### An approximate lower order penalty approach for solving second-order cone linear complementarity problems

Zijun Hao ^{1}

School of Mathematics and Information Science North Minzu University

Yinchuan 750021, China

Chieu Thanh Nguyen ^{2}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan

Jein-Shan Chen ^{3}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan

March 24, 2020

(revised on August 12, 2021)

Abstract Based on a class of smoothing approximations to projection function onto second-order cone, an approximate lower order penalty approach for solving second-order cone linear complementarity problems (SOCLCPs) is proposed, and four kinds of spe- cific smoothing approximations are considered. In light of this approach, the SOCLCP is approximated by asymptotic lower order penalty equations with penalty parameter and smoothing parameter. When the penalty parameter tends to positive infinity and the smoothing parameter monotonically decreases to zero, we show that the solution se- quence of the asymptotic lower order penalty equations converges to the solution of the SOCLCP at an exponential rate under a mild assumption. A corresponding algorithm is constructed and numerical results are reported to illustrate the feasibility of this ap- proach. The performance profile of four specific smoothing approximations is presented, and the generalization of two approximations are also investigated.

1E-mail: zijunhao@126.com. The author’s work is supported by the National Natural Science Foun- dation of China (Nos. 11661002,11871383), the Natural Science Fund of Ningxia (No. 2020AAC03236), the First-class Disciplines Foundation of Ningxia (No. NXYLXK2017B09).

2E-mail: thanhchieu90@gmail.com.

3Corresponding author. E-mail: jschen@math.ntnu.edu.tw. The author’s work is supported by Ministry of Science and Technology, Taiwan.

Keywords Second-order cone; linear complementarity problem; lower order penalty approach; exponential convergence rate

MC codes 90C25; 90C30; 90C33

### 1 Introduction

This paper targets the following second-order cone linear complementarity problem (SO-
CLCP), which is to find x ∈ IR^{n}, such that

x ∈ K, Ax − b ∈ K, x^{T}(Ax − b) = 0, (1)
where A is an n × n matrix, b is a vector in IR^{n}, and K is the Cartesian product of
second-order cones (SOCs), also called Lorentz cones [7,18]. In other words,

K := K^{n}^{1} × · · · × K^{n}^{r} (2)

with r, n_{1}, . . . , n_{r}≥ 1, n_{1}+ · · · + n_{r} = n and

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

where k · k denotes the Euclidean norm and (x_{1}, x_{2}) := (x_{1}, x^{T}_{2})^{T}. Note that K^{1} denotes
the set of nonnegative real numbers IR_{+}. The SOCLCP, as an extension of the linear
complementarity problem (LCP), has a wide range of applications in linear and quadratic
programming problems, computer science, game theory, economics, finance, engineering,
and network equilibrium problems [3, 15, 17,26, 27, 30].

During the past several years, there are many methods proposed for solving the SO-
CLCPs (1)-(2), including the interior-point method [1, 28, 32], the smoothing Newton
method [14, 19, 24], the smoothing-regularization method [23], the semismooth New-
ton method [25, 33], the merit function method [5, 10, 12], and the matrix splitting
method [22,41] etc. Although the effectiveness of some methods has been improved sub-
stantially in recent years, the fact remains that there still have many complementarity
problems require efficient and accurate numerical methods. The penalty methods are
well-known for solving constrained optimization problems which possess many nice prop-
erties. More specifically, the l_{1} exact penalty function method and lower order penalty
function method are known as the approaches which hold many nice properties and at-
tracts much attention [2,20,29,34,39,40]. The smoothing of the exact penalty methods
are also proposed [35, 37,38]. Besides, Wang and Yang [36] focus on the power of lower
order penalty function, and propose a power penalty method for solving LCP based on
approximating the LCP by nonlinear equations. It shows that under some mild assump-
tions, the solution sequence of the nonlinear equations converges to the solution of the
LCP at an exponential rate when the penalty parameter tends to positive infinity. Based

on the method in [36], Hao et al. [21] propose a power penalty method for solving the
SOCLCP with a single K = K^{n}, i.e.,

x ∈ K^{n}, Ax − b ∈ K^{n}, x^{T}(Ax − b) = 0, (3)
where A ∈ IR^{n×n} and b ∈ IR^{n}. In particular, they consider the power penalty equations:

