https://doi.org/10.1007/s12190-019-01262-1
**O R I G I N A L R E S E A R C H**

**Neural network based on systematically generated** **smoothing functions for absolute value equation**

**B. Saheya**^{1}**· Chieu Thanh Nguyen**^{2}**· Jein-Shan Chen**^{2}

Received: 4 January 2019 / Published online: 20 April 2019

© Korean Society for Informatics and Computational Applied Mathematics 2019

**Abstract**

In this paper, we summarize several systematic ways of constructing smoothing func- tions and illustrate eight smoothing functions accordingly. Then, based on these systematically generated smoothing functions, a unified neural network model is pro- posed for solving absolute value equation. The issues regarding the equilibrium point, the trajectory, and the stability properties of the neural network are addressed. More- over, numerical experiments with comparison are presented, which suggests what kind of smoothing functions work well along with the neural network approach.

**Keywords Absolute value equations**· Neural network · Smoothing function
**Mathematics Subject Classifications 65K10**· 93B40 · 26D07

**1 Introduction**

The main target that we tackle with in this paper is the so-called absolute value equation (AVE for short), whose mathematical format is as below. In fact, the original standard AVE is described by

B. Saheya: The author’s work is supported by National Key R&D Program of China (Award No.:

2017YFC1405605) and Foundation of Inner Mongolia Normal University (Award No.: 2017YJRC003) J.-S. Chen: The author’s work is supported by Ministry of Science and Technology, Taiwan.

### B

Jein-Shan Chen jschen@math.ntnu.edu.tw B. Saheyasaheya@imnu.edu.cn Chieu Thanh Nguyen thanhchieu90@gmail.com

1 College of Mathematical Science, Inner Mongolia Normal University, Hohhot 010022, Inner Mongolia, People’s Republic of China

2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan

*Ax− |x| = b,* (1)
*where A*∈ R^{n}^{×n}*and b*∈ R* ^{n}*. Here

*|x| means the componentwise absolute value of*

*vector x*∈ R

*. Another generalized absolute value equation is in the form of*

^{n}*Ax* *+ B|x| = b,* (2)

*where B* ∈ R^{n}^{×n}*, B* *= O. When B = −I , I is the identity matrix, the AVE (2)*
reduces to the special form (1).

The AVE (1) was first introduced by Rohn in [29] and recently has been investigated
by many researchers, for example, Hu and Huang [7], Saheya and Chen [32], Jiang and
Zhang [9], Ketabchi and Moosaei [10], Mangasarian [14–18], Mangasarian and Meyer
[21], Prokopyev [25], and Rohn [31]. In particular, Mangasarian and Meyer [21] show
that the AVE (1) is equivalent to the bilinear program, the generalized LCP (linear
complementarity problem), and the standard LCP provided 1 is not an eigenvalue of
*A. With these equivalent reformulations, they also prove that the AVE (1) is NP-hard*
in its general form and provide existence results. Prokopyev [25] further obtain an
improvement indicating that the AVE (1) can be equivalently recast as (a larger) LCP
*without any assumption on A and B, and also provides a relationship with mixed*
integer programming. It is known that, if solvable, the AVE (1) can have either unique
solution or multiple (e.g., exponentially many) solutions. Indeed, various sufficiency
conditions on solvability and non-solvability of the AVE (1) with unique and multiple
solutions are discussed in [21,25,30]. Some variants of the AVEs including the abso-
lute value equation associated with second-order cone (SOCAVE) and the absolute
value programs, are investigated in [8] and [38], respectively. Furthermore, some other
type of absolute value equation, an extension of the AVE (2), is considered [8,19,20].

Roughly, there have three approaches for dealing with the AVEs (1)–(2). The first one is reformulating the AVEs (1)–(2) as complementarity problem and then solve it accordingly. The second one is to recast the AVEs (1)–(2) as a system of nonsmooth equations and then tackle with the nonsmooth equations by applying nonsmooth New- ton algorithm [26] or smoothing Newton algorithm [27]. The third one is applying the neural network approach. In this paper, we follow the third idea for solving the AVEs (1)–(2). Inspired by our another recent work [24], we will combine various smoothing functions with the neural network approach. Different from [24,32], the smoothing functions studied in this paper are not only constructed from one way, they are gener- ated by several systematic ways. Accordingly, this one can be viewed as a follow-up of [24,32].

Now, we quickly go over neural network approach which is different from tradi- tional optimization methods. To consider this approach, the main reason lies on the real-time solutions of optimization problems, which are sometimes required in prac- tice. It is well known that the neural networks approach is a very promising approach to solving the real-time optimization problem. In general, the neural networks can be implemented using integrated circuits and were first introduced in the 1980s by Hop- field and Tank [6,34] for optimization problems. Since then, significant research results have been achieved for various optimization problems, including linear programming [39], quadratic programming [1], linear complementarity problems [12], nonlinear

complementarity problem [13] and nonlinear programming [5]. In general, the essence of neural network approach is to construct a nonnegative energy function and establish a dynamic system that represents an artificial neural network. A first order differential equation represents the dynamic system. Furthermore, it is expected that the dynamic system will converge to its static state (or an equilibrium point), which corresponds to the solution for the underlying optimization problem, starting from an initial point.

Although similar idea was employed by Wang, Yu and Guo in [36], only one smoothing function was studied therein. In this paper, we present systematical ways about how to construct smoothing functions for AVE (1) and illustrate eight smoothing functions accordingly. After that, we design a gradient descent neural network model by using these eight different smoothing functions. We not only discuss the stability of the neural networks, but also give numerical comparison for these smoothing functions.

In fact, the new upshot of this paper lies on the numerical comparison, which suggest what kind of smoothing functions work well along with the neural network approach.

**2 Preliminaries**

By looking into the mathematical format of the aforementioned AVEs, it is observed
that the absolute value function*|x| is the key component. Indeed, the absolute value*
function also plays an important role in a lot of applications, like machine learning and
image processing, etc. In particular, the absolute value function*|x| is not differentiable*
*at x* = 0, which causes limits in analysis and application. To conquer this hurdle,
researchers consider smoothing methods and construct smoothing functions for it. We
summarize all possible ways to construct smoothing functions for*|x| as below. For*
more details, please refer to [2,4,11,23,28,35].

