to appear in Applied Numerical Mathematics, 2017

### A smoothing Newton method for absolute value equation associated with second-order cone

Xin-He Miao^{1}

Department of Mathematics Tianjin University, China

Tianjin 300072, China

Jian-Tao Yang ^{2}
Department of Mathematics

Tianjin University, China Tianjin 300072, China

B. Saheya^{3}

College of Mathematical Science Inner Mongolia Normal University Hohhot 010022, Inner Mongolia, P. R. China

Jein-Shan Chen ^{4}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan.

May 20, 2016

(1st revision on November 14, 2016) (2nd revision on February 26, 2017)

Abstract In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which

1E-mail: xinhemiao@tju.edu.cn. The author’s work is supported by National Natural Science Foun- dation of China (No. 11471241).

2E-mail: zzlyyjt@163.com.

3E-mail: saheya@imnu.edu.cn. The author’s work is supported by Natural Science Foundation of Inner Mongolia (Award Number: 2014MS0119).

4E-mail:jschen@math.ntnu.edu.tw. The author’s work is supported by Ministry of Science and Tech- nology, Taiwan.

is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

Keywords. Second-order cone, absolute value equations, smoothing Newton algorithm.

## 1 Introduction

The standard absolute value equation (AVE) is in the form of

Ax + B|x| = b, (1)

where A ∈ IR^{n×n}, B ∈ IR^{n×n}, B 6= 0, and b ∈ IR^{n}. Here |x| means the componentwise
absolute value of vector x ∈ IR^{n}. When B = −I, where I is the identity matrix, the AVE
(1) reduces to the special form:

Ax − |x| = b.

It is known that the AVE (1) was first introduced by Rohn in [38] and recently has been investigated by many researchers, for example, Caccetta, Qu and Zhou [1], Hu and Huang [14], Jiang and Zhang [22], Ketabchi and Moosaei [23], Mangasarian [25, 26, 27, 28, 29, 30, 31, 32], Mangasarian and Meyer [34], Prokopyev [35], and Rohn [40].

In particular, Mangasarian and Meyer [34] show that the AVE (1) is equivalent to the bilinear program, the generalized LCP (linear complementarity problem), and the stan- dard LCP provided 1 is not an eigenvalue of A. With these equivalent reformulations, they also show that the AVE (1) is NP-hard in its general form and provide existence results. Prokopyev [35] further improves the above equivalence which indicates that the AVE (1) can be equivalently recast as LCP without any assumption on A and B, and also provides a relationship with mixed integer programming. In general, 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 [34, 35, 39]. Some variants of the AVE, like the absolute value equation associated with second-order cone and the absolute

value programs, are investigated in [16] and [41], respectively.

In this paper, we target another type of absolute value equation which is a natural ex- tension of the standard AVE (1). More specifically the following absolute value equation associated with second-order cones, abbreviated as SOCAVE, as below:

Ax + B|x| = b, (2)

where A, B ∈ IR^{n×n} and b ∈ IR^{n} are the same as those in (1); |x| denotes the absolute
value of x coming from the square root of the Jordan product “◦” of x and x. What is
the difference between the standard AVE (1) and the SOCAVE (2)? Their mathematical
formats look the same. In fact, the main difference is that |x| in the standard AVE (1)
means the componentwise |x_{i}| of each x_{i} ∈ IR, i.e., |x| = (|x_{1}|, |x_{2}|, · · · , |x_{n}|)^{T} ∈ IR^{n};
however, |x| in the SOCAVE (2) denotes the vector satisfying √

x^{2} :=√

x ◦ x associated
with second-order cone under Jordan product. To understand its meaning, we need to
introduce the definition of second-order cone (SOC). The second-order cone in IR^{n} (n ≥
1), also called the Lorentz cone, is defined as

K^{n}:=(x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k ≤ x_{1} ,

where k · k denotes the Euclidean norm. If n = 1, then K^{n} is the set of nonnegative reals
IR_{+}. In general, a general second-order cone K could be the Cartesian product of SOCs,
i.e.,

K := K^{n}^{1} × · · · × K^{n}^{r}.

For simplicity, we focus on the single SOC K^{n} because all the analysis can be carried
over to the setting of Cartesian product. The SOC is a special case of symmetric cones
and can be analyzed under Jordan product, see [11]. In particular, for any two vectors
x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}, the Jordan product of x and y
associated with K^{n} is defined as

x ◦ y :=

x^{T}y
y1x2+ x1y2

.

The Jordan product, unlike scalar or matrix multiplication, is not associative, which is
a main source of complication in the analysis of optimization problems involved SOC,
see [3, 10, 12] and references therein for more details. The identity element under this
Jordan product is e = (1, 0, ..., 0)^{T} ∈ IR^{n}. With these definitions, x^{2} means the Jordan
product of x with itself, i.e., x^{2} := x ◦ x; and √

x with x ∈ K^{n} denotes the unique vector
such that √

x ◦√

x = x. In other words, the vector |x| in the SOCAVE (2) is computed by

|x| :=√ x ◦ x.

As mentioned earlier, the significance of the AVE (1) arises from the fact that the AVE is capable to formulate many optimization problems (also see [26, 30, 32, 34, 35]), such as,

linear programs, quadratic programs, bimatrix games, and so on. Moreover, the absolute value equations is equivalent to the linear complementarity problem [34]. Accordingly, we see that the SOCAVE (2) plays similar role in various optimization problems involved second-order cones. For solving the standard AVE (1), there are many various numerical methods proposed in the literature (see [1, 21, 22, 25, 26, 27, 35, 43]). As for the SO- CAVE (2), Hu, Huang and Zhang [16] propose a generalized Newton method for solving the SOCAVE (2). It is well known that smoothing-type algorithms is a powerful tool for solving many optimization problems, for example, the linear and nonlinear complemen- tarity problems [3, 12, 19, 20, 24], the system of equalities and inequalities [17, 42]. In this paper, we are interested in an smoothing Newton method for solving the SOCAVE (2). Our numerical results also support that the smoothing Newton method is a bet- ter way than the generalized Newton method employed in [16]. That is why we adopt this algorithm as the main tool to do numerical implementations. In addition, we have shown that the proposed smoothing Newton method is locally quadratically convergent under suitable condition. We report some preliminary numerical results to show that the method is efficient. Moreover, numerical comparisons based on various value of p are presented as well.

To close this section, we say a few words about notations and the organization of
this paper. As usual, IR^{n} denotes the space of n-dimensional real column vectors. IR_{+}
and IR_{++} denote the nonnegative and positive reals. For any x, y ∈ IR^{n}, the Euclidean
inner product is denoted hx, yi = x^{T}y, and the Euclidean norm kxk is denoted as kxk =
phx, xi. This paper is organized as follows. In Section 2, we briefly describe some
concepts and properties on second-order cone. Besides, we review Jordan product and
the spectral decomposition for elements x and y in IR^{n}. In Section 3, we introduce
a smoothing function of the absolute value |x|, and study the Jacobian matrix of the
smoothing function. In Section 4, we propose a smoothing Newton algorithm for solving
the SOCAVE (2), and discuss the convergence of the proposed method under suitable
conditions. In Section 5, the preliminary numerical results and numerical comparisons
are given.