Ax − α[x]^{1/k}_{−} = b, (4)

where k ≥ 1 and α ≥ 1 are parameters,

[x]^{1/k}_{−} = [λ_{1}(x)]^{1/k}_{−} u^{(1)}_{x} + [λ_{2}(x)]^{1/k}_{−} u^{(2)}_{x}

with [t]− = max{0, −t} and the spectral decomposition (will be introduced later in (5)).

Under a mild assumption of matrix A, as α → +∞, the solution sequence of (4) converges to the solution of the SOCLCP (3) at an exponential rate.

In this paper, we further enhance improvement and extension of the method and the
problem studied in [21]. We first generalize [x]^{1/k}_{−} in (4) to general lower order penalty
function [x]^{σ}_{−} with σ ∈ (0, 1], then focus on a class of approximate function to [x]^{σ}_{−} for
solving the general SOCLCP (1) instead of the SOCLCP (3) with single SOC constraint.

In addition, we construct a class of functions Φ^{−}(µ, x)^{σ} to approximate [x]^{σ}_{−} as µ → 0^{+}.
Four kinds of specific smoothing approximations are studied. Theoretically, we prove
that the solution sequence of the approximating lower order penalty equations converge
to the solution of the SOCLCP (1) at an exponential rate O(α^{−1/σ}) when α → +∞

and µ → 0^{+}. This generalizes all its counterparts in the literature. Moreover, a cor-
responding algorithm is constructed and numerical results are also reported to examine
the feasibility of the proposed method. The performance profile of those specific smooth-
ing approximations is presented, and the generalization of two approximations are also
investigated.

This paper is organized as follows: In Section 2, we review some properties related to the single SOC which is the basis for our subsequent analysis. In Section 3, a class of approximation functions for lower order penalty function is constructed, and four kinds of specific smoothing approximations are investigated. In Section 4, we study the ap- proximating lower order penalty equations for solving the SOCLCP (1), and prove the convergence analysis. In Section 5, a corresponding algorithm is constructed and the pre- liminary numerical experiments are presented. The performance profiles of the considered four specific smoothing approximations and the generalization of two approximations are also considered. Finally, we draw the conclusion in Section 6.

For simplicity, we denote the interior of single SOC K^{n} by int(K^{n}). For any x, y in
IR^{n}, we write x K^{n} y if x − y ∈ K^{n} and write x K^{n} y if x − y ∈ int(K^{n}). In other
words, we have x K^{n} 0 if and only if x ∈ K^{n}, and x K^{n} 0 if and only if x ∈ int(K^{n}).

We usually denote (x, y) := (x^{T}, y^{T})^{T} for the concatenation of two column vectors x, y
for simplicity. The notation k · k_{p} denotes the usual l_{p}-norm on IR^{n} for any p ≥ 1. In
particular, it is Euclidean norm k · k when p = 2.

### 2 Preliminary results

In this section, we first recall some basic concepts and preliminary results related to a
single SOC K = K^{n} that will be used in the subsequent analysis. All of the analysis are
then carried over to the general structure K (2). For any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}, y =
(y_{1}, y_{2}) ∈ IR × IR^{n−1}, their Jordan product [7,18] is defined as

x ◦ y := (hx, yi , y_{1}x_{2}+ x_{1}y_{2}).