**1. Smoothing by the convex conjugate**

*Let X be a real topological vector space, and let X*^{∗}*be the dual space to X . For*
*any function f* *: dom f → R, its convex conjugate f*^{∗}*: (dom f )*^{∗}→ R is defined in
terms of the supremum by

*f*^{∗}*(y) := sup*

*x**∈dom f*

*x*^{T}*y− f (x)*
*.*

*In light of this, one can build up smooth approximation of f , denoted by f** _{μ}*, by adding

*strongly convex component to its dual g:= f*

^{∗}, namely,

*f*_{μ}*(x) = sup*

*z**∈domg*

*z*^{T}*x− g(x) − μd(z)*

*= (g + μd)*^{∗}*(x),*

*for some 1-strongly convex and continuous function d(·) (called proximity function).*

*Here, d(·) is 1-strongly convex which satisfies*

*d((1 − t)x + ty) ≤ (1 − t)d(x) + td(y) −*1

2*t(1 − t)x − y*^{2}*,*

*for all x, y and t ∈ (0, 1). Note that |x| = sup*_{|z|≤1}*zx. If we take d(z) := z*^{2}*/2, then*
the constructed smoothing function via conjugation leads to

*φ*1*(μ, x) = sup*

*|z|≤1*

*zx*−*μ*

2*z*^{2}

=

⎧⎨

⎩

*x*^{2}

2*μ**,* if *|x| ≤ μ,*

*|x| −μ*

2*, otherwise.* (3)

which is the traditional Huber function.

It is also possible to consider another expression:

*|x| = sup*

*z*_{1}*+z*2=1
*z*_{1}*,z*20

*(z*1*− z*2*)x.*

*Under this case, if we take d(z) := z*1*log z*1*+ z*2*log z*2+ log 2, the constructed
smoothing function by conjugation becomes

*φ*2*(μ, x) = μ log*

cosh

*x*
*μ*

*,* (4)

where cosh(x) := *e*^{x}*+ e*^{−x}

2 *. Alternatively, choosing d(y) := 1 −*

1*− y*^{2} gives
another smoothing function:

*φ*3*(μ, x) = sup*

*−1≤y≤1*

*x y+ μ*

1*− y*^{2}*− μ*

=

*x*^{2}*+ μ*^{2}*− μ.* (5)

**2. The Moreau proximal smoothing**

Suppose that *E is an Euclidean space and f : E → (−∞, ∞] is a closed and*
proper convex function. One natural tool for generating an approximate smoothing
function is through the use of the so-called proximal map introduced by Moreau [22].

The Moreau proximal approximation yields a family of approximations*{ f*_{μ}^{px}}* _{μ>0}*as
below:

*f*_{μ}^{px}*(x) = inf*

*u*∈E

*f(u) +* 1

2*μu − x*^{2}

*.* (6)

*It is known that the Moreau proximal approximation f*_{μ}^{px}*(x) is convex continuous,*
finite-valued, and differentiable with gradient *∇ f**μ*^{px} which is Lipschitz continuous
with constant _{μ}^{1}, see [22]. When applying the Moreau proximal smoothing way to
construct the smoothing function for the absolute value function*|x|, it also yields the*
Huber smoothing function*φ*1*(μ, x) by using the Moreau envelope [2].*

**3. Nesterov’s smoothing**

There is a class of nonsmooth convex functions considered in [23], which is given by

*q(x) = max{u, Ax − φ(u) | u ∈ Q}, x ∈ E,*

where*E, V are finite dimensional vector spaces, Q ⊆ V*^{∗}is compact and convex,*φ*
*is a continuous convex function on Q, and A* *: E → V is a linear map. The smooth*
*approximation of q suggested in [23] is described by the convex function*

*q*_{μ}*(x) = max{u, Ax − φ(u) − μd(u) | u ∈ Q}, x ∈ E,* (7)
*where d(·) is a prox-function for Q. It was proved in [*23, Theorem 1] that the convex
*function q*_{μ}*(x) is C*^{1}^{,1}*(E). More specifically, its gradient mapping is Lipschitz contin-*
*uous with constant L** _{μ}*=

*A*

^{2}

*σμ* and the gradient is described by*∇q**μ**(x) = Au**μ**(x),*
*where u*_{μ}*(x) is the unique minimizer of (7).*

*For the absolute value function q(x) = |x| with x ∈ R*^{1}*. Let A* *= 1, b = 0,*
E = R^{1}*, Q* *= {u ∈ R*^{1}*| |u| ≤ 1} and taking d(u) :=* ^{1}_{2}*u*^{2}. Then, we have

*q*_{μ}*(x) = max*

*u* *{Ax − b, u − μd(u) | u ∈ Q}*

= max

*u*

*xu*−*μ*

2*u*^{2}

=
*x*^{2}

2*μ**,* if*|x| ≤ μ,*

*|x| −*^{μ}_{2}*, otherwise.*

As we see, it also yields the Huber smoothing function*φ*1*(μ, x) defined by (3) through*
this approximation way.

**4. The infimal-convolution smoothing technique**

Suppose that*E is a finite vector space and f , g : E → (−∞, ∞]. The infimal*
*convolution of f and g, f* * g : E → [−∞, +∞] is defined by*

*( f g)(x) = inf*

*y*∈E*{ f (y) + g(x − y)} .*

In light of the concept of infimal convolution, one can also construct smoothing approx-
*imation functions. More specifically, we consider f* *: E → (−∞, ∞] which is a*
closed proper convex function and let*ω : E → R be a C*^{1}* ^{,1}* convex function with
Lipschitz gradient constant 1/σ (σ > 0). Suppose that for any μ > 0 and any x ∈ E,
the following infimal convolution is finite:

*f*_{μ}^{ic}*(x) = inf*

*u*∈E

*f(u) + μω*

*x− u*
*μ*

*= ( f ω*_{μ}*)(x),* (8)

where *ω*_{μ}*(·) = μω*

*μ*·

*. Then, f*_{μ}^{ic} is called the infimal-convolution *μ-smooth*
*approximation of f . In particular, whenμ ∈ R*++ *and p* *∈ (1, +∞), the infimal*
convolution of a convex function and a power of the norm function is obtained as
below:

*f*

1
*μp* · ^{p}

= inf

*u*∈E

*f(u) +*

1

*μpx − u*^{p}

*.* (9)

*For the absolute value function, it can be verified that f*_{μ}*(x) = (| · |) *

*μ∗p*1 | · |* ^{p}*
is

*the Huber function of order p, i.e.,*

**Fig. 1** *|x| and Huber function of order p (μ = 0.3)*

*f*_{μ}*(x) =*

⎧⎨

⎩

*|x| −* ^{p}^{−1}_{p}*μ*^{p}^{1}^{−1}*, if |x| > μ*^{p}^{−1}^{1} *,*

*|x|*^{p}

*μp**,* if *|x| ≤ μ*^{p−1}^{1} *.* (10)

*Note that when p*= 2 in the above expression (10), the Huber function of order p
reduces to the Huber function*φ*1*(μ, x) as shown in (3). Figure*1depicts the Huber
*function of order p with various value of p. To the contrast, plugging p*= 2 into infimal
convolution formula (9) yields the Moreau approximation (6). For more details about
infimal convolution and its induces approximation functions, please refer to [2,3].

