t*0)*− x*^{∗}*∥ ∀t ≥ t*0*,*
*where c > 0 and ω > 0 are constants independent of the initial point.*

**3** **Neural network model**

We now discuss properties of the neural network model introduced in (10). First, from Lemma 2.2(a), we obtain the following result.

**Proposition 3.1 Let Ψ*** _{p}* : IR

^{n}*→ IR*+

*be deﬁned as in (7). Then, Ψ*

_{p}*(x)*

*≥ 0 for all*

*x∈ IR*

^{n}*and Ψ*

_{p}*(x) = 0 if and only if x solves the NCP.*

**Proposition 3.2 Let Ψ*** _{p}* : IR

^{n}*→ IR*+

*be given by (7). Then, the following results hold.*

**(a) The function Ψ**_{p}*is continuously diﬀerentiable everywhere with*

*∇Ψ**p**(x) = V** ^{T}*Φ

_{p}*(x)*

*for any V*

*∈ ∂Φ*

*p*

*(x)*(14)

*or*

*∇Ψ**p**(x) =* *∇**a**ψ*_{p}*(x, F (x)) +∇F (x)∇**b**ψ*_{p}*(x, F (x))* (15)
*with*

*∇**a**ψ*_{p}*(x, F (x)) := [∇**a**ψ*_{p}*(x*_{1}*, F*_{1}*(x)), . . . ,∇**a**ψ*_{p}*(x*_{n}*, F*_{n}*(x))]*^{T}*,*

*∇**b**ψ**p**(x, F (x)) := [∇**b**ψ**p**(x*1*, F*1*(x)), . . . ,∇**b**ψ**p**(x**n**, F**n**(x))]*^{T}*.*

**(b) If F is an P**_{0}*-function, then every stationary point of (9) is a global minimizer of*
Ψ_{p}*(x), and it consequently solves the NCP.*

**(c) If F is a uniform P -function, then the level sets***L(Ψ**p**, γ) :={x ∈ IR*^{n}*| Ψ**p**(x)≤ γ}*

*are bounded for all γ* *∈ IR.*

**(d) Ψ***p**(x(t)) is nonincreasing with respect to t.*

**Proof. The ﬁrst equality in (a) follows from Lemma 2.2 (c) and [5, Theorem 2.6.6].**

The second one follows from the chain rule. Part (b) is the result of [3, Proposition 3.4],
and part (c) is the result of [4, Proposition 3.5]. It remains to show part (d). By the
deﬁnition of Ψ*p**(x) and (10), it is not diﬃcult to compute*

*dΨ**p**(x(t))*

*dt* =*∇Ψ**p**(x(t))*^{T}*dx(t)*

*dt* = *∇Ψ**p**(x(t))** ^{T}* (

*−ρ∇Ψ*

*p*

*(x(t)))*

= *−ρ∥∇Ψ**p**(x(t))∥*^{2} *≤ 0.* (16)
Therefore, Ψ_{p}*(x(t)) is a monotonically decreasing function with respect to t.* *2*

Proposition 3.2(a) provides two ways to compute *∇Ψ**p**(x), which is needed in the*
network (10). One is to use formula (14), for which we give an algorithm (see Algorithm
*3.1 below), to evaluate an element V* *∈ ∂Φ**p**(x). The other is to adopt formula (15).*

**Algorithm 3.1 (The procedure to evaluate an element V***∈ ∂Φ**p**(x))*

*(S.0) Let x∈ IR*^{n}*be given, and let V*_{i}*denote the i-th row of a matrix V* *∈ IR*^{n}* ^{×n}*.

*(S.1) Set I(x) :={i ∈ {1, 2, . . . , n}| x*

*i*

*= F*

_{i}*(x) = 0}.*

*(S.2) Set z* *∈ IR*^{n}*such that z**i* *= 0 for i /∈ I(x), and z**i* *= 1 for i∈ I(x).*

*(S.3) For i∈ I(x), let u**i* =

[*|z**i**|*^{p}^{−1}* ^{p}* +

*|∇F*

*i*

*(x)*

^{T}*z|*

^{p}

^{−1}*]*

^{p}

^{p}

^{−1}*, and*

_{p}*V** _{i}* =
(

*z*

_{i}*u*_{i}*− 1*
)

*e*^{T}* _{i}* +

(*∇F**i**(x)*^{T}*z*
*u*_{i}*− 1*

)

*∇F**i**(x)*^{T}*.*

*(S.4) For i /∈ I(x), set*
*V**i* =

(*sgn(x** _{i}*)

*· |x*

*i*

*|*

^{p}

^{−1}*∥(x**i**, F*_{i}*(x))∥*^{p−1}*p*

*− 1*
)