We write x + y to mean the usual componentwise addition of vectors and x^{2} to mean
x ◦ x. The identity element under this product is e = (1, 0, . . . , 0)^{T} ∈ IR^{n}. It is known
that x^{2} ∈ K^{n} for all x ∈ IR^{n}. Moreover, if x ∈ K^{n}, then there is a unique vector in K^{n},
denoted by x^{1}^{2}, such that (x^{1}^{2})^{2} = x^{1}^{2}◦ x^{1}^{2} = x. For any x ∈ IR^{n}, we define x^{0} = e if x 6= 0.

For any integer k ≥ 1, we recursively define the powers of element as x^{k} = x ◦ x^{k−1}, and
define x^{−k} = (x^{k})^{−1} if x ∈ int(K^{n}). The Jordan product is not associative for n > 2, but
it is power associated, i.e., x ◦ (x ◦ x) = (x ◦ x) ◦ x. Thus, for any positive integer p, the
form x^{p} is definite, and x^{m+n} = x^{m} ◦ x^{n} for all positive integer m and n. Note that K^{n}
is not closed under the Jordan product for n > 2.

In the following, we recall the expression of the spectral decomposition of x with
respect to SOC, see [5,6, 7, 8, 10, 11, 12,18, 19, 33]. For x = (x1, x2) ∈ 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} , (5)
where for i = 1, 2,

λ_{i}(x) = x_{1}+ (−1)^{i}kx_{2}k, u^{(i)}_{x} =
( 1

2(1, (−1)^{i x}_{kx}^{2}

2k) if kx2k 6= 0,

1

2(1, (−1)^{i}w) if kx2k = 0, (6)
with w ∈ IR^{n−1} being any unit vector. 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 x2 6= 0.

Some basic properties of the spectral decomposition in the Jordan algebra associated with SOC are stated as below, whose proofs can be found in [6, 7,18, 19].

Proposition 2.1. For any x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} with the spectral values λ_{1}(x), λ_{2}(x)
and spectral vectors u^{(1)}x , u^{(2)}x given as (6), we have:

(a) u^{(1)}x ◦ u^{(2)}x = 0 and u^{(i)}x ◦ u^{(i)}x = u^{(i)}x , ku^{(i)}x k^{2} = 1/2 for i = 1, 2.

(b) λ_{1}(x), λ_{2}(x) are nonnegative (positive) if and only if x ∈ K^{n} (x ∈ int(K^{n})).

(c) For any x ∈ IR^{n}, x K^{n} 0 if and only if hx, yi ≥ 0 for all y K^{n} 0.

The spectral decomposition (5)-(6) and the Proposition 2.1 indicate that x^{k} can be
described as x^{k} = λ^{k}_{1}(x)u^{(1)}x + λ^{k}_{2}(x)u^{(2)}x . For any x ∈ IR^{n}, let [x]_{+} denote the projection
of x onto 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, by
Proposition 2.1, the dual cone of K^{n} being itself, i.e., (K^{n})^{∗} = K^{n}. Due to the special
structure of K^{n}, the explicit formula of projection of x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} onto K^{n}
is obtained in [14,17, 19] as below

[x]_{+}=

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

where u =

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

2

x1+kx2k 2

x2

kx2k

# .

Similarly, the expression of [x]− can be written out as

[x]− =

0 if x ∈ K^{n},

−x if x ∈ −K^{n},
v otherwise,

where v =

"

−^{x}^{1}^{−kx}_{2} ^{2}^{k}

x1−kx_{2}k
2

x2

kx2k

# .

It is easy to verify that x = [x]+− [x]− and

[x]_{+}= [λ_{1}(x)]_{+}u^{(1)}_{x} + [λ_{2}(x)]_{+}u^{(2)}_{x} , [x]−= [λ_{1}(x)]−u^{(1)}_{x} + [λ_{2}(x)]−u^{(2)}_{x} ,

where [α]_{+} = max{0, α} and [α]− = max{0, −α} for α ∈ IR. Thus, it can be seen that
[x]_{+}, [x]− ∈ K^{n} and [x]_{+}◦ [x]− = 0.