**5. The convolution smoothing technique**

The smoothing approximation via convolution for the absolute value function is a
popular way, which can be found in [4,11,28,35]. Its construction idea is described
as follows. First, one constructs a smoothing approximation for the plus function
*(x)*_{+}*= max{0, x}. To this end, we consider the piecewise continuous function d(x)*
with finite number of pieces which is a density (kernel) function, that is, it satisfies

*d(x) ≥ 0 and*

_{+∞}

−∞ *d(x)dx = 1.*

Next, define *ˆs(μ, x) :=* _{μ}^{1}*d*

*x*
*μ*

, where *μ is a positive parameter. Suppose that*

_{+∞}

−∞ *|x| d(x)dx < +∞, then a smoothing approximation (denoted by ˆp(μ, x)) for*
*(x)*_{+}is obtained as below:

*ˆp(μ, x) =*

_{+∞}

−∞ *(x − s)*_{+}*ˆs(μ, s)ds =*

*x*

−∞*(x − s)ˆs(μ, s)ds.*

The following are four well-known smoothing functions for the plus function [4,28]:

*ˆφ*1*(μ, x) = x + μ log*

1*+ e*^{−}^{x}^{μ}

*.* (11)

*ˆφ*2*(μ, x) =*

⎧⎨

⎩

*x* if *x* ≥ ^{μ}_{2}*,*

1
2*μ*

*x*+^{μ}_{2}2

if −^{μ}_{2} *< x <* ^{μ}_{2}*,*

0 if *x* ≤ −^{μ}_{2}*.*

(12)

*ˆφ*3*(μ, x) =*

4μ^{2}*+ x*^{2}*+ x*

2 *.* (13)

*ˆφ*4*(μ, x) =*

⎧⎨

⎩

*x*−^{μ}_{2} if *x> μ,*

*x*^{2}

2*μ* if 0*≤ x ≤ μ,*
0 if *x< 0.*

(14)

where their corresponding kernel functions are

*d*1*(x) =* *e*^{−x}*(1 + e*^{−x}*)*^{2}*,*
*d*2*(x) =*

1 if −^{1}_{2} *≤ x ≤* ^{1}_{2}*,*
0 otherwise,
*d*3*(x) =* 2

*(x*^{2}*+ 4)*^{3}^{2}*,*
*d*4*(x) =*

1 if 0*≤ x ≤ 1,*
0 otherwise.

Using the fact that *|x| = (x)*+*+ (−x)*_{−}. Then, the smoothing function of *|x| via*
convolution can be written as

*ˆp(μ, |x|) = ˆp(μ, x) + ˆp(μ, −x) =*

_{+∞}

−∞ *|x − s| ˆs(μ, s)ds.*

Analogous to (11)–(14), we reach the following smoothing functions for*|x|:*

*φ*4*(μ, x) = μ*
log

1*+ e*^{−}^{μ}* ^{x}*
+ log

1*+ e*^{x}^{μ}

*.* (15)

*φ*5*(μ, x) =*

⎧⎨

⎩

*x* if *x*≥ ^{μ}_{2}*,*

*x*^{2}

*μ* +^{μ}_{4} if −^{μ}_{2} *< x <* ^{μ}_{2}*,*

*−x* if *x*≤ −^{μ}_{2}*.*

(16)

*φ*6*(μ, x) =*

4μ^{2}*+ x*^{2}*.* (17)

as well as the Huber function (3). If we take a Epanechnikov kernel function

*K(x) =*
_{3}

4*(1 − x*^{2}*) if |x| ≤ 1,*

0 otherwise*,*

we achieve the smoothing function for*|x|:*

*φ*7*(μ, x) =*

⎧⎪

⎨

⎪⎩

*x* if *x> μ,*

−_{8}^{x}_{μ}^{4}3 +^{3x}_{4}_{μ}^{2} +^{3}_{8}* ^{μ}* if

*−μ ≤ x ≤ μ,*

*−x* if *x< −μ.*

(18)

*Moreover, if we take a Gaussian kernel function K(x) =* ^{√}^{1}_{2}_{π}*e*^{−}^{x2}^{2} *for all x* ∈ R.

Then, it yields

*ˆs(μ, x) :=* 1
*μK*

*x*
*μ*

= 1

2πμ^{2}*e*^{−}

*x2*
2*μ2**.*

Hence, we obtain the below smoothing function [35] for*|x|:*

*φ*8*(μ, x) = x erf*

*x*

√2μ

+

2

*πμe*^{−}^{2}^{x2}^{μ2}*.* (19)
where the error function is defined by

erf*(x) =* 2

√*π*

_{x}

0

*e*^{−u}^{2}*du, ∀x ∈ R.*

To sum up, we have eight smoothing functions in total through the above construc-
tions. Figure2depicts the graphs of all the aforementioned smoothing functions*φ**i*,
*i* *= 1, . . . , 8 and the absolute value equation. Not only from the geometric view, φ**i*,
*i* *= 1, . . . , 8 are clearly smoothing functions of |x|, it can be also verified theoretically*
in Proposition2.1.

*φ*1*(μ, x) = sup*

*|z|≤1*

*zx*−*μ*

2*z*^{2}

=

⎧⎨

⎩

*x*^{2}

2*μ**,* if *|x| ≤ μ,*

*|x| −μ*

2*, otherwise.*

*φ*2*(μ, x) = μ log*

cosh

*x*
*μ*

*.*
*φ*3*(μ, x) = sup*

*−1≤y≤1*

*x y+ μ*

1*− y*^{2}*− μ*

=

*x*^{2}*+ μ*^{2}*− μ.*

*φ*4*(μ, x) = μ*
log

1*+ e*^{−}^{x}* ^{μ}*
+ log

1*+ e*^{μ}^{x}

*.*
*φ*5*(μ, x) =*

⎧⎨

⎩

*x* if *x*≥ ^{μ}_{2}*,*

*x*^{2}

*μ* +^{μ}_{4} if −^{μ}_{2} *< x <*^{μ}_{2}*,*

*−x* if *x*≤ −^{μ}_{2}*.*
*φ* *(μ, x) =*

4*μ*^{2}*+ x*^{2}*.*

**Fig. 2 The graphs of***|x| and the smoothing functions φ**i**, i = 1, . . . , 8 (μ = 0.3)*

*φ*7*(μ, x) =*

⎧⎪

⎨

⎪⎩

*x* if *x> μ,*

−_{8}^{x}_{μ}^{4}3 +^{3x}_{4}_{μ}^{2} +^{3}_{8}* ^{μ}* if

*−μ ≤ x ≤ μ,*

*−x* if *x< −μ.*

*φ*8*(μ, x) = x erf*

*x*

√2μ

+

2

*πμe*^{−}^{2}^{x2}^{μ2}*.*

From Fig.2, we see that the local behavior of all eight smoothing functions can be described as