*e*^{T}* _{i}* +

(*sgn(F*_{i}*(x))· |F**i**(x)|*^{p}^{−1}

*∥(x**i**, F*_{i}*(x))∥*^{p−1}*p*

*− 1*
)

*∇F**i**(x)*^{T}*.*

The above procedure is a traditional way of obtaining *∇Ψ**p**(x(t)).* For example,
the neural network in [25] uses (14) and a similar algorithm to evaluate an element of
*V* *∈ ∂Φ*FB*(x). We propose a simpler way of obtaining* *∇Ψ**p**(x(t)) which is to compute*

*∇Ψ**p**(x(t)) by using formula (15) rather than formula (14). Formula (15) also provides an*
indication on how the neural network (10) can be implemented on hardware; see Figure
1 below.

Figure 1: A simpliﬁed block diagram for the neural network (10).

To close this section, we claim that Ψ*p* provides a global error bound for the solution
*of the NCP. This result is important and will be used to analyze the inﬂuence of p on the*
*convergence rate of the trajectory x(t) of the neural network (10) in the next section.*

**Proposition 3.3 Suppose F is a uniform P -function with modulus κ > 0 and Lipschitz***continuous with constant L > 0. Then, the NCP has a unique solution x*^{∗}*, and*

*∥x − x*^{∗}*∥*^{2} *≤* *4L*^{2}

*κ*^{2}(2*− 2** ^{1/p}*)

^{2}Ψ

_{p}*(x)*

*∀x ∈ IR*

^{n}*.*

* Proof. Since F is a uniform P -function, by Proposition 3.2(c), there exists a global*
minimizer of Ψ

_{p}*(x) which says the NCP has a solution. Assume that the NCP has two*

*diﬀerent solutions x*

^{∗}*and y*

*, then by Deﬁnition 2.1(d) we have*

^{∗}*κ∥x*^{∗}*− y*^{∗}*∥*^{2} *≤ max*

1*≤i≤m**(x*^{∗}_{i}*− y*^{∗}*i**)(F*_{i}*(x** ^{∗}*)

*− F*

*i*

*(y*

*))*

^{∗}= max

1*≤i≤m*

{*− x*^{∗}*i**F*_{i}*(y** ^{∗}*)

*− y*

*i*

^{∗}*F*

_{i}*(x*

*) }*

^{∗}*≤ 0*

*where the equality is due to the fact that x*^{∗}_{i}*F*_{i}*(x*^{∗}*) = y*^{∗}_{i}*F*_{i}*(y*^{∗}*) = 0 for i = 1, 2, . . . , n*
*(note that x*^{∗}*and y** ^{∗}* are the solutions to the NCP), and the last inequality holds since

*x*

^{∗}*, y*

^{∗}*≥ 0 and F (x*

^{∗}*), F (y*

*)*

^{∗}*≥ 0. This leads to a contradiction. Hence, the NCP has a*unique solution.

*For any x* *∈ IR*^{n}*, let r(x) := (r*_{1}*(x), . . . , r*_{n}*(x))*^{T}*with r*_{i}*(x) = min{x**i**, F*_{i}*(x)} for*
*i = 1, . . . , n. Since F is Lipschitz continuous with constant L > 0, by [21, Lemma 7.4]*

we have

*(x*_{i}*− x*^{∗}_{i}*)(F*_{i}*(x)− F**i**(x** ^{∗}*))

*≤ 2L|r*

*i*

*(x)|∥x − x*

^{∗}*∥,*

*for all x* *∈ IR*^{n}*and i = 1, 2, . . . , n. On the other hand, since F is a uniform P -function*
*with modulus κ > 0, from Deﬁnition 2.1(d) it follows that*

*κ∥x − x*^{∗}*∥*^{2} *≤ max*

1*≤i≤n**(x**i**− x*^{∗}*i**)(F**i**(x)− F**i**(x** ^{∗}*))

*for any x∈ IR*

*. Combining the last two equations yields*

^{n}*∥x − x*^{∗}*∥ ≤ (2L/κ) max*

1*≤i≤n**|r**i**(x)| ∀x ∈ IR*^{n}*.*
This together with Lemma 2.2(f) implies

*∥x − x*^{∗}*∥ ≤* *2L*

*κ(2− 2** ^{1/p}*) max

1*≤i≤n**|ϕ**p**(x*_{i}*, F*_{i}*(x))| ≤* *2L*

*κ(2− 2** ^{1/p}*)

*∥Φ*

*p*

*(x)∥.*

Consequently, we obtain the desired result. *2*

**4** **Convergence and stability of the trajectory**

This section focuses on issues of convergence and stability of the neural network (10).