## 2 Preliminaries

In this section, we recall some basic concepts and background materials regarding the
second-order cone, which will be extensively used in the subsequent analysis. More
details can be found in [3, 10, 11, 12, 16]. First, we recall the expression of the spectral
decomposition of x with respect to SOC. For x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}, the spectral
decomposition of x with respect to SOC is given by

x = λ_{1}(x)u^{(1)}_{x} + λ_{2}(x)u^{(2)}_{x} , (3)

where λ_{i}(x) = x_{1}+ (−1)^{i}kx_{2}k for i = 1, 2 and

u^{(i)}_{x} =

1 2

1, (−1)^{i x}_{kx}^{T}^{2}

2k

T

if kx_{2}k 6= 0,

1

2 1, (−1)^{i}ω^{T}T

if kx_{2}k = 0,

(4)

with ω ∈ IR^{n−1} being any vector satisfying kωk = 1. The two scalars λ_{1}(x) and λ_{2}(x)
are called spectral values of x; while the two vectors u^{(1)}x and u^{(2)}x are called the spectral
vectors of x. Moreover, it is obvious that the spectral decomposition of x ∈ IR^{n} is unique
if x_{2} 6= 0.

Lemma 2.1. For any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} with the spectral decomposition given as
in (3)-(4), the following results hold.

(a) u^{(1)}x ◦ u^{(2)}x = 0 and u^{(i)}x ◦ u^{(i)}x = u^{(i)}x for i = 1, 2;

(b) ku^{(1)}x k^{2} = ku^{(2)}x k^{2} = ^{1}_{2} and kxk^{2} = ^{1}_{2}(λ^{2}_{1}(x) + λ^{2}_{2}(x)).

Proof. The property can be verified directly or can be found in [3, 11, 12, 16, 10]. 2

In the next content, we talk about the projection onto second-order cone. We let x+be
the projection of x onto SOC K^{n}, and x_{−}be the projection of −x onto the dual cone (K^{n})^{∗}
of K^{n}, where the dual cone (K^{n})^{∗} is defined by (K^{n})^{∗} := {y ∈ IR^{n}| hx, yi ≥ 0, ∀x ∈ K^{n}}.

In fact, the dual cone of K^{n}is itself, i.e., (K^{n})^{∗} = K^{n}. Due to the special structure of SOC
K^{n}, the explicit formula of projection of x = (x1, x2) ∈ IR × IR^{n−1} onto K^{n} is obtained in
[3, 10, 11, 12, 13] as below:

x_{+} =

x if x ∈ K^{n},
0 if x ∈ −K^{n},
u otherwise,
where

u =

" _{x}_{1}_{+kx}_{2}k

_{x} 2

1+kx2k 2

x2

kx2k

# . Similarly, the expression of x− is in the form of

x− =

0 if x ∈ K^{n},

−x if x ∈ −K^{n},
w otherwise,
where

w =

"

−^{x}^{1}^{−kx}_{2} ^{2}^{k}

x1−kx_{2}k
2

x2

kx2k

# .

Together with the spectral decomposition of x, it is shown that x = x_{+}+ x− and the
expression of x+ has the form:

x_{+}= (λ_{1}(x))_{+}u^{(1)}_{x} + (λ_{2}(x))_{+}u^{(2)}_{x} ,
and

x− = (−λ_{1}(x))_{+}u^{(1)}_{x} + (−λ_{2}(x))_{+}u^{(2)}_{x} ,
where (α)_{+}= max{0, α} for α ∈ IR.

Next, we talk about the expression of |x| associated with SOC. There is an alternative
way via the so-called SOC-function to obtain the expression of |x|, which can be found
in [2, 4]. More specifically, for any x ∈ IR^{n}, we define the absolute value |x| of x with
respect to SOC as |x| := x_{+}+ x−. In fact, in the setting of SOC, the form |x| = x_{+}+ x−

is equivalent to the form |x| =√

x ◦ x. Combining the above expression of x+ and x−, it cab be verified that the expression of the absolute value |x| is in the form of

|x| = (λ1(x))++ (−λ1(x))+u^{(1)}_{x} +(λ2(x))++ (−λ2(x))+u^{(2)}_{x}

=
λ_{1}(x)

u^{(1)}_{x} +
λ_{2}(x)

u^{(2)}_{x} .

To end this section, we point out the relation between SOCAVE and SOCLCP
(second-order cone linear complementarity problem). In [16], it was shown that SO-
CAVE (2) is equivalent to the following SOCLCP: find x, y ∈ IR^{n} such that

M x + P y = c, and x ∈ K^{n}, y ∈ K^{n}, hx, yi = 0,

where M, P ∈ IR^{n×n} are matrices and c ∈ IR^{n}. However, the above is not a standard
SOCLCP because there exists the equations M x + P y = c therein. As below, we show
that the SOCAVE (2) can be further converted into a standard SOCLCP.

Theorem 2.1. The SOCAVE (2) can be reduced to the second-order cone linear comple- mentarity problem (SOCLCP):

v ∈ K^{n}× K^{n}× K^{n}, w = Qv + q ∈ K^{n}× K^{n}× K^{n} and hv, wi = 0, (5)
where

Q :=

−I 2I 0

A B − A 0

−A A − B 0

, v :=

2x_{+}

|x|

0

and q :=

0

−b b

. (6)

Proof. By looking into (6), we have

w = Qv + q =

2x_{−}
Ax + B|x| − b

−Ax − B|x| + b

.

Plugging this into SOCLCP (18) implies that

Ax + B|x| − b ∈ K^{n} and − Ax − B|x| + b ∈ K^{n}.

Since K^{n} is pointed, it follows that Ax + B|x| − b = 0. On the other hand, the above
argument is reversible. Thus, we show that SOCAVE (2) is equivalent to second-order
cone linear complementarity problem. 2

Remark 2.1. From Theorem 2.1, it follows that we can also solve the SOCAVE (2) by employing many efficient algorithms for solving SOCLCP (18). Nonetheless, when we apply the Newton method to solve SOCLCP, it still needs reformulate it as smooth equations or nonsmooth equations. This means that we need twice reformulations if we follow this way. In view of this, in this paper, we reformulate the SOCAVE (2) directly as the smooth equations, and solve the equations by smoothing Newton method.

## 3 Smoothing functions associate with SOCAVE

In this paper, we employ the smoothing Newton method for solving the SOCAVE (2).

To this end, we need to adopt a smoothing function. Due to the non-differentiability of

|α| for α ∈ IR, we consider a class of smoothing functions for the absolute value function

|α|. More specifically, we define the function φ_{p}(·, ·) : IR^{2} → IR as

φ_{p}(a, b) :=p|a|^{p} ^{p}+ |b|^{p}, p > 1. (7)
This class of functions is extracted from the so-called generalized Fischer-Burmeister
function φ_{p}(a, b) = p|a|^{p} ^{p}+ |b|^{p} − (a + b), which is heavily studied in many references
[5, 6, 7, 8, 9, 15]. For convenience, we still use the notation φ_{p} even it is no longer exactly
the same as the generalized Fischer-Burmeister function.