Putting these analyses into a single SOC K^{n}^{i}, i = 1, . . . , r in (2), we can extend them
to the general case K = K^{n}^{1} × · · · × K^{n}^{r}. More specifically, for any x = (x_{1}, . . . , x_{r}) ∈
IR^{n}^{1} × · · · × IR^{n}^{r}, y = (y_{1}, . . . , y_{r}) ∈ IR^{n}^{1} × · · · × IR^{n}^{r}, their Jordan product is defined as

x ◦ y := (x1◦ y1, . . . , xr◦ yr).

Let [x]_{+}, [x]− respectively denote the projection of x onto K and the projection of −x
onto the dual cone K^{∗} = K, then

[x]_{+}:= ([x_{1}]_{+}, . . . , [x_{r}]_{+}), [x]−:= ([x_{1}]−, . . . , [x_{r}]−), (7)
where [x_{i}]_{+}, [x_{i}]_{−} for i = 1, . . . , r respectively denote the projection of x_{i} onto the single
SOC K^{n}^{i} and the projection of −x_{i} onto (K^{n}^{i})^{∗}.

### 3 Approximation of projection function with power

This section is devoted to presenting a way to generate smoothing functions for the plus
function [t]_{+} = max{0, t} and minus function [t]− = max{0, −t} via convolution which
was proposed by Chen and Mangasarian [4]. First, we consider the piecewise continuous

function d(t) with finite number of pieces, which is a density (kernel) function. In other words, it satisfies

d(t) ≥ 0 and

Z +∞

−∞

d(t)dt = 1. (8)

Next, we define ˆs(µ, t) := ^{1}_{µ}d

t µ

, where µ is a positive parameter. If R+∞

−∞ |t| d(t)dt <

+∞, then a smoothing approximation for [t]_{+} is formed. In particular,
φ^{+}(µ, t) =

Z +∞

−∞

(t − s)+ˆs(µ, s)ds = Z t

−∞

(t − s)ˆs(µ, s)ds ≈ [t]+. (9)
The following proposition states the properties of φ^{+}(µ, t), whose proofs can be found in
[4, Proposition 2.2].

Proposition 3.1. Let d(t) be a density function satisfying (8) and ˆs(µ, t) = ^{1}_{µ}d

t µ

with positive parameter µ. If d(t) is piecewise continuous with finite number of pieces and R+∞

−∞ |t| d(t)dt < +∞. Then, the function φ^{+}(µ, t) defined by (9) possesses the following
properties.

(a) φ^{+}(µ, t) is continuously differentiable.

(b) −D_{2}µ ≤ φ^{+}(µ, t) − [t]_{+} ≤ D_{1}µ, where
D_{1} =

Z 0

−∞

|t|d(t)dt and D_{2} = max

Z +∞

−∞

td(t)dt, 0

.

(c) _{∂t}^{∂}φ^{+}(µ, t) is bounded satisfying 0 ≤ _{∂t}^{∂}φ^{+}(µ, t) ≤ 1.

From Proposition 3.1(b), we have lim

µ→0^{+}

φ^{+}(µ, t) = [t]_{+}

under the assumptions of this proposition. Applying the above way of generating smooth- ing function to approximate [t]− = max{0, −t}, which appears in equation (4), we also achieve a smoothing approximation as follows:

φ^{−}(µ, t) =
Z −t

−∞

(−t − s)ˆs(µ, −s)ds = Z +∞

t

(s − t)ˆs(µ, s)ds ≈ [t]−. (10)
Similar to Proposition3.1, we have the below properties for φ^{−}(µ, t).

Proposition 3.2. Let d(t) and ˆs(µ, t) be as in Proposition3.1with the same assumptions.

Then, the function φ^{−}(µ, t) defined by (10) possesses the following properties.

(a) φ^{−}(µ, t) is continuously differentiable.