*φ*3*≤ φ*2*≤ φ*1*≤ |x| ≤ φ*5*≤ φ*7*≤ φ*8*≤ φ*4*≤ φ*6*.* (20)
In particular, three smoothing function*φ*1,*φ*2,*φ*3approach to*|x| from below with*
*φ*1*≥ φ*2*≥ φ*3. To the contrast, the other five smoothing functions*φ*4,*φ*5,*φ*6,*φ*7,*φ*8

appraoch to*|x| from above with φ*5*≤ φ*7*≤ φ*8*≤ φ*4*≤ φ*6. Apparently, the smoothing
function*φ*1and*φ*5are closest to*|x| among these smoothing functions.*

Besides the geometric observation, we also provide algebraic analysis for (20).

Noting that each function*φ**i**(μ, x), for i = 1, 2, . . . , 8, is symmetric, so we only need*
to prove (20) with x *≥ 0. To proceed, for fixed μ > 0, we let y =* _{μ}* ^{x}*. The verifications
consist of seven parts.

Part (1):*φ*3*(μ, x) ≤ φ*2*(μ, x). To verify this inequality, we consider*

*f(y) = log*

*e*^{y}*+ e** ^{−y}*
2

−

*y*^{2}*+ 1 + 1.*

*Then, we compute the derivation of f(y) as below:*

*f*^{}*(y) =* *e*^{y}*− e*^{−y}

*e*^{y}*+ e** ^{−y}* −

*y*

*y*

^{2}+ 1

= *e** ^{2y}*− 1

*e** ^{2y}*+ 1−

*y*

*y*

^{2}+ 1

= 1 − 2

*e** ^{2y}*+ 1 − 1 +

*y* ^{2}*+ 1 − y*
*y*^{2}+ 1

= 1

*y*^{2}+ 1

*y*^{2}*+ 1 + y* − 2
*e** ^{2y}*+ 1

= *e*^{2y}*− 1 − 2y*^{2}*− 2y*
*y*^{2}+ 1

*e** ^{2y}*+ 1

*y*^{2}+ 1

*y*^{2}*+ 1 + y.*

*For convenience, we denote g(y) = e*^{2y}*− 1 − 2y*^{2}*− 2y*

*y*^{2}+ 1. It is known that the
*function e** ^{x}*can be expressed as

*e** ^{x}* =

^{∞}

*n*=0

*x*^{n}

*n!,* (21)

*which indicates e*^{x}*− 1 ≥ x +*^{x}_{2}^{2} +^{x}_{6}^{3}. Then, it follows that

*g(y) ≥ 2y +(2y)*^{2}

2 +*(2y)*^{3}

6 *− 2y*^{2}*− 2y*
*y*^{2}+ 1

= *4y*^{3}
3 *+ 2y*

1−

*y*^{2}+ 1

= *4y*^{3}

3 − *2y*^{3}

1+
*y*^{2}+ 1

*= 2y*^{3}

2

3 − 1

1+
*y*^{2}+ 1

*≥ 2y*^{3}

2 3−1

2

*≥ 0, ∀y ≥ 0.*

*This implies that f*^{}*(y) ≥ 0 for all y ≥ 0. Thus, f is monotonically nondecreasing*
*which yields f(y) ≥ f (0) = 0. Then, we verify the assertion that φ*3*(μ, x) ≤*
*φ*2*(μ, x).*

Part (2): *φ*2*(μ, x) ≤ φ*1*(μ, x). In order to prove this inequality, we discuss two*
cases.

(i) For 0*≤ x ≤ μ, this implies that 0 ≤ y ≤ 1. Considering*

*f(y) =* *y*^{2}

− log

*e*^{y}*+ e*^{−y}

yields that

*f*^{}*(y) = y −e** ^{2y}*− 1

*e** ^{2y}*+ 1 =

*ye*

^{2y}*+ y − e*

*+ 1*

^{2y}*e*

*+ 1*

^{2y}*.*

*By denoting g(y) := ye*

^{2y}*+ y − e*

*+ 1 and using (21) leads to*

^{2y}*g(y) = ye*^{2y}*+ y − e** ^{2y}*+ 1

*= y*^{∞}

*n*=0

*(2y)*^{n}*n*! −^{∞}

*n*=0

*(2y)*^{n}

*n*! *+ y + 1*

*= y*^{∞}

*n*=0

*(2y)*^{n}*n!* −

_{∞}

*n*=0

*(2y)*^{n}^{+1}
*(n + 1)!*+ 1

*+ y + 1*

*= y*

_{∞}

*n*=0

*(2y)*^{n}*n!*

1− 2

*n*+ 1

*+ y*

*≥ 0, ∀y ∈ [0, 1].*

*Therefor, we obtain that f*^{}*(y) ≥ 0 for all y ∈ [0, 1].*

*(ii) For x* *> μ, this implies that y > 1. Considering*

*f(y) = y −*1
2− log

*e*^{y}*+ e** ^{−y}*
2

gives

*f*^{}*(y) = 1 −e** ^{2y}*− 1

*e*

*+ 1*

^{2y}*> 0.*

*To sum up, we obtain that f*^{}*(y) ≥ 0 for all y ∈ [0, 1] in both cases. Following the*
same arguments as in part(1), we conclude that*φ*2*(μ, x) ≤ φ*1*(μ, x).*

Part (3):*φ*1*(μ, x) ≤ |x| and |x| ≤ φ*5*(μ, x). It is easy to verify these inequalities.*

We omit the verification.

Part (4):*φ*5*(μ, x) ≤ φ*7*(μ, x). We will prove this inequality by discussing three*
cases.