Lemma 3.1. Let φ_{p} : IR^{2} → IR be defined as in (7). Then, the following hold.

(a) φp(a, 0) = |a| and φp(0, b) = |b|;

(b) φ_{p}(·, ·) is Lipschitz continuous on IR^{2};
(c) φ_{p}(·, ·) is strongly semismooth on IR^{2};

(d) φ_{p}(a, b) is continuously differentiable for any (a, b) 6= (0, 0) ∈ IR^{2} with

∂φ_{p}(a, b)

∂a = sgn(a)|a|^{p−1}

(φ_{p}(a, b))^{p−1} and ∂φ_{p}(a, b)

∂b = sgn(b)|b|^{p−1}
(φ_{p}(a, b))^{p−1},

where the function sgn(·) is defined by sgn(α) :=

1 if α > 0, 0 if α = 0,

−1 if α < 0.

Proof. Please refer to [5, 6, 7, 8, 9, 15] for a proof. 2

According to Lemma 3.1, it follows that for any a ∈ IR and a → 0, we have φ_{p}(a, b) →

|b|. Therefore, combining the spectral decomposition of x and the function φp, we define
a vector-valued smoothing function Φ_{p} : IR × IR^{n}→ IR^{n} as

Φp(µ, x) = φp(µ, λ1(x))u^{(1)}_{x} + φp(µ, λ2(x))u^{(2)}_{x}

= p|µ|^{p} ^{p}+ |λ_{1}(x)|^{p}u^{(1)}_{x} +p|µ|^{p} ^{p}+ |λ_{2}(x)|^{p}u^{(2)}_{x} ,

where µ ∈ IR is a parameter, and λ_{1}(x), λ_{2}(x) are the spectral values of x. From Lemma
3.1, it is easy to verify that

µ→0limΦ_{p}(µ, x) = |λ_{1}(x)| u^{(1)}_{x} + |λ_{2}(x)| u^{(2)}_{x} = |x|.

In other words, the function Φ_{p}(µ, x) is a uniformly smoothing function of |x| associated
with SOC. With this function, for the SOCAVE (2), we further define a function H(µ, x) :
IR × IR^{n} → IR × IR^{n} by

H(µ, x) =

µ

Ax + BΦ_{p}(µ, x) − b

, ∀µ ∈ IR, x ∈ IR^{n}. (8)
Then, we observe that

H(µ, x) = 0 ⇐⇒ µ = 0 and Ax + BΦ_{p}(µ, x) − b = 0

⇐⇒ Ax + B|x| − b = 0 and µ = 0.

This indicates that x is a solution to the SOCAVE (2) if and only if (µ, x) is a solution
to the equation H(µ, x) = 0. In fact, we often choose µ ∈ IR_{++}. Applying Lemma 3.1
again, it is not difficult to show that the function H(µ, x) is continuously differentiable
on IR_{++}× IR^{n}. From direct calculation, we can also obtain the explicit formula of the
Jacobian matrix for the function H as below:

H^{0}(µ, x) =

"

1 0

B^{∂Φ}^{p}_{∂µ}^{(µ,x)} A + B^{∂Φ}^{p}_{∂x}^{(µ,x)}

#

(9)
for all (µ, x) ∈ IR++× IR^{n} with x = (x1, x2) ∈ IR × IR^{n−1}, where

∂Φp(µ, x)

∂µ = ∂φp(µ, λ1(x))

∂µ u^{(1)}_{x} + ∂φp(µ, λ2(x))

∂µ u^{(2)}_{x}

= µ^{p−1}

[φ_{p}(µ, λ_{1}(x))]^{p−1}u^{(1)}_{x} + µ^{p−1}

[φ_{p}(µ, λ_{2}(x))]^{p−1}u^{(2)}_{x}
and

∂Φ_{p}(µ, x)

∂x =

sgn(x1)|x1|^{p−1}
h√p

µ^{p}+|x1|^{p}ip−1I if x_{2} = 0,

"

b c_{kx}^{x}^{T}^{2}

2k

c_{kx}^{x}^{2}

2k aI + (b − a)_{kx}^{x}^{2}^{x}^{T}^{2}

2k^{2}

#

if x_{2} 6= 0,

with

a = φ_{p}(µ, λ_{2}(x)) − φ_{p}(µ, λ_{1}(x))
λ2(x) − λ1(x) ,
b = 1

2

sgn(λ_{2}(x))|λ_{2}(x)|^{p−1}

[φ_{p}(µ, λ_{2}(x))]^{p−1} + sgn(λ_{1}(x))|λ_{1}(x)|^{p−1}
[φ_{p}(µ, λ_{1}(x))]^{p−1}

, (10)

c = 1 2

sgn(λ_{2}(x))|λ_{2}(x)|^{p−1}

[φ_{p}(µ, λ_{2}(x))]^{p−1} − sgn(λ_{1}(x))|λ_{1}(x)|^{p−1}
[φ_{p}(µ, λ_{1}(x))]^{p−1}

.

## 4 Smoothing Newton method

In this section, we investigate the smoothing algorithm based on the smoothing function
Φ_{p}(µ, x) for solving the SOCAVE (2), and show the convergence properties of the con-
sidered algorithm. First, we present the generic framework of the smoothing algorithm.

Algorithm 4.1. (A Smoothing Newton Algorithm)

Step 0 Choose δ ∈ (0, 1), σ ∈ (0, 1), and µ_{0} ∈ IR_{++}, x^{0} ∈ IR^{n}. Set z^{0} := (µ_{0}, x^{0}),
e := (1, 0) ∈ IR × IR^{n−1}. Choose β > 1 satisfying min{1, kH(z^{0})k^{2}} ≤ βµ_{0}. Set
k := 0.

Step 1 If kH(z^{k})k = 0, stop. Otherwise, set τ_{k}:= min{1, kH(z^{k})k}.

Step 2 Compute 4z^{k} = (4µ_{k}, 4x^{k}) ∈ IR × IR^{n} by
H(z^{k}) + H^{0}(z^{k})4z^{k} = 1

βτ_{k}^{2}e, (11)

where H^{0}(z^{k}) denotes the Jacobian matrix of H(z^{k}) at (µ_{k}, x^{k}) given by (9).

Step 3 Let α_{k} be the maximum of the values 1, δ, δ^{2}, · · · such that
kH(z^{k}+ α_{k}4z^{k})k ≤

1 − σ(1 − 1
β)α_{k}

kH(z^{k})k. (12)

Step 4 Set z^{k+1} := z^{k}+ α_{k}4z^{k} and k := k + 1. Go to Step 1.