(b) −D_{2}µ ≤ φ^{−}(µ, t) − [t]− ≤ D_{1}µ, where
D_{1} =

Z +∞

0

|t|d(t)dt and D_{2} = max

Z +∞

−∞

td(t)dt, 0

.

(c) _{∂t}^{∂}φ^{−}(µ, t) is bounded satisfying −1 ≤ _{∂t}^{∂}φ^{−}(µ, t) ≤ 0.

Similar to Proposition 3.1, we also obtain lim_{µ→0}^{+}φ^{−}(µ, t) = [t]−. Therefore, in view
of Proposition 3.1 and 3.2, we know that φ^{+}(µ, t) defined by (9) and φ^{−}(µ, t) defined
by (10), are the smoothing functions of [t]_{+} and [t]−, respectively. Accordingly, using
the continuity of compound function and φ^{+}(µ, t) ≥ 0, φ^{−}(µ, t) ≥ 0, we can generate
approximate function (not necessarily smooth) for [t]^{σ}_{+} and [t]^{σ}_{−}, see below lemma.

Lemma 3.1. Under the assumptions of Proposition 3.1, let φ^{+}(µ, t), φ^{−}(µ, t) be the
smoothing functions of [t]_{+}, [t]−, defined by (9) and (10) respectively. Then, for any
σ > 0, we have

(a) lim

µ→0^{+}φ^{+}(µ, t)^{σ} = [t]^{σ}_{+},
(b) lim

µ→0^{+}

φ^{−}(µ, t)^{σ} = [t]^{σ}_{−}.

By modifying the smoothing functions used in [4,9,31], we have four specific smooth-
ing functions for [t]_{−} as well:

φ^{−}_{1}(µ, t) = −t + µ ln
1 + e^{µ}^{t}

, (11)

φ^{−}_{2}(µ, t) =

0 if t ≥ ^{µ}_{2},

1

2µ −t + ^{µ}_{2}2

if − ^{µ}_{2} < t < ^{µ}_{2},

−t if t ≤ −^{µ}_{2},

(12)

φ^{−}_{3}(µ, t) = p4µ^{2}+ t^{2}− t

2 , (13)

φ^{−}_{4}(µ, t) =

0 if t > 0,

t^{2}

2µ if − µ ≤ t ≤ 0,

−t − ^{µ}_{2} if t < −µ,

(14)

where the corresponding kernel functions are
d_{1}(t) = e^{t}

(1 + e^{t})^{2},

d_{2}(t) = 1 if − ^{1}_{2} ≤ t ≤ ^{1}_{2},
0 otherwise,
d_{3}(t) = 2

(t^{2}+ 4)^{3}^{2},

d_{4}(t) = 1 if − 1 ≤ t ≤ 0,
0 otherwise.

Figure 1: Graphs of [t]− and φ^{−}_{i} (µ, t), i = 1, 2, 3, 4 with µ = 0.1.

For those specific functions (11)-(14), they certainly obey Proposition3.2and Lemma
3.1. The graphs of [t]− and φ^{−}_{i} (µ, t), i = 1, 2, 3, 4 with µ = 0.1 are depicted in Figure 1.

From Figure 1, we see that, for a fixed µ > 0, the function φ^{−}_{2}(µ, t) seems the one
which best approximate the function [t]_{−} among all φ^{−}_{i} (µ, t), i = 1, 2, 3, 4. Indeed, for a
fixed µ > 0 and all t ∈ IR, we have

φ^{−}_{3}(µ, t) ≥ φ^{−}_{1}(µ, t) ≥ φ^{−}_{2}(µ, t) ≥ [t]− ≥ φ^{−}_{4}(µ, t). (15)
Furthermore, we shall show that φ^{−}_{2}(µ, t) is the function closest to [t]− in the sense of the
infinite norm. For any fixed µ > 0, it is clear that

|t|→∞lim

φ^{−}_{i} (µ, t) − [t]−

= 0, i = 1, 2, 3.