We analyze the behavior of the solution trajectory of (10) including the existence and convergence, and establish three kinds of stability for an isolated equilibrium point. We ﬁrst state the relationships between an equilibrium point of (10) and a solution to the NCP.

**Proposition 4.1 (a) Every solution to the NCP is an equilibrium point of (10).**

**(b) If F is an P**_{0}*-function, then every equilibrium point of (10) is a solution to the NCP.*

* Proof. (a) Suppose that x is a solution to the NCP. Then, from Proposition 3.1, it is*
clear that Φ

_{p}*(x) = 0. Using Lemma 2.2 (d) and (15), we then have*

*∇Ψ*

*p*

*(x) = 0. This,*

*by Deﬁnition 2.2, shows that x is an equilibrium point of (10).*

(b) This is a direct consequence of Proposition 3.2 (b). *2*

The following proposition establishes the existence of the solution trajectory of (10).

**Proposition 4.2 For any ﬁxed p**≥ 2, the following hold.

**(a) For any initial state x**_{0} *= x(t*_{0}*), there exists exactly one maximal solution x(t) with*
*t∈ [t*0*, τ (x*_{0}*)) for the neural network (10).*

**(b) If the level set***L(x*0) = *{x ∈ IR*^{n}*| Ψ**p**(x)* *≤ Ψ**p**(x*_{0})*} is bounded or F is Lipschitz*
*continuous, then τ (x*_{0}) = +*∞.*

**Proof. (a) Since F is continuously diﬀerentiable,***∇F (x) is continuous, and therefore,*

*∇F (x) is bounded on a local compact neighborhood of x. On the other hand, ∇**a**ψ** _{p}* and

*∇**b**ψ** _{p}* are Lipschitz continuous by Lemma 2.2 (e). These two facts together with formula
(15) show that

*∇Ψ*

*p*

*(x) is locally Lipschitz continuous. Thus, applying Lemma 2.3 leads*to the desired result.

(b) We proceed the arguments by the two cases as shown below.

Case (i): The level set*L(x*0) is bounded. We prove the result by contradiction. Suppose
*τ (x*_{0}*) <∞. Then, by Lemma 2.4, lim*

*t**↑τ(x*0)*∥x(t)∥ = ∞. Let L*^{c}*(x*_{0}) := IR^{n}*\L(x*0) and
*τ*_{0} := inf*{s ≥ 0 | s < τ(x*0*), x(s)∈ L*^{c}*(x*_{0})*} < ∞.*

*We know that x(τ*_{0}) lies on the boundary of*L(x*0) and*L*^{c}*(x*_{0}). Moreover,*L(x*0) is compact
since it is bounded by assumption and it is also closed because of the continuity of Ψ_{p}*(x).*

*Therefore, we have x(τ*_{0})*∈ L(x*0*) and τ*_{0} *< τ (x*_{0}), implying that

Ψ_{p}*(x(s)) > Ψ*_{p}*(x*_{0}*) > Ψ*_{p}*(x(τ*_{0})) *for some s∈ (τ*0*, τ (x*_{0}*)).* (17)
However, Proposition 3.2(d) says that Ψ*p**(x(·)) is nonincreasing on [t*0*, τ (x*0)), which
contradicts (17). This completes the proof of Case (i).

*Case (ii): F is Lipschitz continuous. From the proof of part (a), we know that* *∇Ψ**p**(x)*
*is Lipschitz continuous. Thus, by Lemma 2.3, we have τ (x*_{0}) =*∞.* *2*

Next, we investigate the convergence of the solution trajectory of (10).

* Theorem 4.1 (a) Let x(t) with t∈ [t*0

*, τ (x*

_{0}

*)) be the unique maximal solution to (10).*

*If τ (x*_{0}) =*∞ and {x(t)} is bounded, then lim*

*t**→∞**∇Ψ**p**(x(t)) = 0.*

* (b) If F is strongly monotone or a uniform P -function, thenL(x*0

*) is bounded and every*

*accumulation point of the trajectory x(t) is a solution to the NCP.*

**Proof. With Proposition 3.2 (b) and (d) and Proposition 4.2, the arguments are exactly**
the same as those for [25, Corollary 4.3]. Thus, we omit them. *2*

*From Proposition 4.1 (a), every solution x** ^{∗}* to the NCP is an equilibrium point of the

*neural network (10). If, in addition, x*

*is an isolated equilibrium point of (10), then we*

^{∗}*can show that x*

*is not only Lyapunov stable but also asymptotically stable.*

^{∗}**Theorem 4.2 Let x**^{∗}*be an isolated equilibrium point of the neural network (10). Then,*
*x*^{∗}*is Lyapunov stable for (10), and furthermore, it is asymptotically stable.*