In order to explain that Algorithm 4.1 is well defined, we have to prove that the system of Newton equation (11) is solvable, and the line search (12) is well-defined. To this end, we need the next two technical lemmas.

Lemma 4.1. For any M, N ∈ IR^{n×n}, σmin(M ) > σmax(N ) if and only if σmin(M^{T}M ) >

σ_{max}(N^{T}N ). In addition, if σ_{min}(M^{T}M ) > σ_{max}(N^{T}N ), then M^{T}M − N^{T}N is positive
definite. Here σ_{min}(M ) denotes the minimum singular value of M , and σ_{max}(N ) denotes
the maximum singular value of N .

Proof. The proof is straightforward or can be found in usual textbook of matrix analysis, so we omit it here. 2

Lemma 4.2. Let A, S ∈ IR^{n×n} and A be symmetric. Suppose that the eigenvalues of
A and SS^{T} are arranged in non-increasing order. Then, for each k = 1, 2, · · · , n, there
exists a nonnegative real number θ_{k} such that

λmin(SS^{T}) ≤ θk≤ λmax(SS^{T}) and λk(SAS^{T}) = θkλk(A).

Proof. Please see [18, Corollary 4.5.11] for a proof. 2

In order to show that the Jacobian matrix H^{0}(µ, x) in Newton equation (11) is non-
singular for any µ > 0. we need the following assumption:

Assumption 4.1. For the SOCAVE (2), it holds σ_{min}(A) > σ_{max}(B).

In fact, under the condition of Assumption 4.1, The SOCAVE (2) has a unique solu- tion, which is verified in [33].

Theorem 4.1. Let H be defined as in (8). Suppose that Assumption 4.1 holds. Then,
the Jacobian matrix H^{0}(µ, x) in Newton equations (11) is nonsingular for any µ > 0.

Proof. From the expression of H^{0}(µ, x) given as in (9), we know that H^{0}(µ, x) is non-
singular if and only if the matrix A + B ^{∂Φ(µ,x)}_{∂x} is nonsingular. Thus, it suffices to show
that the matrix A + B ^{∂Φ(µ,x)}_{∂x} is nonsingular. Suppose not, i.e., there exists a vector
0 6= v ∈ IR^{n} such that

A + B∂Φ(µ, x)

∂x

v = 0.

This implies that

v^{T}A^{T}Av = v^{T} ∂Φ(µ, x)

∂x

T

B^{T}B ∂Φ(µ, x)

∂x v. (13)

For convenience, we denote C := ^{∂Φ(µ,x)}_{∂x} . Then, it follows that v^{T}A^{T}Av = v^{T}C^{T}B^{T}BCv.

By Lemma 4.2, there exists a constant ˆθ such that

λ_{min}(C^{T}C) ≤ ˆθ ≤ λ_{max}(C^{T}C) and λ_{max}(C^{T}B^{T}BC) = ˆθλ_{max}(B^{T}B).

Note that if we can prove that 0 ≤ λ_{min}(C^{T}C) ≤ λ_{max}(C^{T}C) ≤ 1, we have λ_{max}(C^{T}B^{T}BC) ≤
λ_{max}(B^{T}B). Then, by the assumption that the minimum singular value of A strictly ex-
ceeds the maximum singular value of B, and applying Lemma 4.1, we obtain v^{T}A^{T}Av >

v^{T}C^{T}B^{T}BCv. This contradicts the formula (13), which shows the Jacobian matrix
H^{0}(µ, x) in Newton equations (11) is nonsingular for µ > 0.

Thus, as discussed above, we only need to prove 0 ≤ λ_{min}(C^{T}C) ≤ λ_{max}(C^{T}C) ≤ 1.

For x_{2} = 0, we compute that C = h^{sgn(x}^{1}^{)|x}^{1}^{|}^{p−1}

√p

µ^{p}+|x1|^{p}ip−1I. Then, it is clear that 0 < λ(C^{T}C) < 1
for µ > 0. For x_{2} 6= 0, using the fact that the matrix M^{T}M is always positive semidefinite
for any matrix M ∈ IR^{m×n}, we see that the inequality λ_{min}(C^{T}C) ≥ 0 always holds. In
order to prove that λ_{max}(C^{T}C) ≤ 1, we need to further prove that the matrix I − C^{T}C
is positive semidefinite. To see this, note that

I − C^{T}C =

"

1 − b^{2}− c^{2} −2bc_{kx}^{x}^{T}^{2}

2k

−2bc_{kx}^{x}^{2}

2k (1 − a^{2})I + (a^{2} − b^{2}− c^{2})_{kx}^{x}^{2}^{x}^{T}^{2}

2k^{2}

# .

Because b^{2}+ c^{2} = 1
2

|λ_{2}(x)|^{2(p−1)}

[φ_{p}(µ, λ_{2}(x))]^{2(p−1)} + |λ_{1}(x)|^{2(p−1)}
[φ_{p}(µ, λ_{1}(x))]^{2(p−1)}

< 1

2· 2 = 1 for µ > 0, we
have 1 − b^{2}− c^{2} > 0. Moreover, the Schur complement of 1 − b^{2}− c^{2} has the form of

(1 − a^{2})I + (a^{2}− b^{2} − c^{2})x_{2}x^{T}_{2}

kx_{2}k^{2} − 4b^{2}c^{2}
1 − b^{2}− c^{2}

x_{2}x^{T}_{2}
kx_{2}k^{2}

= (1 − a^{2})

I − x_{2}x^{T}_{2}
kx_{2}k^{2}

+

1 − b^{2}− c^{2}− 4b^{2}c^{2}
1 − b^{2}− c^{2}

x_{2}x^{T}_{2}

kx_{2}k^{2}. (14)
On the other hand, |λ_{i}(x)| < φ_{p}(µ, λ_{i}(x)) (i = 1, 2) for µ > 0, we have

|φ_{p}(µ, λ_{2}(x)) − φ_{p}(µ, λ_{1}(x))|

=

|λ_{2}(x)|^{p}− |λ_{1}(x)|^{p}

p

X

i=1

[φ_{p}(µ, λ_{2}(x))]^{p−i}[φ_{p}(µ, λ_{1}(x))]^{i−1}

=

(|λ_{2}(x)| − |λ_{1}(x)|)

p

X

i=1

|λ_{2}(x)|^{p−i}|λ_{1}(x)|^{i−1}

p

X

i=1

[φ_{p}(µ, λ_{2}(x))]^{p−i}[φ_{p}(µ, λ_{1}(x))]^{i−1}

< ||λ_{2}(x)| − |λ_{1}(x)||

≤ |λ2(x) − λ1(x)|.