The functions φ^{−}_{i} (µ, t) − [t]−, i = 1, 3 have no stable point but unique non-differentiable
point t = 0, and φ^{−}_{2}(µ, t) − [t]− is non-zero only on the interval (−µ/2, µ/2) with
maxt∈(−µ/2,µ/2)

φ^{−}_{2}(µ, t) − [t]_{−}

= φ^{−}_{2}(µ, 0). These imply that
max

t∈IR

φ^{−}_{i} (µ, t) − [t]_{−}
=

φ^{−}_{i} (µ, 0)

, i = 1, 2, 3.

Since φ^{−}_{1}(µ, 0) = (ln 2)µ ≈ 0.7µ, φ^{−}_{2}(µ, 0) = µ/8, φ^{−}_{3}(µ, 0) = µ, we obtain
kφ^{−}_{1}(µ, t) − [t]−k∞ = (ln 2)µ,

kφ^{−}_{2}(µ, t) − [t]−k∞ = µ/8,
kφ^{−}_{3}(µ, t) − [t]−k∞ = µ.

On the other hand, it is obvious that max_{t∈IR}

φ^{−}_{4}(µ, t) − [t]−

= µ/2, which says
kφ^{−}_{4}(µ, t) − [t]−k∞ = µ/2.

Figure 2: Graphs of φ^{−}_{i} (µ, t), i = 1, 2, 3, 4 with different µ.

In summary, we have

kφ^{−}_{3}(µ, t) − [t]−k∞> kφ^{−}_{1}(µ, t) − [t]−k∞ > kφ^{−}_{4}(µ, t) − [t]−k∞ > kφ^{−}_{2}(µ, t) − [t]−k∞. (16)
The orderings of (15) and (16) indicate the behavior of φ^{−}_{i} (µ, t), i = 1, 2, 3, 4 for fixed
µ > 0. When taking µ → 0^{+}, we know lim_{µ→0}^{+}φ^{−}_{i} (µ, t) = [t]−, i = 1, 2, 3, 4 and φ^{−}_{2}(µ, t)
is the closest to [t]−, which can be verified by geometric views depicted as in Figure 2.

Remark 3.1. For any µ > 0, σ > 0 and continuously differentiable φ^{−}(µ, t) defined
by (10), it can be easily seen that, φ^{−}(µ, t)^{σ} is continuous function about t, but may
not be differentiable. For example, φ^{−}_{1}(µ, t)^{σ}, φ^{−}_{3}(µ, t)^{σ} are continuously differentiable,
but φ^{−}_{2}(µ, t)^{σ}, φ^{−}_{4}(µ, t)^{σ} are not continuously differentiable for σ = 1/2 since the non-
differentiable points are t = µ/2 and t = 0 respectively. Their geometric views are
depicted in Figure 3.

With the aforementioned discussions, for any x = (x_{1}, . . . , x_{r}) ∈ IR^{n}^{1} × · · · × IR^{n}^{r},
we are ready to show how to construct a smoothing function for vectors [x]_{+} and [x]_{−}
associated with K = K^{n}^{1}× · · · × K^{n}^{r}. We start by constructing a smoothing function for
vectors [x_{i}]_{+}, [x_{i}]− on a single SOC K^{n}^{i}, i = 1, . . . , r since [x]_{+} and [x]− are shown as (7).

First, given smoothing functions φ^{+}, φ^{−} in (9),(10) and x_{i} ∈ IR^{n}^{i}, i = 1, . . . , r, we define
vector-valued function Φ^{+}_{i} , Φ^{−}_{i} : IR_{++}× IR^{n}^{i} → IR^{n}^{i}, i = 1, . . . , r as

Φ^{+}_{i} (µ, x_{i}) := φ^{+}(µ, λ_{1}(x_{i})) u^{(1)}_{x}

i + φ^{+}(µ, λ_{2}(x_{i})) u^{(2)}_{x}

i, (17)

Figure 3: Graphs of φ^{−}_{i} (µ, t)^{σ}, i = 1, 2, 3, 4 with different µ and σ = 1/2.