*(i) For x> μ, it is easy to see that φ*5*(μ, x) = φ*7*(μ, x) = x.*

(ii) For ^{μ}_{2} *≤ x ≤ μ, it means* ^{1}_{2} *≤ y ≤ 1. Considering*

*f(y) = −y*^{4}
8 +*3y*^{2}

4 +3
8 *− y*

= *−y*^{4}*+ 6y*^{2}*− 8y + 6*
8

= *−(y*^{2}*− 1)*^{2}*+ 4(y − 1)*^{2}+ 3
8

= *(y − 1)*^{2}

4*− (y + 1)*^{2}
+ 3
8

ad using the facts of ^{1}_{2}*≤ y ≤ 1 and*^{9}_{4} *≤ (y + 1)*^{2}*≤ 4, it follows that f (y) ≥ 0.*

(iii) For 0*≤ x <* ^{μ}_{2}, 0*≤ y <*^{1}_{2}. Considering

*f(y) = −y*^{4}
8 +*3y*^{2}

4 +3

8*− y*^{2}−1
4

= −*y*^{4}
8 −*y*^{2}

4 +1 8

= *−y*^{4}*− 2y*^{2}+ 1
8

= 2*− (y*^{2}*+ 1)*^{2}
8

and applying the facts 0 *≤ y <* ^{1}_{2} and 1 *≤ (y*^{2}*+ 1)*^{2} ≤ ^{25}_{16}, it follows that
*f(y) > 0. From all the above, we achieve that φ*5*(μ, x) ≤ φ*7*(μ, x).*

Part (5):*φ*7*(μ, x) ≤ φ*8*(μ, x). To proceed this assertion, we discuss two cases.*

(i) For 0*≤ x ≤ μ, this implies that 0 ≤ y ≤ 1. Consider*

*f(y) = yerf*

*y*

√2

+

2

*πe*^{−}^{y2}^{2} +*y*^{4}
8 −*3y*^{2}

4 −3
8*.*

By applying [35, Lemma 2.5], we have erf

√*y*
2

≥

1*− e*^{−}^{y2}^{2}

^{1}_{2}

. Then, it implies that

*f(y) ≥ y*

1*− e*^{−}^{y2}^{2}

^{1}

2 +

2

*πe*^{−}^{y2}^{2} + *y*^{4}
8 −*3y*^{2}

4 −3

8 *:= g(y).*

*It is easy to verify g(y) is monotonically decreasing on [0, 1] which indicates that*
*g(y) ≥ g(1) > 0. Hence, we obtain f (y) > 0 and φ*7*(μ, x) ≤ φ*8*(μ, x) is proved.*

*(ii) For x* *> μ, it means y > 1. Consider*

*f(y) = yerf*

*y*

√2

+

2

*πe*^{−}^{y2}^{2} *− y,*

*which yields f*^{}*(y) = erf*

√*y*
2

*− 1 < 0. Hence, f (y) is monotonically decreasing*
on*[1, +∞). Moreover, using [*35, Lemma 2.5] gives

*f(y) ≥ y*

1*− e*^{−}^{y2}^{2}

^{1}

2 +

2

*e*^{−}^{y2}^{2} *− y.*

Taking the limit in this inequality, we obtain lim*y*→∞ *f(y) ≥ 0. Therefore, f (y) ≥*
lim*y*→∞ *f(y) ≥ 0, which show the assertion φ*7*(μ, x) ≤ φ*8*(μ, x).*

Part (6):*φ*8*(μ, x) ≤ φ*4*(μ, x). Consider*

*f(y) = log*

1*+ e** ^{−y}*
+ log

1*+ e*^{y}

*− yerf*

*y*

√2

−

2
*πe*^{−}^{y2}^{2} *.*
Then, we have

*f*^{}*(y) =* *e** ^{y}*− 1

*e*

*+ 1− erf*

^{y} *y*

√2

= 1 −

2

*e** ^{y}*+ 1+ erf

*y*

√2

*:= 1 − g(y),*

which says that

*g*^{}*(y) = −* *2e*^{y}*e** ^{y}*+ 1+ 2

√*π*

√1
2*e*^{−}^{y2}^{2}

= 2

− *e*^{y}*e** ^{y}*+ 1

√2
2*πe*^{−}^{y2}^{2}

= 2

⎛

⎝−

√2πe√^{y}*+ (1 + e*^{y}*)e*^{−}^{y2}^{2}
2*π(1 + e*^{y}*)*

⎞

⎠

≤ 2

−√

2πe^{y}*+ 1 + e*^{y}

√2π(1 + e^{y}*)*

*, as y ≥ 0,*

= 2

*(1 −*√

2π)e* ^{y}*+ 1

√2π(1 + e^{y}*)*

*≤ 2(1 −*√

2*π + 1) < 0, as e*^{y}*≥ 1, 1 −*√
2*π < 0.*

*Hence, g(y) < g(0) = 1 which leads to f*^{}*(y) > 0 for all y ≥ 0. Then, it follows that*
*f(y) > f (0) = 2 log 2 −*

*π*2 *> 0, and hence φ*8*(μ, x) ≤ φ*4*(μ, x) is proved.*

Part (7):*φ*4*(μ, x) ≤ φ*6*(μ, x). Consider*

*f(y) =*

4*+ y*^{2}−
log

1*+ e** ^{−y}*
+ log

1*+ e*^{y}*,*
which gives

*f*^{}*(y) =* *y*

4*+ y*^{2}−*e** ^{y}*− 1

*e*

*+ 1*

^{y}= *y*

4*+ y*^{2}− 1 + 2
*e** ^{y}*+ 1

= *y*−
4*+ y*^{2}
4*+ y*^{2} + 2

*e** ^{y}*+ 1

=*−4(1 + e*^{y}*) + 2*

4*+ y*^{2}
*y*+

4*+ y*^{2}
4*+ y*^{2}

*y*+
4*+ y*^{2}

*(1 + e*^{y}*)* *.*

For convenience, we denote
*g(y) := −2(1 + e*^{y}*) +*

4*+ y*^{2}

*y*+

4*+ y*^{2}

*= −2e*^{y}*+ 2 + y*^{2}*+ y*
4*+ y*^{2}*.*

*Because e*^{y}*> 1 + y +* ^{y}_{2}^{2}, it yields
*g(y) < −2y + y*

4*+ y*^{2}*= y*

2−

4*+ y*^{2}

*≤ 0, ∀y ≥ 0.*

*This means that f*^{}*(y) < 0, i.e., f (y) is monotonically decreasing on [0, +∞). On*
the other hand, we know that

*y*lim→∞ *f(y) = lim*

*y*→∞

4*+ y*^{2}−
log

1*+ e** ^{−y}*
+ log

1*+ e*^{y}

= lim

*y*→∞

4*+ y*^{2}*− y + y −*
log

1*+ e** ^{−y}*
+ log

1*+ e*^{y}

= lim

*y*→∞

4*+ y*^{2}*− y + lim*

*y*→∞*y*−
log

1*+ e** ^{−y}*
+ log

1*+ e*^{y}

= lim

*y*→∞*y*− log
1*+ e*^{y}

= lim

*y*→∞log *e** ^{y}*
1

*+ e*

^{y}*= 0.*

*Thus, f(y) ≥ lim**y*→∞ *f(y) = 0 which implies that φ*4*(μ, x) ≤ φ*6*(μ, x).*

From Parts (1)–(7), the proof of (20) is complete.