This together with (10) implies that 1 − a^{2} > 0 for any µ > 0. In addition, for any µ > 0,
we observe that

(1 − b^{2}− c^{2})^{2}− 4b^{2}c^{2}

= (1 − (b − c)^{2})(1 − (b + c)^{2})

=

"

1 − |λ_{1}(x)|^{2(p−1)}
[φp(µ, λ1(x))]^{2(p−1)}

#

·

"

1 − |λ_{2}(x)|^{2(p−1)}
[φp(µ, λ2(x))]^{2(p−1)}

#

> 0,

where the inequality holds due to |λ_{i}(x)| < φ_{p}(µ, λ_{i}(x)) for i = 1, 2 and µ > 0. With
all of these, we see that the Schur complement of 1 − b^{2}− c^{2} given as in (14) is a linear
positive combination of the matrices

I − _{kx}^{x}^{2}^{x}^{T}^{2}

2k^{2}

and _{kx}^{x}^{2}^{x}^{T}^{2}

2k^{2}, which yields that the Schur
complement (14) of 1 − b^{2}− c^{2} is positive semidefinite. Hence, the matrix I − C^{T}C is also
positive semidefinite, which is equivalent to saying 0 ≤ λ_{min}(C^{T}C) ≤ λ_{max}(C^{T}C) ≤ 1.

Thus, the proof is complete. 2

Theorem 4.1 indicates the Newton equation (11) in Algorithm 4.1 is solvable. It paves a way to show that the linear search (12) in Algorithm 4.1 is well-defined which is given in Theorem 4.2 as below. Indeed, the proof is very similar to the one in [17, Remark 2.1 (v)], we only present it here and omit its proof.

Theorem 4.2. Suppose that Assumption 4.1 holds. Then, for 4z ∈ IR × IR^{n} given by
(11), the linear search (12) is well-defined.

Next, we discuss the convergence of Algorithm 4.1. To this end, we need the following results whose arguments are similar to the ones in [17, Remark 2.1].

Theorem 4.3. Let H be defined as in (8). Suppose that Assumption 4.1 holds and that
the sequence {z^{k}} is generated by Algorithm 4.1. Then, the following results are hold.

(a) The sequences {kH(z^{k})k} and {τk} are monotonically non-increasing.

(b) βµ_{k} ≥ τ_{k}^{2} for all k.

(c) The sequence {µ_{k}} is monotonically non-increasing and µ_{k} > 0 for all k.

(d) The sequence {z^{k}} is bounded.

Proof. (a) From definition of the line search in (12) and τ_{k} := min{1, kH(z^{k})k}, it is
clear that {kH(z^{k})k} and {τ_{k}} are monotonically non-increasing.

(b) We prove this conclusion by induction. First, by Algorithm 4.1, it is clear that
τ_{0}^{2} ≤ βµ0 with τ0, β and µ0 chosen in Algorithm 4.1. Secondly, we suppose that τ_{k}^{2} ≤ βµk

for some k. Then, for k + 1, we have
µ_{k+1}−τ_{k+1}^{2}

β = µ_{k}+ α_{k}4µ_{k}− τ_{k+1}^{2}
β

= (1 − α_{k})µ_{k}+ α_{k}τ_{k}^{2}

β −τ_{k+1}^{2}
β

≥ (1 − α_{k})τ_{k}^{2}

β + α_{k}τ_{k}^{2}

β − τ_{k+1}^{2}
β

≥ 0,

where the second equality holds due to the Newton equation (11), and the second in-
equality holds due to part (a). Hence, it follows that βµk ≥ τ_{k}^{2} for all k.

(c) From the iterative scheme z^{k+1} = z^{k}+ α_{k}∆z^{k}, we know µ_{k+1} = µ_{k}+ α_{k}4µ_{k}. By the
Newton equations (11) and the line search as in (12) again, it follows that

µ_{k+1}= (1 − α_{k})µ_{k}+ α_{k}τ_{k}^{2}

β ≥ (1 − α_{k})τ_{k}^{2}

β + α_{k}τ_{k}^{2}
β > 0
for all k. On the other hand, we have

µ_{k+1} = (1 − α_{k})µ_{k}+ α_{k}τ_{k}^{2}

β ≤ (1 − α_{k})µ_{k}+ α_{k}µ_{k}≤ µ_{k},

where the first inequality holds due to part (b). Hence, the sequence {µ_{k}} is monotoni-
cally non-increasing and µk > 0 for all k.

(d) From part (a), we know the sequence {kH(z^{k})k} is bounded. Thus, there is a constant
C such that kH(z^{k})k ≤ C. In addition, since

4

λ_{1}(x^{k})u^{(1)}_{x} + λ_{2}(x^{k})u^{(2)}_{x}

2−

√p

4

4 |λ_{1}(x^{k})| + |λ_{2}(x^{k})|2

= 1 4

h

(8 − 2√^{p}

4)(|λ_{1}(x^{k})|^{2}+ |λ_{2}(x^{k})|^{2}) +√^{p}

4(|λ_{1}(x^{k})| − |λ_{2}(x^{k})|)^{2}i

> 0 (∀p > 1), it follows that

kH(z^{k})k

≥

Ax^{k}+ BΦ_{p}(µ_{k}, x^{k}) − b

≥
Ax^{k}

−

BΦ_{p}(µ_{k}, x^{k})
− kbk

= p

(x^{k})^{T}A^{T}Ax^{k}−
q

[Φ_{p}(µ_{k}, x^{k})]^{T}B^{T}BΦ_{p}(µ_{k}, x^{k}) − kbk

≥ p

λ_{min}(A^{T}A)kx^{k}k −
q

λ_{max}(B^{T}B)kΦ_{p}(µ_{k}, x^{k})k^{2}− kbk

= p

λ_{min}(A^{T}A)kx^{k}k −
r

λ_{max}(B^{T}B)

φ_{p}(µ_{k}, λ_{1}(x^{k}))u^{(1)}x + φ_{p}(µ_{k}, λ_{2}(x^{k}))u^{(2)}x

2

− kbk

= p

λ_{min}(A^{T}A)kx^{k}k −
r

λ_{max}(B^{T}B)h

φ^{2}_{p}(µ_{k}, λ_{1}(x^{k}))ku^{(1)}x k^{2}+ φ^{2}_{p}(µ_{k}, λ_{2}(x^{k}))ku^{(2)}x k^{2}i

− kbk

= p

λ_{min}(A^{T}A)kx^{k}k −
s

λ_{max}(B^{T}B) 1
2

p

q

(µ^{p}_{k}+ |λ_{1}(x^{k})|^{p})^{2}+ ^{p}
q

(µ^{p}_{k}+ |λ_{2}(x^{k})|^{p})^{2}

− kbk

≥ p

λ_{min}(A^{T}A)kx^{k}k −p

λ_{max}(B^{T}B)

· s 1

2

(µ^{2}_{k}+ |λ_{1}(x^{k})|^{2}+√^{p}

2µ_{k}|λ_{1}(x^{k})|) + (µ^{2}_{k}+ |λ_{2}(x^{k})|^{2} +√^{p}

2µ_{k}|λ_{2}(x^{k})|)

− kbk

= p

λ_{min}(A^{T}A)kx^{k}k

−p

λ_{max}(B^{T}B)
s

µ^{2}_{k}+1

2|λ_{1}(x^{k})|^{2}+ 1

2|λ_{2}(x^{k})|^{2}+

√p

2

2 µ_{k}(|λ_{1}(x^{k})| + |λ_{2}(x^{k})|) − kbk

≥ p

λ_{min}(A^{T}A)kx^{k}k

−p

λ_{max}(B^{T}B)
r

µ^{2}_{k}+1

2|λ_{1}(x^{k})|^{2}+ 1

2|λ_{2}(x^{k})|^{2}+ 2µ_{k}kλ_{1}(x^{k})u^{(1)}x + λ_{2}(x^{k})u^{(2)}x k − kbk

= p

λ_{min}(A^{T}A)kx^{k}k −p

λ_{max}(B^{T}B)µ_{k}+ kλ_{1}(x^{k})u^{(1)}_{x} + λ_{2}(x^{k})u^{(2)}_{x} k − kbk

= p

λ_{min}(A^{T}A) −p

λ_{max}(B^{T}B)

kx^{k}k −p

λ_{max}(B^{T}B)µ_{k}− kbk.