**Proof. Since x*** ^{∗}* is a solution to the NCP, Ψ

*p*

*(x*

^{∗}*) = 0. In addition, since x*

*is an isolated equilibrium point of (10), there exists a neighborhood Ω*

^{∗}

^{∗}*⊆ IR*

^{n}*of x*

*such that*

^{∗}*∇Ψ**p**(x*^{∗}*) = 0, and* *∇Ψ**p**(x)̸= 0 ∀x ∈ Ω*^{∗}*\{x*^{∗}*}.*

Next, we argue that Ψ_{p}*(x) is indeed a Lyapunov function at x** ^{∗}* over the set Ω

*for (10) by showing that the conditions in (13) are satisﬁed. First, notice that Ψ*

^{∗}

_{p}*(x)≥ 0. Suppose*that there is an ¯

*x*

*∈ Ω*

^{∗}*\{x*

^{∗}*} such that Ψ*

*p*(¯

*x) = 0. Then, by formula (15) and Lemma*2.2(d), we have

*∇Ψ(¯x) = 0, i.e., ¯x is also an equilibrium point of (10), which clearly*

*contradicts the assumption that x*

*is an isolated equilibrium point in Ω*

^{∗}*. Thus, we prove that Ψ*

^{∗}

_{p}*(x) > 0 for any x*

*∈ Ω*

^{∗}*\{x*

^{∗}*}. This together with (16) shows that the*conditions in (13) are satisﬁed, and hence Ψ

_{p}*(x) is a Lyapunov function at x*

*over the set Ω*

^{∗}

^{∗}*for (10). Therefore, x*

*is Lyapunov stable by Lemma 2.5(a).*

^{∗}*Now, we show that x*^{∗}*is asymptotically stable. Since x** ^{∗}* is isolated, from (16) we have

*dΨ*

_{p}*(x(t))*

*dt* *< 0,* *∀ x(t) ∈ Ω*^{∗}*\{x*^{∗}*}.*

*This, by Lemma 2.5 (b), implies that x** ^{∗}* is asymptotically stable.

*2*

Furthermore, using the same arguments we can prove that the neural network (10) is
*also exponentially stable if x*^{∗}*is a regular solution to the NCP. Recall that x** ^{∗}* is a regular

*solution to the NCP if every element V*

*∈ ∂Φ*

*p*

*(x*

*) is nonsingular.*

^{∗}**Theorem 4.3 If x**^{∗}*is a regular solution of the NCP, then it is exponentially stable.*

**Remark 4.1 (a) Using arguments similar to those used in Proposition 3.2 of [13], we***can prove that x*^{∗}*is regular if* *∇F**αα* *is nonsingular and the Schur complement of*

*∇F**αα* *in* (

*∇F**αα**(x** ^{∗}*)

*∇F*

*αβ*

*(x*

*)*

^{∗}*∇F**βα**(x** ^{∗}*)

*∇F*

*ββ*

*(x*

*) )*

^{∗}*is an P -matrix, where α :=* *{i | x*^{∗}*i* *> 0} and β := {i | x*^{∗}*i* *= F**i**(x** ^{∗}*) = 0

*}. Clearly,*

*if*

*∇F is positive deﬁnite, then the conditions hold true.*

**(b) From Deﬁnition 2.6, if an isolated equilibrium point x**^{∗}*is exponentially stable, then*
*there exists a δ > 0 such that x(t) with x*_{0} *= (t*_{0}*), and* *∥x(t*0)*− x*^{∗}*∥ < δ satisﬁes*

*∥x(t) − x*^{∗}*∥ ≤ ce*^{−ωt}*∥x(t*0)*− x*^{∗}*∥ ∀t ≥ t*0*,*
*which together with Proposition 3.3 implies that*

*∥x(t) − x*^{∗}*∥ ≤* *2cL*
*κ(2− 2** ^{1/p}*)

√

Ψ_{p}*(x*_{0}*)e*^{−ωt}*∀t ≥ t*0*.* (18)
*Since the strong monotonicity of F implies that F is a uniform P -function and*
*that* *∇F is positive deﬁnite, from (18) we obtain that the neural network (10) can*
*yield a trajectory with an exponential convergence rate under the condition that F*
*is strongly monotone and Lipschitz continuous.*

**(c) We observe from (18) that, when p increases, the coeﬃcient of e**^{−ωt}*in the right hand*
*side term becomes smaller, which in turn implies that a larger p yields a better*
*convergence rate. This agrees with the result obtained by [2] for a descent-type*
*method based on Ψ*_{p}*. In addition, from (18) we notice that the energy of the initial*
*state, i.e., Ψ*_{p}*(x*_{0}*) also has an inﬂuence on the convergence rate. A higher initial*
*energy will lead to a worse convergence rate.*

**5** **Simulation results**

In this section, we test four well-known nonlinear complementarity problems by our neural
network model (10). For each test problem, we also compare the numerical performance
*of the proposed neural network with various values of p and various initial states x(t*_{0}).