Φ^{−}_{i} (µ, x_{i}) := φ^{−}(µ, λ_{1}(x_{i})) u^{(1)}_{x}

i + φ^{−}(µ, λ_{2}(x_{i})) u^{(2)}_{x}

i, (18)

where µ ∈ IR_{++}is a parameter, λ_{1}(x_{i}), λ_{2}(x_{i}) are the spectral values, and u^{(1)}xi , u^{(2)}xi are the
spectral vectors of x_{i}. Consequently, Φ^{+}_{i} (µ, x_{i}), Φ^{−}_{i} (µ, x_{i}) are also smooth on IR_{++}× IR^{n}^{i}
[8]. Moreover, it is easy to assert that

lim

µ→0^{+}Φ^{+}_{i} (µ, x_{i}) = [λ_{1}(x_{i})]_{+}u^{(1)}_{x}_{i} + [λ_{2}(x_{i})]_{+}u^{(2)}_{x}_{i} = [x_{i}]_{+}, (19)
lim

µ→0^{+}

Φ^{−}_{i} (µ, x_{i}) = [λ_{1}(x_{i})]−u^{(1)}_{x}

i + [λ_{2}(x_{i})]−u^{(2)}_{x}

i = [x_{i}]−, (20)

which means each function Φ^{+}_{i} (µ, x_{i}), Φ^{−}_{i} (µ, x_{i}) serves as a smoothing function of [x_{i}]_{+}, [x_{i}]−

associated with single SOC K^{n}^{i}, i = 1, . . . , r, respectively. Due to Lemma 3.1, Remark
3.1 and from definition of Φ^{+}_{i} (µ, x_{i}), Φ^{−}_{i} (µ, x_{i}) in (17), (18), it is not difficult to verify
that for any σ > 0, the below two functions

Φ^{+}_{i} (µ, x_{i})^{σ} := φ^{+}(µ, λ_{1}(x_{i}))^{σ}u^{(1)}_{x}

i + φ^{+}(µ, λ_{2}(x_{i}))^{σ}u^{(2)}_{x}

i , (21)

Φ^{−}_{i} (µ, x_{i})^{σ} := φ^{−}(µ, λ_{1}(x_{i}))^{σ}u^{(1)}_{x}

i + φ^{−}(µ, λ_{2}(x_{i}))^{σ}u^{(2)}_{x}

i (22)

are continuous functions approximate to [x_{i}]^{σ}_{+} and [x_{i}]^{σ}_{−}, respectively. In other words,
lim

µ→0^{+}

Φ^{+}_{i} (µ, x_{i})^{σ} = [λ_{1}(x_{i})]^{σ}_{+}u^{(1)}_{x}

i + [λ_{2}(x_{i})]^{σ}_{+}u^{(2)}_{x}

i = [x_{i}]^{σ}_{+},
lim

µ→0^{+}Φ^{−}_{i} (µ, x_{i})^{σ} = [λ_{1}(x_{i})]^{σ}_{−}u^{(1)}_{x}_{i} + [λ_{2}(x_{i})]^{σ}_{−}u^{(2)}_{x}_{i} = [x_{i}]^{σ}_{−}.