This together with kH(z^{k})k ≤ C implies

kx^{k}k ≤ C +pλmax(B^{T}B)µk+ kbk
pλ_{min}(A^{T}A) −pλ_{max}(B^{T}B)
holds for all k. Thus, the sequence {x^{k}} is bounded. 2

Theorem 4.4. Suppose that Assumption 4.1 holds and that {z^{k}} is generated by Algo-
rithm 4.1. Then, any accumulation point of {z^{k}} is a solution to the SOCAVE (2).

Proof. From Theorem 4.3 (d), we know the sequence {z^{k}} is bounded. Hence, there
exists at least a accumulation point for the sequence {z^{k}}. Without loss of generality,
let lim_{k→∞}z^{k} := z^{?} = (µ_{?}, x^{?}). Then, it follows that H^{?} := H(z^{?}) = lim_{k→∞}H(z^{k}) and
τ_{?} := min{1, kH^{?}k} = lim_{k→∞}min{1, kH(z^{k})k}. Now, we will show H^{?} = 0. Suppose
not, i.e., kH^{?}k > 0. To proceed, we discuss two cases according to whether lim_{k→∞}α_{k} = 0
or α_{k}≥ ˆα > 0 with ˆα ∈ IR_{++}.

Case 1: lim_{k→∞}α_{k} = 0. Then, from the line search (12), for the number α_{k} := ^{α}_{δ}^{k} with
all sufficiently large k, we have

kH(z_{k}+ α_{k}4z_{k})k > [1 − σ(1 − 1

β)α_{k}]kH(z_{k})k.

Furthermore, this leads to

kH(z^{k}+ α_{k}4z^{k})k − kH(z^{k})k
αk

> −σ(1 − 1

β)kH(z^{k})k. (15)

Besides, from Theorem 4.3 (c) again, we know µ^{?} ≥ 0. It follows that the function H is
continuously differentiable at the point z^{?}. Taking k → ∞ in the formula (15), we have

hH(z^{?}), H^{0}(z^{?})4z^{?}i

kH(z^{?})k ≥ −σ(1 − 1

β)kH(z^{?})k. (16)

This combining the Newton equations (11) yields
hH(z^{?}), H^{0}(z^{?})4z^{?}i

kH(z^{?})k = (τ_{?})^{2}

βkH(z^{?})khH(z^{?}), ei − kH(z^{?})k

≤ (τ_{?})^{2}kH(z^{?})k

βkH(z^{?})k − kH(z^{?})k

≤ τ_{?}

β − kH(z^{?})k

≤ (1

β − 1)kH(z^{?})k, (17)

where the first inequality holds due to the H¨older inequality hH(z^{?}), ei ≤ kH(z^{?})kkek =
kH(z^{?})k, the second and third inequality hold due to τ_{?} = min{1, kH(z^{?})k}. Putting
(16) and (17) together gives _{β}^{1} − 1 ≥ −σ(1 −_{β}^{1}). This contradicts σ ∈ (0, 1) and β > 1.

Case 2: α_{k} ≥ ˆα > 0 for all k. From the line search (12), we have
kH(z^{k+1})k ≤

1 − σ(1 − 1 β) ˆα

kH(z^{k})k = kH(z^{k})k − σ(1 − 1

β) ˆαkH(z^{k})k.

Then, it follows from the boundedness of kH(z^{k})k that P∞

k=0ασ(1 −ˆ _{β}^{1})kH(z^{k})k is
bounded. Moreover, we have lim_{k→∞}kH(z^{k})k = 0, i.e., kH^{?}k = 0. This contradicts
kH^{?}k > 0.

Hence, from all the above, we show H(z^{?}) = 0. That is, the element x^{?} is a solution of
the SOCAVE (2). Then, the proof is complete. 2

Now, we show the local quadratic convergence of Algorithm 4.1. In fact, we can achieve the following result by similar arguments as those in [37, Theorem 8]. For com- pleteness, we also provide a detailed proof.

Theorem 4.5. Let H be defined as in (8) and z^{?} be the unique solution to SOCAVE (2).

Suppose that Assumption 4.1 holds and that all V ∈ ∂H(z^{?}) are nonsingular. Then, the
whole sequence {z^{k}} converges to z^{?}, and kz^{k+1}− z^{?}k = O(kz^{k}− z^{?}k^{2}).

Proof. Since z^{?} is the solution to SOCAVE (2), using Assumption 4.1 and applying
Theorem 4.1 yield that the Jacobian matrix H^{0}(z^{k}) is nonsingular for all z^{k} sufficiently
close to z^{?}. On the other hand, applying the condition that all V ∈ ∂H(z^{?}) are nonsin-
gular and from [36, Proposition 3.1], we have kH^{0}(z^{k})^{−1}k = O(1) for all z^{k} sufficiently

close to z^{?}. Because z^{?} is the solution to SOCAVE (2), it is clear that z^{?} is a solution of
H(z) = 0. In addition, the function H is strongly semismooth, it follows that

kH(z^{k}) − H(z^{?}) − H^{0}(z^{k})(z^{k}− z^{?}))k = O(kz^{k}− z^{?}k^{2}).

Thus, we have
z^{k}+ 4z^{k}− z^{?}

=

z^{k}+ H^{0}(z^{k})^{−1}

−H(z^{k}) + 1
βτ_{k}^{2}e

− z^{?}

≤

H^{0}(z^{k})^{−1} −H(z^{k}) + H^{0}(z^{k})(z^{k}− z^{?})
+

H^{0}(z^{k})^{−1}1
βτ_{k}^{2}e

≤

H^{0}(z^{k})^{−1} −H(z^{k}) + H^{0}(z^{k})(z^{k}− z^{?})

+ O(1)
1
βτ_{k}^{2}e

= O(kH(z^{k}) − H(z^{?}) − H^{0}(z^{k})(z^{k}− z^{?})k) + O(kH(z^{k})k^{2})

= O(kz^{k}− z^{?}k^{2}) + O(kz^{k}− z^{?}k^{2})

= O(kz^{k}− z^{?}k^{2})

where the first equality holds due to the Newton equation (11), and the third equality
holds since the function H is locally Lipschitz continuous near z^{k}. Then, the proof is
complete. 2

## 5 Numerical Results

This section is devoted to the numerical results. First, we show the numerical comparison between the smoothing Newton algorithm and generalized Newton method. This provides the numerical evidence about why we adopt the smoothing Newton algorithm, not the generalized Newton algorithm, in this paper. Secondly, we use the performance profile to depict the comparison among different values of p. This shows that the smoothing Newton algorithm is not regularly affected when p is perturbed. Moreover, a suitable smoothing function from the class of smoothing functions is suggested in view of the numerical comparisons.