The test instances are described below.

**Example 5.1 [32, Example 2] Consider the NCP, where F : IR**^{5} *→ IR*^{5} *is given by*

*F (x) =*

*x*1*+ x*2*x*3*x*4*x*5*/50*
*x*_{2}*+ x*_{1}*x*_{3}*x*_{4}*x*_{5}*/50− 3*
*x*_{3}*+ x*_{1}*x*_{2}*x*_{4}*x*_{5}*/50− 1*
*x*_{4}*+ x*_{1}*x*_{2}*x*_{3}*x*_{5}*/50 + 1/2*

*x*5*+ x*1*x*2*x*3*x*4*/50*

*.*

*The NCP has only one solution x*^{∗}*= (0, 3, 1, 0, 0).*

**Example 5.2 [31, Watson] Consider the NCP, where F : IR**^{5} *→ IR*^{5} *is given by*

*F (x) = 2 exp*
( _{5}

∑

*i=1*

*(x*_{i}*− i + 2)*^{2}
)

*x*_{1}+ 1
*x*_{2}
*x*_{3}*− 1*
*x*_{4}*− 2*
*x*_{5}*− 3*

*.*

*Note that F is not a P*_{0}-function on IR^{n}*. The solution to this problem is x*^{∗}*= (0, 0, 1, 2, 3).*

**Example 5.3 [24, Kojima-Shindo] Consider the NCP, where F : IR**^{4} *→ IR*^{4} *is given by*

*F (x) =*

*3x*^{2}_{1} *+ 2x*_{1}*x*_{2}*+ 2x*^{2}_{2}*+ x*_{3}*+ 3x*_{4}*− 6*
*2x*^{2}_{1}*+ x*1*+ x*^{2}_{2}*+ 3x*3*+ 2x*4*− 2*
*3x*^{2}_{1} *+ x*_{1}*x*_{2}*+ 2x*^{2}_{2}*+ 2x*_{3}*+ 3x*_{4}*− 1*

*x*^{2}_{1}*+ 3x*^{2}_{2} *+ 2x*_{3}*+ 3x*_{4}*− 3*

* .*

*This is a non-degenerate NCP and the solution is x** ^{∗}* = (

*√*

*6/2, 0, 0, 1/2).*

**Example 5.4 [24, Kojima-Shindo] Consider the NCP, where F : IR**^{4} *→ IR*^{4} *is given by*

*F (x) =*

*3x*^{2}_{1} *+ 2x*_{1}*x*_{2}*+ 2x*^{2}_{2}*+ x*_{3}*+ 3x*_{4}*− 6*
*2x*^{2}_{1}*+ x*_{1}*+ x*^{2}_{2}*+ 10x*_{3}*+ 2x*_{4}*− 2*
*3x*^{2}_{1} *+ x*1*x*2*+ 2x*^{2}_{2}*+ 2x*3*+ 9x*4*− 9*

*x*^{2}_{1}*+ 3x*^{2}_{2} *+ 2x*_{3}*+ 3x*_{4}*− 3*

* .*

*This is a degenerate NCP and has two solutions x** ^{∗}* = (

*√*

*6/2, 0, 0, 1/2) and x*^{∗}*= (1, 0, 3, 0).*

The numerical implementation is coded by Matlab 7.0 and the ordinary diﬀerential
*equation solver adopted is ode23, which uses an Runge-Kutta (2, 3) formula. We ﬁrst*
*test the inﬂuence of the parameter p on the value of* *∥x(t) − x*^{∗}*∥. Figures 2–5 in the*
appendix describe how*∥x(t)−x*^{∗}*∥ varies with p for these instances with the initial states*
*x*_{0} = (10^{−2}*, 1, 0.5, 10*^{−2}*, 10** ^{−2}*)

^{T}*, x*

_{0}= (10

^{−2}*, 10*

^{−2}*, 0.5, 0.5, 0.5)*

^{T}*, x*

_{0}

*= (2, 10*

^{−2}*, 10*

^{−2}*, 0.1)*

*,*

^{T}*and x*

_{0}= (10

^{−3}*, 10*

^{−3}*, 10*

^{−3}*, 10*

*)*

^{−3}

^{T}*, respectively. In the tests, the design parameter ρ in*

*the neural network (10) is set to be 1000. From Figures 2–5, we see that, when p = 1.1,*the neural network (10) generates the slowest decrease of

*∥x(t)−x*

^{∗}*∥ for all test instances,*

*whereas when p = 20 it generates the fastest decrease of*

*∥x(t) − x*

^{∗}*∥. This veriﬁes the*analysis of Remark 4.1 (c). We should emphasize that the conclusion in Remark 4.1

*(c) requires the initial state x*

_{0}to be suﬃciently close to an equilibrium point. If this

condition is not satisﬁed, we cannot draw such conclusion; see Figure 6.