**Proposition 2.1 Let**φ*i* : R^{2} *→ R for i = 1, . . . , 6 be defined as in (3)–(5) and*
(15)–(19), respectively. Then, we have

**(a)** *φ**i* *is continuously differentiable at(μ, x) ∈ R*_{++}*× R;*

**(b) lim**_{μ↓0}*φ**i**(μ, x) = |x|.*

**Proof The proof is straightforward and we omit it.**

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

*˙w(t) = H(w(t)), w(t*0*) = w*0∈ R^{n}*,* (22)
*where H* : R* ^{n}* → R

*is a mapping. We also introduce three kinds of stability that will be discussed later. These materials can be found in usual ODE textbooks. A point*

^{n}*w*^{∗} *= w(t*^{∗}*) is called an equilibrium point or a steady state of the dynamic system*
(22) if H*(w*^{∗}*) = 0. If there is a neighborhood *^{∗}⊆ R* ^{n}*of

*w*

^{∗}

*such that H(w*

^{∗}

*) = 0*

*and H(w) = 0 ∀w ∈*

^{∗}

*\{w*

^{∗}

*}, then w*

^{∗}is called an isolated equilibrium point.

* Lemma 2.1 Suppose that H* : R

*→ R*

^{n}

^{n}*is a continuous mapping. Then, for any*

*t*0

*> 0 and w*0 ∈ R

^{n}*, there exists a local solutionw(t) to (22) with t*

*∈ [t*0

*, τ) for*

*someτ > t*0

*. If, in addition, H is locally Lipschitz continuous at x*0

*, then the solution*

*is unique; if H is Lipschitz continuous in*R

^{n}*, thenτ can be extended to ∞.*

Let*w(t) be a solution to dynamic system (22). An isolated equilibrium pointw*^{∗}
is Lyapunov stable if for any*w*0 *= w(t*0*) and any ε > 0, there exists a δ > 0 such*
that*w(t) − w*^{∗}* < ε for all t ≥ t*0and*w(t*0*) − w*^{∗}* < δ. An isolated equilibrium*
point*w*^{∗} is said to be asymptotic stable if in addition to being Lyapunov stable, it
has the property that*w(t) → w*^{∗}*as t* *→ ∞ for all w(t*0*) − w*^{∗}* < δ. An isolated*
equilibrium point*w*^{∗}is exponentially stable if there exists a*δ > 0 such that arbitrary*
point*w(t) of (22) with the initial conditionw(t*0*) = w*0and*w(t*0*) − w*^{∗}* < δ is well*
defined on*[0, +∞) and satisfies*

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

Let* ⊆ R** ^{n}*be an open neighborhood of

*¯w. A continuously differentiable function*

*V*: R

^{n}*→ R is said to be a Lyapunov function at the state ¯w over the set for*Eq. (22) if

*V( ¯w) = 0, V (w) > 0, ∀w ∈ \{ ¯w},*

*˙V (w) ≤ 0, ∀w ∈ \{ ¯w}.*

The Lyapunov stability and asymptotical stability can be verified by using Lyapunov function, which is a useful tool for analysis.

**Lemma 2.2 (a) An isolated equilibrium point**w^{∗}*is Lyapunov stable if there exists a*
*Lyapunov function over some neighborhood*^{∗}*ofw*^{∗}*.*

**(b) An isolated equilibrium point**w^{∗}*is asymptotically stable if there exists a Lya-*
*punov function over some neighborhood*^{∗}*ofw*^{∗}*such that ˙V(w) < 0, ∀w ∈*

^{∗}*\{w*^{∗}*}.*

**3 Neural network model for AVE**

In order to design a suitable neural network for absolute value Eq. (1), the key step is to
*construct an appropriate energy function E(x) for which the global minimization x*^{∗}
is simultaneously a solution of the AVE (1). One approach to constructing a desired
energy function is the merit function method. The basic idea in this approach is to
transform the AVE (1) into an unconstrained problem.

*To this end, we define H**i* : R^{n}^{+1}→ R^{n}^{+1}as
*H**i**(μ, x) =*

!*μ*

*Ax+ B**i**(μ, x) − b*

"

*, for μ ∈ R, and x ∈ R*^{n}*,* (23)

where*i* : R^{n}^{+1}→ R* ^{n}*is given by

*i**(μ, x) :=*

⎡

⎢⎢

⎢⎣

*φ**i**(μ, x*1*)*
*φ**i**(μ, x*2*)*
*...φ**i**(μ, x**n**)*

⎤

⎥⎥

⎥⎦*, for μ ∈ R, and x ∈ R*^{n}*,* (24)

with various smoothing functions*φ**i* : R^{2} → R that is introduced in Sect.2. Then,
the AVE (1) can be transformed into an unconstrained optimization problem:

min*(μ, x) =* 1

2*H**i**(μ, x)*^{2}*.* (25)

Let*w = (μ, x), the AVE (*1) is equivalent to H*i**(μ, x) = 0. It is clear that if w*^{∗} ∈
R++× R^{n}*solves H**i**(w) = 0, then w*^{∗}solves*∇(w) = 0. Applying the gradient*
approach to the minimization of the energy function (25), we obtain the system of
differential equation:

*du(t)*

*dt* *= −ρ ∇(v(t), u(t)) = −ρ∇ H**i**(v(t), u(t))*^{T}*H**i**(v(t), u(t)),*

*u(t*0*) = u*0*.* (26)

*where u*0 *= x*0 ∈ R^{n}*, v(t) = μ*0*e** ^{−t}*,

*ρ > 0 is a time scaling factor. In fact, let-*ting

*τ = ρt leads to*

^{du}

_{dt}

^{(t)}*= ρ*

^{du}

_{d}

^{(τ)}*. Hence, it follows from (26) that*

_{τ}

^{du}

_{d}

^{(τ)}*=*

_{τ}*−∇(*^{1}_{2}*H**i**(w*^{∗}*)*^{2}*). In view of this, we set ρ = 1 in the subsequent analysis.*