### 5.1 Smoothing Newton algorithm vs Generalized Newton method

In this subsection, for fixed p = 2, we provide some numerical examples to evaluate the efficiency of Algorithm 4.1. In our tests, we choose parameters

µ_{0} = 0.1, x_{0} = rand(n, 1), δ = 0.5, σ = 10^{−5} and β = max(1, 1.01 ∗ τ_{0}^{2}/µ).

We stop the iterations when kH(z_{k})k ≤ 10^{−6}or the number of iterations exceeds 100. All
the experiments are done on a PC with Intel(R) CPU of 2.40GHz and RAM of 4.00GHz,

and all the program codes are written in Matlab and run in Matlab environment. We consider the following four problems, and compute these problems by using Smoothing Newton Algorithm (SN for short) 4.1 and Generalized Newton method (GN for short) which introduced in [16], respectively. Illustrative examples further demonstrate the superiority of our proposed algorithm.

Problem 5.1. Consider the SOCAVE (2) which is generated in the following way: first
choose two random matrices B, C ∈ IR^{n×n} from a uniformly distribution on [−10, 10]

for every element. We compute the maximal singular value σ_{1} of B and the minimal
singular value σ_{2} of C, and let σ := min{1, σ_{2}/σ_{1}}. Next, we divide C by σ multiplied
by a random number in the interval [0, 1], and the resulting matrix is denoted as A.

Accordingly, the minimum singular values of A exceeds the maximal singular value of
B. We choose randomly b ∈ IR^{n} on [0, 1] for every element. By Algorithm 4.1 in this
paper, the resulting SOCAVE (2) is solvable. The initial point is chosen in the range
[0, 1] entry-wisely. Note that a similar way to construct the problem was given in [16].

Table 1: Numerical results for Problem 5.1

SN GN

n ares itn time maxi mini fails ares itn time maxi mini fails 100 8.618e-08 2.8 0.078 3 2 0 9.992e-08 2.8 0.349 3 2 0 200 4.901e-08 2.6 0.051 3 2 0 6.904e-10 2.9 0.134 3 2 0 300 1.574e-08 2.7 0.122 3 2 0 3.779e-09 2.9 0.231 3 2 0 400 3.041e-09 2.7 0.232 3 2 0 9.155e-08 2.7 0.326 3 2 0 500 1.778e-07 2.2 0.300 3 2 0 1.445e-07 2.6 0.421 3 2 0 600 1.385e-07 2.5 0.498 3 2 0 5.626e-08 2.8 0.844 3 2 0 700 2.578e-07 2.4 0.668 3 2 0 1.527e-08 2.6 1.334 3 2 0 800 2.356e-07 2.1 0.771 3 2 0 6.846e-08 2.6 1.905 3 2 0 900 2.420e-08 2.5 1.031 3 2 0 1.272e-09 2.7 2.685 3 2 0 1000 4.718e-08 2.5 1.193 3 2 0 1.135e-07 2.7 3.691 3 2 0 1500 2.027e-07 2.3 1.919 3 2 0 6.417e-08 2.6 13.369 3 2 0 2000 3.121e-08 2.2 3.892 3 2 0 1.015e-07 2.5 32.982 3 2 0 2500 1.565e-07 2.1 6.625 3 2 0 3.940e-08 2.5 53.510 3 2 0 3000 1.028e-07 2.3 12.340 3 2 0 1.293e-07 2.5 87.910 3 2 0

Problem 5.2. Consider the SOCAVE (2) which is generated in the following way:

choose two random matrices C, D ∈ IR^{n×n} from a uniformly distribution on [−10, 10]

for every element, and compute their singular value decompositions C := U_{1}S_{1}V_{1}^{T} and
D := U_{2}S_{2}V_{2}^{T} with diagonal matrices S_{1} and S_{2}; unitary matrices U_{1}, V_{1}, U_{2} and V_{2}.

Then, we choose randomly b, c ∈ IR^{n} on [0, 10] for every element. Next, we take a ∈ IR^{n}
by setting ai = ci + 10 for all i ∈ {1, . . . , n}, so that a ≥ b. Set A := U1Diag(a)V_{1}^{T}
and B := U_{2}Diag(b)V_{2}^{T}, where Diag(x) denotes a diagonal matrix with its i-th diagonal
element being x_{i}. The gap between the minimal singular value of A and the maximal
singular value of B is limited and can be very small. We choose randomly b ∈ IR^{n} in
[0, 10]. The initial point is chosen in the range [0, 1] entry-wisely.

Table 2: Numerical results for Problem 5.2

SN GN

n ares itn time maxi mini fails ares itn time maxi mini fails 100 2.884e-07 4.2 0.050 5 4 0 1.920e-07 4.4 0.134 5 4 0 200 4.556e-07 4.3 0.067 5 4 0 2.637e-07 4.6 0.346 5 4 0 300 2.805e-07 4.5 0.172 5 4 0 3.522e-07 4.4 0.615 5 4 0 400 2.453e-07 4.6 0.312 5 4 0 2.617e-07 4.6 0.863 5 4 0 500 1.809e-13 5.0 0.516 5 5 0 1.037e-07 4.8 1.440 5 4 0 600 1.870e-07 4.8 0.680 5 4 0 3.414e-12 5.0 2.346 5 5 0 700 2.550e-13 5.0 0.880 5 5 0 6.571e-08 4.9 3.535 5 4 0 800 2.868e-13 5.0 1.083 5 5 0 1.606e-07 4.8 5.317 5 4 0 900 7.559e-08 4.9 1.201 5 4 0 2.485e-07 4.7 7.596 5 4 0 1000 3.595e-13 5.0 1.572 5 5 0 1.662e-07 4.8 10.552 5 4 0 1500 5.412e-13 5.0 4.196 5 5 0 1.782e-11 5.0 34.400 5 5 0 2000 7.230e-13 5.0 8.962 5 5 0 2.851e-11 5.0 79.108 5 5 0 2500 8.893e-13 5.0 17.207 5 5 0 4.451e-11 5.0 146.769 5 5 0 3000 1.054e-12 5.0 29.175 5 5 0 6.119e-11 5.0 247.029 5 5 0

Problem 5.3. Consider the SOCAVE (2) which is generated in the following way: choose
two random matrices A, B ∈ IR^{n×n} from a uniformly distribution on [−10, 10] for every
element. In order to ensure that the SOCAVE (2) is solvable, we update the matrix A
by the following: let [U SV ] = svd(A). If min{S(i, i)} = 0 for i = 0, 1, · · · , n, we make
A = U (S + 0.01E)V , and then A = ^{λ}^{max}_{λ} ^{(B}^{T}^{B)+0.01}

min(A^{T}A) A. We choose randomly b ∈ IR^{n} on
[0, 10] for every element. The initial point is chosen in the range [0, 1] entry-wisely.