Example 5.1 shows how the value of *∥x(t) − x*^{∗}*∥ varies with initial state x*0. Figure 7
describes the convergence behavior of*∥x(t) − x*^{∗}*∥ with initial states x*^{(1)}0 *= (1, 1, 1, 1, 1)** ^{T}*,

*x*

^{(2)}

_{0}

*= (5, 5, 5, 5, 5)*

^{T}*, and x*

^{(3)}

_{0}

*= (10, 10, 10, 10, 10)*

*. Notice that the initial energies cor- responding to these three states are Ψ*

^{T}

_{p}*(x*

^{(1)}

_{0}

*) = 5.814, Ψ*

_{p}*(x*

^{(2)}

_{0}

*) = 39.367, and Ψ*

_{p}*(x*

^{(3)}

_{0}) =

*226.333, respectively. In the tests, we choose p = 1.8 and ρ = 1000. Figure 7, shows that*a larger initial energy yields a slower decrease of the error

*∥x(t) − x*

^{∗}*∥ if the initial state*is close to the solution of the NCP. This agrees with the analysis in Remark 4.1(c).

*The convergence behavior of x(t) from several initial states with a ﬁxed p and ρ = 1000*
*for each example is shown in Figures 8–12. The transient behavior of x(t) for Example*
5.4 is depicted in Figure 11 and Figure 12 since there are two solutions for this problem.

More speciﬁcally, we test 12 random initial points for the NCP, 9 of which converge to
(*√*

*6/2, 0, 0, 1/2); the remaining 3 converge to (1, 0, 3, 0). When ﬁnding the solution tra-*
*jectory x(t), we employ* *∥∇Ψ**p**(x(t))∥ ≤ 10** ^{−5}* as the stopping criterion.

To sum up, the neural network (10) is a better alternative for the network based on
*the FB function ϕ*_{FB} *if an appropriate p is chosen. Based on the analysis of Remark 4.1*
(c) and the above numerical simulations, we see that, to obtain a better convergence rate
*of the trajectory x(t), the parameter p cannot be set too small. In addition, we should*
*emphasize that the initial state x(t*_{0}) has a great inﬂuence on the convergence behavior
of *∥x(t) − x*^{∗}*∥.*

To end this section, we answer a natural question: are there advantages of our pro- posed neural network compared to the existing ones? To answer this, we summarize what we have observed from numerical experiments and theoretical results as below.

*• We compare our neural network model with some existing models which also work*
for NCP, for instance, the ones used in [6, 32, 33]. At ﬁrst glance, the neural network
models based on projection in [6, 32, 33] look having lower complexity. However,
we observe that the diﬀerence of the numerical performance is very marginal by
testing MCPLIB benchmark problems.

*• Our proposed model seems having better properties from theoretical view. Note*
*that there requires monotonicity (strong monotonicity) of F to guarantee the*
Lyapunov stability (exponential stability) of the neural network models used in
[6, 32, 33]. In contrast, such conditions are not needed for our neural network
*model. In fact, it can be veriﬁed that all F ’s are non-monotone in previous exam-*
ples except Example 5.2 (by checking the positive semi-deﬁniteness of their Jacobian
matrices).

*• For the following special NCP:*

*x = (x*_{1}*, x*_{2}*, x*_{3})*≥ 0, F (x) = (x*1*,−x*2*,−x*3)*≥ 0, ⟨x, F (x)⟩ = x*^{2}_{1}*− x*^{2}_{2}*− x*^{2}_{3} *= 0,*
*it is easy to verify that the unique solution is (0, 0, 0) which can be solved easily*
by our neural network model. But, the solution trajectory diverges by using the
model in [32].

*• Changing initial points may not having much eﬀect for our neural network model,*
whereas it does for other existing models. For instance, choosing

*x*_{0} *= (12,−12, 12, −12, 12) as the initial point in Example 5.1 causes the divergence*
of solution trajectory solved by the neural network model used in [32], while it does
not aﬀect anything by our neural network model.