**Assumption 3.1 The minimal singular value of the matrix A is strictly greater than***the maximal singular value of the matrix B.*

* Proposition 3.1 The AVE (2) is uniquely solvable for any b*∈ R

^{n}*if Assumption*3.1

*is*

*satisfied.*

**Proof Please see [9, Proposition 2.3] for a proof.**

**Proposition 3.2 Let***i**(μ, x) for i = 1, . . . , 8 be defined as in (*24). Then, we have
**(a) H***i**(μ, x) = 0 if and only if x solves the AVE (2);*

**(b) H***i* *is continuously differentiable on*R^{n}**\ {0} with the Jacobian matrix given by**

*∇ H**i**(μ, x) := [A + B ∇**i**(μ, x)] ,* (27)
*where*

*∇**i**(μ, x) :=*

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

*∂φ**i**(μ, x*1*)*

*∂x*1

0 · · · 0

0 *∂φ**i**(μ, x*2*)*

*∂x*2 · · · 0

*...* *...* *... ...*

0 · · · 0 *∂φ**i**(μ, x**n**)*

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦
*.*

**Proof The arguments are straightforward and we omit them.**

**Proposition 3.3 Let H***i* *and∇ H**i* *be given as in (23) and (27), respectively. Suppose*
*that Assumption*3.1*holds. Then,∇ H**i**(μ, x) is invertible at any x ∈ R*^{n}*andμ > 0.*

* Proof The result follows from Proposition*3.2immediately.

**Proposition 3.4 Let** : R* ^{n}*→ R

_{+}

*be given by (25). Then, the following results hold.*

**(a)** *(x) ≥ 0, ∀(μ, t) ∈ R*++*× R and (μ, x) = 0 if and only if x solve the AVE*
(2).

**(b) The function**(x) is continuously differentiable on R^{n}^{+1}**\ {0} with**

*∇(μ, x) = ∇ H*^{T}*H(μ, x),*
*where∇ H is the Jacobian of H(μ, x).*

**(c) The function**(w(t)) is nonincreasing with respect to t.

* Proof Parts (a)–(b) follow from Proposition*3.2immediately.

For part (c), we observe that
*d(w(t))*

*dt* =

)*dw*

*dt* *, ∇(μ, x)*

*

*= −ρ ∇(μ, x), ∇(μ, x)*

*= −ρ ∇(μ, x)*^{2}*< 0,*

*for all x∈ \{x*^{∗}}. Then, the desired result follows.

**4 Stability and existence**

In this section, we first adress the relation between the solution of AVE (1) and the equilibrium point of neural network (26). Then, we discuss the issues of the stability and the the solution trajectory of the neural network (26).

**Lemma 4.1 Let x**^{∗}*be a equilibrium of the neural network (26) and suppose that the*
*singular values of A*∈ R^{n}^{×n}*exceed 1. Then x*^{∗}*solves the system (1).*

**Proof Since ∇ ((w**^{∗}*)) = ∇ H**i*^{T}*H**i**(w*^{∗}*) and from the Proposition*3.3obtain*∇ H is*
nonsingular. It is clear to see that

∇
*(w*^{∗}*)*

*= 0,*

*if and only if H**i**(w*^{∗}*) = 0.*

* Theorem 4.1 (a) For any initial pointw*0

*= w(t*0

*), there exists a unique continuously*

*maximal solutionw(t) with t ∈ [t*0

*, τ) for the neural network (*26).

* (b) If the level setL(w*0

*) :=*+

*w | H**i**(w)*^{2}*≤ H(w*0*)*^{2},

*is bounded, thenτ can*
*be extended to∞.*

* Proof This proof is exactly the same as the one in [*33, Proposition 3.4], so we omit it

here.

*Now, we are going to analyze the stability of an isolated equilibrium x*^{∗}of the neural
network (26), which is to assert that*∇(x*^{∗}*) = 0 and ∇(x) = 0 for x ∈ \{x*^{∗}},

* is a neighborhood of x*^{∗}.

* Theorem 4.2 If the singular values of A* ∈ R

^{n}

^{×n}*exceed 1, then the isolated equi-*

*librium x*

^{∗}

*of the neural network (26) is asymptotically stable, and hence Lyapunov*

*stable.*

* Proof We consider the Lyapunov function (w) : → R defined by (25). First, it*
is clear that

*(x) ≥ 0 and from (a) of Proposition*3.4we have

*(·) is continuously*

*differentiable. Considering the singular values of A exceed 1 and Proposition*3.3we obtain

*∇ H(w*

^{∗}

*) is nonsingular. Then applying ∇ H(x*

^{∗}

*) and Lemma*4.1, we have

*H(w*

^{∗}

*) = 0 and (w*

^{∗}

*) = 0. Furthermore, if (w) = 0 on , then H(w) = 0 and*hence

*∇ = 0 on . This yields that w = w*

^{∗}since

*w*

^{∗}is isolated.

Secondly, consider the (b) of Proposition3.4and Lemma2.2, it says the isolated
*equilibrium x*^{∗}is asymptotically stable, and hence is Lyapunov stable.

* Theorem 4.3 If the singular values of A*∈ R

^{n}

^{×n}*exceed 1, then the isolated equilib-*

*rium x*

^{∗}

*of the neural network (26) is exponentially stable.*

* Proof The proof is routine and similar to that in the literature. For completeness, we*
include it again. Let

*= R*

_{++}×R

^{n}*, it is clear that H(·) is continuously differentiable,*which implies

*H(w) = H(w*^{∗}*) + ∇ H(w*^{∗}*)*^{T}*(w − w*^{∗}*) + o(w − w*^{∗}*), ∀ w ∈ .* (28)
*Let g(t) :=* ^{1}_{2}*w(t) − w*^{∗}^{2}*and we compute the derivative of g(t) as below:*

*dg(t)*
*dt* =

*dw*
*dt*

*T*

*(w(t) − w*^{∗}*) = −ρ*

2*∇(H(w)*^{2}*)*^{T}*(w(t) − w*^{∗}*)*

*= −ρ (∇ H(w) · H(w))*^{T}*(w(t) − w*^{∗}*)*

*= −ρ H(w)*^{T}*∇ H(w)*^{T}*(w(t) − w*^{∗}*)*

*= −ρ (w(t) − w*^{∗}*)*^{T}*∇ H(w*^{∗}*)∇ H(w)*^{T}*(w(t) − w*^{∗}*)*

*−ρ o(w − w*^{∗}*)*^{T}*∇ H(w)*^{T}*(w(t) − w*^{∗}*),*

where the last equality is due to (28). To proceed, we claim two assertions. First,
we claim that *(w − w*^{∗}*)*^{T}*∇ H(w*^{∗}*)∇ H(w)*^{T}*(w − w*^{∗}*) ≥ κ||w − w*^{∗}||^{2}*, for some*
*κ. To see this, from the Propositions* 3.3 and 3.4, we know *∇ H(w) is nonsin-*
*gular and H is a continuously differentiable function, which implies the matrix*