Problem 5.4. We consider the SOCAVE (2) which is generated the same as Problem
5.1. But, here the SOC is given by K := K^{n}^{1} × · · · × K^{n}^{r}, where n_{1} = · · · = n_{r} = ^{n}_{r}.

The above problems 5.1–5.4 are both generated randomly. Below, as suggested by the reviewer, we consider a real application problem. It is well known that the second-order cone linear complementarity problem (SOCLCP) has various applications in engineering,

Table 3: Numerical results for Problem 5.3

SN GN

n ares itn time maxi mini fails ares itn time maxi mini fails 100 7.928e-10 3.0 0.048 3 3 0 2.085e-08 3.0 0.075 3 3 0 200 9.461e-10 3.0 0.062 3 3 0 4.297e-09 3.0 0.108 3 3 0 300 2.388e-10 3.0 0.122 3 3 0 5.843e-08 2.9 0.237 3 2 0 400 5.780e-11 3.0 0.236 3 3 0 3.841e-08 2.8 0.379 3 2 0 500 1.133e-08 2.9 0.360 3 2 0 1.183e-09 2.9 0.501 3 2 0 600 2.655e-08 2.9 0.566 3 2 0 1.225e-10 3.0 0.627 3 3 0 700 2.202e-11 3.0 0.807 3 3 0 2.525e-10 3.0 0.978 3 3 0 800 8.893e-08 2.8 0.975 3 2 0 2.563e-10 3.0 1.576 3 3 0 900 1.818e-08 2.9 1.240 3 2 0 2.505e-10 3.0 2.374 3 3 0 1000 6.951e-10 3.0 1.502 3 3 0 3.247e-10 3.0 3.367 3 3 0 1500 4.225e-08 2.9 2.482 3 2 0 4.245e-10 3.0 11.625 3 3 0 2000 6.979e-08 2.6 4.683 3 2 0 1.705e-09 3.0 27.704 3 3 0 2500 9.459e-10 2.9 9.441 3 2 0 1.376e-09 3.0 53.306 3 3 0 3000 5.624e-08 2.9 15.765 3 2 0 1.943e-08 2.8 91.226 3 2 0

control, finance, robust optimization and combinatorial optimization since the KKT sys-
tem of a second-order cone programming can be recastas an SOCLCP. In general, the
SOCLCP is to find x, y ∈ R^{n} such that

M x + P y = c, x ∈ K, y ∈ K, x^{T}y = 0, (18)
where M, P ∈ R^{n×m} are given matrices and c ∈ R^{n} is given vector. From [16, Theorem
1.1], we know that the SOCLCP (18) is equivalent to the SOCAVE (2). In view of this,
the next experiment is on this case.

Problem 5.5. Consider the SOCLCP with P = −I, which is generated in the following
way: First, we generate a matrix B and a vector b as those given in Problem 5.1. Then,
let d be a random number in [0, 1]. We set M := BB^{T}+(1+d)I and c := 0.5(M (b+|b|)+

|b| − b) to ensure the solvability of the SOCLCP. We test the above SOCLCP by casting it into an SOCAVE according to [16, Theorem 1.1], i.e., we implement the corresponding SOCAVE with A = M + I, B = M − I and b = 2c. Moreover, the initial point is chosen in the range [0, 1] entry-wisely.

In our experiments, every set of the simulations for every problem is randomly gener-
ated ten times, and the numerical results are listed in Tables 1–5, respectively. In Tables
1–5, n denotes the size of testing problem; ares denotes the average value of kH(z^{k})k
when the test stops; itn denotes the average value of the iteration numbers; time denotes

Table 4: Numerical results for Problem 5.4

SN GN

n r ares itn time maxit minit fails ares itn time maxit minit fails 2 9.933e-08 2.4 1.318 3 2 0 2.995e-10 2.9 3.627 3 2 0 4 1.174e-07 2.5 1.245 3 2 0 1.594e-08 2.6 3.106 3 2 0 1000 5 1.056e-07 2.4 1.293 3 2 0 9.657e-08 2.7 3.115 3 2 0 10 3.380e-13 5.0 1.791 5 5 0 3.971e-08 2.5 3.218 3 2 0 20 3.360e-13 5.0 2.103 5 5 0 5.291e-08 2.7 3.181 3 2 0 2 1.971e-08 2.6 5.084 3 2 0 2.494e-08 2.6 28.888 3 2 0 4 1.047e-07 2.3 4.270 3 2 0 5.363e-08 2.6 29.002 3 2 0 2000 5 1.257e-07 2.5 4.813 3 2 0 1.360e-08 2.8 29.055 3 2 0 10 6.689e-13 5.0 10.463 5 5 0 1.360e-08 2.8 29.055 3 3 0 20 6.653e-13 5.0 11.255 5 5 0 1.360e-08 2.8 29.055 3 4 0 2 1.560e-07 2.1 12.312 3 2 0 2.496e-07 2.5 90.699 3 2 0 4 1.162e-07 2.5 14.457 3 2 0 1.609e-07 2.3 89.813 3 2 0 3000 5 3.156e-07 2.2 12.995 3 2 0 6.872e-0 2.4 88.921 3 2 0 10 9.922e-13 5.0 32.011 5 5 0 1.688e-07 2.4 90.041 3 2 0 20 1.016e-12 5.0 33.877 5 5 0 1.411e-08 2.5 88.949 3 2 0

the average value of the CPU time in seconds; maxit and minit denote the maximal value and the minimal value of the iteration numbers, respectively; and f ails denotes that the times of test is failed. From the numerical results that are presented in Tables 1–5, it is easy to see that the proposed smoothing Newton method is effective for solving all the simulated SOCAVE problems. For the SOCLCP, although the smoothing New- ton method performs slightly less than the generalized Newton method, the difference is marginal. To sum up, both approaches are competitive and can be employed to solve SOCAVE.

### 5.2 Numerical Comparisons with different values of p

In this subsection, we observe the numerical comparison of Algorithm 4.1 with different
values of p. In particular, we consider the performance profile which is introduced in
[44] as a means. In other words, we regard Algorithm 4.1 corresponding to different
p = 1.1, 2, 3, 10, 20, 80 as a solver, and assume that there are n_{s} solvers and n_{q} test
problems from the test set P which is generated randomly. We are interested in using
the computing time as performance measure for Algorithm 4.1 with different p. For each
problem q and solver s, let

f_{q,s} = computing time required to solve problem q by solver s.

We employ the performance ratio

r_{q,s}:= f_{q,s}

min{f_{q,s}: s ∈ S},