**6** **Conclusions**

In this paper, we have studied a (class of) neural network based on the generalized
*FB function ϕ** _{p}* deﬁned as in (5). We establish the Lyapunov stability, the asymptotic
stability, and the exponential stability for the neural network. In addition, we also analyze

*the inﬂuence of the parameter p on the convergence rate of the trajectory (or the local*convergence behavior of the error

*∥x(t)−x*

^{∗}*∥) and obtain that a larger p leads to a better*convergence rate. This agrees with the result obtained by [2] for a descent-type method

*based on ϕ*

*p*

*, which also indicates how to choose a suitable p in practice. Numerical*experiments verify the obtained theoretical results. The advantages of our proposed neural network compared to other existing neural networks are reported as well. One future topic is to modify the proposed neural network model for various optimization problems and establish its related stability accordingly.

**References**

*[1] J.-S. Chen (2006), The semismooth-related properties of a merit function and a*
*descent method for the nonlinear complementarity problem, Journal of Global Opti-*
mization, vol. 36, 565–580.

*[2] J.-S. Chen, H.-T. Gao and S.-H. Pan (2009), A derivative-free R-linearly conver-*
*gent algorithm based on the generalized Fischer-Burmeister merit function, Journal*
of Computational and Applied Mathematics, vol. 232, 455–471.

*[3] J.-S. Chen and S.-H. Pan (2008), A family of NCP functions and a descent method*
*for the nonlinear complementarity problem, Computational Optimization and Appli-*
cations, vol. 40, 389–404.

*[4] J.-S. Chen and S.-H. Pan (2008), A regularization semismooth Newton method*
*based on the generalized Fischer-Burmeister function for P*_{0}*-NCPs, Journal of Com-*
putational and Applied Mathematics, vol. 220, 464–479.

*[5] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.*

*[6] C. Dang, Y. Leung, X. Gao, and K. Chen (2004), Neural networks for nonlinear*
*and mixed complementarity problems and their applications, Neural Networks, vol. 17,*
271–283.

*[7] S. Effati, A. Ghomashi, and A. R. Nazemi (2007), Application of projection*
*neural network in solving convex programming problems, Applied Mathematics and*
Computation, vol. 188, 1103–1114.

*[8] S. Effati and A. R. Nazemi (2006), Neural network and its application for solving*
*linear and quadratic programming problems, Applied Mathematics and Computation,*
vol. 172, 305–331.

*[9] M. C. Ferris, O. L. Mangasarian, and J.-S. Pang, editors, Complementarity:*

*Applications, Algorithms and Extensions, Kluwer Academic Publishers, Dordrecht,*
2001.

*[10] A. Fischer (1992), A special Newton-type optimization methods, Optimization, vol.*

24, 269–284.

*[11] A. Fischer (1997), Solution of the monotone complementarity problem with locally*
*Lipschitzian functions, Mathematical Programming, vol. 76, 513–532.*

*[12] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and*
*Complementarity Problems, Volumes I and II, Springer-Verlag, New York, 2003.*

*[13] F. Facchinei and J. Soares (1997), A new merit function for nonlinear comple-*
*mentarity problems and a related algorithm, SIAM Journal on Optimization, vol. 7,*
225–247.

*[14] C. Geiger, and C. Kanzow (1996), On the resolution of monotone complemen-*
*tarity problems, Computational Optimization and Applications, vol. 5, 155–173.*

*[15] Q. Han, L.-Z. Liao, H. Qi, and L. Qi (2001), Stability analysis of gradient-based*
*neural networks for optimization problems, Journal of Global Optimization, vol. 19,*
363–381.

*[16] J. J. Hopfield and D. W. Tank (1985), Neural computation of decision in*
*optimization problems, Biological Cybernetics, vol. 52, 141–152.*

*[17] X. Hu and J. Wang (2006), Solving pseudomonotone variational inequalities and*
*pseudoconvex optimization problems using the projection neural network, IEEE Trans-*
actions on Neural Networks, vol. 17, 1487–1499.

*[18] X. Hu and J. Wang (2007), A recurrent neural network for solving a class of gen-*
*eral variational inequalities, IEEE Transactions on Systems, Man, and Cybernetics-B,*
vol. 37, 528–539.

*[19] H. Jiang (1996), Unconstrained minimization approaches to nonlinear complemen-*
*tarity problems, Journal of Global Optimization, vol. 9, 169–181.*

*[20] C. Kanzow (1996), Nonlinear complementarity as unconstrained optimization,*
Journal of Optimization Theory and Applications, vol. 88, 139–155.

*[21] C. Kanzow and M. Fukushima (1996), Equivalence of the generalized comple-*
*mentarity problem to diﬀerentiable unconstrained minimization, Journal of Optimiza-*
tion Theory and Applications, vol. 90, pp. 581–603.