*∇ H(w*^{∗}*)∇ H(w*^{∗}*)** ^{T}* is symmetric and positive semi-definite. Hence, we have

*(w −*

*w*

^{∗}

*)*

^{T}*∇ H(w*

^{∗}

*)∇ H(w*

^{∗}

*)*

^{T}*(w − w*

^{∗}

*) ≥ κ*1

*w − w*

^{∗}

^{2}

*> 0 over \{w*

^{∗}} for some

*κ*1

*≥ 0. Then, by the continuity of ∇ H(·), we can conclude that*

*(w − w*^{∗}*)*^{T}*∇ H(w*^{∗}*)∇ H(w)*^{T}*(w − w*^{∗}*) ≥ κw − w*^{∗}^{2}*> 0, for some κ ≥ 0.*

Secondly, we claim that

*−ρ o(w − w*^{∗}*)*^{T}*∇ H(w)*^{T}*(w(t) − w*^{∗}*) ≤ εw − w*^{∗}^{2}*, for some ε > 0.*

This is because that

*-- − ρo(w − w*^{∗}*)*^{T}*∇ H(w)*^{T}*(w(t) − w*^{∗}*)*--

*w − w*^{∗}^{2} *≤ ρ∇ H(w)*

*o(w − w*^{∗}*)*

*w − w*^{∗}

*,*

where the right-hand side vanishes eventually. Thus, it yields that

*−ρ o(w − w*^{∗}*)*^{T}*∇ H(w)*^{T}*(w(t) − w*^{∗}*) ≤ εw − w*^{∗}^{2}*, for some ε > 0.*

*Now, from the above two assertions and noting that g(t) =* ^{1}_{2}*w(t) − w*^{∗}^{2}, we have
*dg(t)*

*dt* *≤ 2(−ρκ + ε)g(t),*
which gives

*g(t) ≤ e*^{2}^{(−ρκ+ε)t}*g(t*0*).*

Thus, we have

*w(t) − w*^{∗}* ≤ e*^{(−ρκ+ε)t}*w(t*0*) − w*^{∗}*,*

which says*w*^{∗}is exponentially stable as we can set*ρ larger enough such that −ρκ +*

*ε < 0. Then, the proof is complete.*

**5 Numerical results**

In order to demonstrate the effectiveness of the proposed neural network, we test several examples for our neural network (26) in this section. The numerical imple- mentation is coded by Mathematica 11.3 and the ordinary differential equation solver adopted is NDSolve[ ], which uses an Runge–Kutta (2,3) formula. The initial point of each problems are selected by randomly and the initial point is same for different smoothing functions. The results are collected together in Tables1,2,3and4, where

*φ**i* Denotes the smoothing functions*φ**i**, i = 1, . . . , 8*
*N* Denotes the number of iterations

*t* Denotes the time when algorithm terminates

*Er* Denotes the value of*x(t) − x*^{∗} when algorithm terminates

*H**(x**t**)* Denotes the value of*H(x(t)) = Ax − |x| − b when algorithm terminates*
*C T* Denotes the CPU time in seconds

**Table 1 Computing results of Example**5.1(dt= 0.2)

Function *N* *t* *Er* *H**(x*0*)* CT

*φ*1 34 6.8 9*.3686 × 10*^{−7} 0.0000136037 1*.5090863*

*φ*2 36 7.2 8*.70587 × 10*^{−7} 0.0000126414 0*.7760444*

*φ*3 38 7.6 8*.41914 × 10*^{−7} 0.000012225 0*.4980285*

*φ*4 2 0.4 2*.90785 × 10*^{−15} 1*.59872 × 10*^{−14} 0*.0340019*

*φ*5 2 0.4 1*.11772 × 10*^{−12} 8*.41527 × 10*^{−12} 0*.0740043*

*φ*6 10 2.0 7*.52691 × 10*^{−7} 0.0000109295 0*.1150066*

*φ*7 2 0.4 1*.29976 × 10*^{−12} 8*.61527 × 10*^{−12} 0*.0730042*

*φ*8 34 6.8 9*.3686 × 10*^{−7} 0.0000136037 0*.6880393*

**Table 2 Computing results of Example**5.2(dt= 0.2)

Function *N* *T* *Er* *H**(x*0*)* CT

*φ*1 55 11 9*.84216 × 10*^{−7} 2*.03995 × 10*^{−7} 3.1531804

*φ*2 57 11.4 9*.14593 × 10*^{−7} 1*.89565 × 10*^{−7} 1.3690783

*φ*3 59 11.8 8*.84473 × 10*^{−7} 1*.83322 × 10*^{−7} 0.6920396

*φ*4 2 0.4 2*.67859 × 10*^{−9} 4*.86096 × 10*^{−10} 0.0370021

*φ*5 2 0.4 2*.68658 × 10*^{−9} 4*.87548 × 10*^{−10} 0.1510086

*φ*6 18 3.6 9*.56666 × 10*^{−7} 2*.57215 × 10*^{−7} 0.2210127

*φ*7 2 0.4 2*.16413 × 10*^{−9} 3*.92736 × 10*^{−10} 0.1600091

*φ*8 55 11 9*.84216 × 10*^{−7} 2*.03995 × 10*^{−7} 1.5320877

**Example 5.1 Consider the following absolute value equation where**

*A*=

⎛

⎜⎜

⎝

10 1 2 0

1 11 3 1

0 2 12 1

1 7 0 13

⎞

⎟⎟

*⎠ , b =*

⎛

⎜⎜

⎝ 12 15 14 20

⎞

⎟⎟

*⎠ .*

*We can verify that one solution of the absolute value equations is x*^{∗}*= (1, 1, 1, 1).*

The parameter*ρ is set to be 1, time step is set to be dt = 0.2 and the initial point is*
generated by randomly. Table1summarizes the computing results for Example5.1.

From Table1, we see that nor matter from the trajectory convergence time, the error, or
the computation time, the smoothing function*φ*4,*φ*5,*φ*7perform significantly better
than other functions. Figure3depicts the norm error*x(t) − x*^{∗} with various time.

This figure indicates the smoothing functions*φ*4,*φ*5,*φ*7also outperform than others
(it follows by*φ*6).