*[22] H. K. Khalil (1996), Nonlinear System, Upper Saddle River, NJ: Prentice Hall.*

*[23] M. P. Kennedy and L. O. Chua (1988), Neural network for nonlinear program-*
*ming, IEEE Tansaction on Circuits and Systems, vol. 35, 554–562.*

*[24] M. Kojima and S. Shindo (1986), Extensions of Newton and quasi-Newton meth-*
*ods to systems of P C*^{1} *equations, Journal of Operations Research Society of Japan,*
vol. 29, 352–374.

*[25] L.-Z. Liao, H. Qi, and L. Qi (2001), Solving nonlinear complementarity problems*
*with neural networks: a reformulation method approach, Journal of Computational*
and Applied Mathematics, vol. 131, 342–359.

*[26] R. K. Miller and A. N. Michel (1982), Ordinary Diﬀerential Equations, Aca-*
demic Press.

*[27] S-K. Oh, W. Pedrycz, and S-B. Roh (2006), Genetically optimized fuzzy polyno-*
*mial neural networks with fuzzy set-based polynomial neurons, Information Sciences,*
vol. 176, 3490–3519.

*[28] A. Shortt, J. Keating, L. Monlinier, and C. Pannell (2005), Optical im-*
*plementation of the Kak neural network, Information Sciences, vol. 171, 273–287.*

*[29] D. W. Tank and J. J. Hopfield (1986), Simple neural optimization networks:*

*an A/D converter, signal decision circuit, and a linear programming circuit, IEEE*
Transactions on Circuits and Systems, vol. 33, 533–541.

*[30] P. Tseng (1996), Global behaviour of a class of merit functions for the nonlinear*
*complementarity problem, Journal of Optimization Theory and Applications, vol. 89,*
17–37.

*[31] L. T. Watson (1979), Solving the nonlinear complementarity problem by a homo-*
*topy method, SIAM Journal on Control and Optimization, vol. 17, 36–46.*

*[32] Y. Xia, H. Leung, and J. Wang (2002), A projection neural network and its*
*application to constrained optimization problems, IEEE Transactions on Circuits and*
Systems-I, vol. 49, 447–458.

*[33] Y. Xia, H. Leung, and J. Wang (2004), A genarl projection neural network for*
*solving monotone variational inequalities and related optimization problems, IEEE*
Transactions on Neural Networks, vol. 15, 318–328.

*[34] Y. Xia, H. Leung, and J. Wang (2005), A recurrent neural network for solving*
*nonlinear convex programs subject to linear constraints, IEEE Transactions on Neural*
Networks, vol. 16, 379–386.

*[35] M. Yashtini and A. Malek (2007), Solving complementarity and variational*
*inequalities problems using neural networks, Applied Mathematics and Computation,*
vol. 190, 216–230.

*[36] S. H. Zak, V. Upatising, and S. Hui (1995), Solving linear programming prob-*
*lems with neural networks: a comparative study, IEEE Transactions on Neural Net-*
works, vol. 6, 94–104.

*[37] G. Zhang (2007), A neural network ensemble method with jittered training data for*
*time series forecasting, Information Sciences, vol. 177, 5329–5340.*

**Appendix**

0 5 10 15 20 25 30 35 40 45 50
10^{−7}

10^{−6}
10^{−5}
10^{−4}
10^{−3}
10^{−2}
10^{−1}
10^{0}
10^{1}

Time (ms)

||x(t)−x*||

p=1.1 p=1.5 p=3 p=20

Figure 2: Convergence behavior of the error*∥x(t) − x*^{∗}*∥ in Example 5.1 with given x*0.

0 5 10 15 20 25 30 35 40 45 50

10^{−7}
10^{−6}
10^{−5}
10^{−4}
10^{−3}
10^{−2}
10^{−1}
10^{0}
10^{1}

Time (ms)

||x(t)−x*||

p=1.1 p=1.5 p=3 p=20

Figure 3: Convergence behavior of the error*∥x(t) − x*^{∗}*∥ in Example 5.2 with given x*0.

0 5 10 15 20 25 30 35 40 45 50
10^{−7}

10^{−6}
10^{−5}
10^{−4}
10^{−3}
10^{−2}
10^{−1}
10^{0}
10^{1}

Time (ms)

||x(t)−x*||

p=1.1 p=1.5 p=3 p=20

Figure 4: Convergence behavior of the error*∥x(t) − x*^{∗}*∥ in Example 5.3 with given x*0.

0 5 10 15 20 25 30 35 40 45 50

10^{−7}
10^{−6}
10^{−5}
10^{−4}
10^{−3}
10^{−2}
10^{−1}
10^{0}
10^{1}

Time (ms)

||x(t)−x*||

p=1.1 p=1.5 p=3 p=20

Figure 5: Convergence behavior of the error*∥x(t) − x*^{∗}*∥ in Example 5.4 with given x*0.