Now we construct smoothing function for vectors [x]_{+}and [x]−associated with general
cone (2). To this end, we define vector-valued function Φ^{+}, Φ^{−} : IR_{++}× IR^{n}→ IR^{n} as

Φ^{+}(µ, x) := Φ^{+}_{1}(µ, x_{1}), . . . , Φ^{+}_{r}(µ, x_{r}) , (23)
Φ^{−}(µ, x) := Φ^{−}_{1}(µ, x_{1}), . . . , Φ^{−}_{r}(µ, x_{r}) , (24)
where Φ^{+}_{i} (µ, xi), Φ^{−}_{i} (µ, xi), i = 1, . . . , r are defined by (17), (18), respectively. Therefore,
from (19), (20) and (7), Φ^{+}(µ, x), Φ^{−}(µ, x) serves as a smoothing function for [x]_{+}, [x]−

associated with K = K^{n}^{1} × · · · × K^{n}^{r}, respectively. At the same time, from (21), (22),
Φ^{+}(µ, x)^{σ} := Φ^{+}_{1}(µ, x_{1})^{σ}, . . . , Φ^{+}_{r}(µ, x_{r})^{σ} , (25)
Φ^{−}(µ, x)^{σ} := Φ^{−}_{1}(µ, x_{1})^{σ}, . . . , Φ^{−}_{r}(µ, x_{r})^{σ}

(26)
are continuous functions approximate to [x]^{σ}_{+} and [x]^{σ}_{−}, respectively.

In light of this idea, we establish an approximating lower order penalty equations for solving SOCLCP (1), which will be described in next section. To end this section, we present a technical lemma for subsequent needs.

Lemma 3.2. Suppose that Φ^{+}(µ, x) and Φ^{−}(µ, x) are defined by (23), (24), respectively,
and Φ^{+}(µ, x)^{σ} and Φ^{−}(µ, x)^{σ} are defined for any σ > 0 as in (25), (26), respectively.

Then, the following results hold.

(a) Both Φ^{+}(µ, x) and Φ^{−}(µ, x) belong to K,
(b) Both Φ^{+}(µ, x)^{σ} and Φ^{−}(µ, x)^{σ} belong to K .

Proof. (a) For any x_{i} ∈ IR^{n}^{i}, i = 1, . . . , r, since φ^{+}(µ, λ_{k}(x_{i})) ≥ 0, φ^{−}(µ, λ_{k}(x_{i})) ≥ 0 for
k = 1, 2 from (9), (10), we have Φ^{+}_{i} (µ, x_{i}), Φ^{−}_{i} (µ, x_{i}) ∈ K^{n}^{i} according to the definition
(17), (18). Therefore, the conclusion holds due to the definitions (23), (24) and (2).

(b) From part (a) and knowing σ > 0, we have φ^{+}(µ, λk(xi))^{σ} ≥ 0, φ^{−}(µ, λk(xi))^{σ} ≥ 0,
k = 1, 2. Applying (25) and (26), the desired result follows. 2

### 4 Approximate lower order penalty approach and convergence analysis

In this section, we propose an approximate lower order penalty approach for solving SOCLCP (1). To this end, we consider the approximate lower order penalty equations (LOPEs):

Ax − αΦ^{−}(µ, x)^{σ} = b, (27)

where σ ∈ (0, 1] is a given power parameter, α ≥ 1 is a penalty parameter and Φ^{−}(µ, x)^{σ}
is defined as (26). Throughout this section, x_{µ,α} means the solution of (27), and corre-
sponding to the structure of (2), we denote

x_{µ,α} = ((x_{µ,α})_{1}, . . . , (x_{µ,α})_{r}) ∈ IR^{n}^{1} × · · · × IR^{n}^{r}. (28)
For simplicity and without causing confusion, we always denote the spectral values and
spectral vectors of (x_{µ,α})_{i}, i = 1, . . . , r as λ_{k} := λ_{k}((x_{µ,α})_{i}), u^{(k)} := u^{(k)}_{(x}

µ,α)i for k = 1, 2.

Accordingly, [λ_{k}]− := [λ_{k}((x_{µ,α})_{i})]− and φ^{−}(µ, λ_{k}) := φ^{−}(µ, λ_{k}((x_{µ,α})_{i})), k = 1, 2 for
instance. Note that for special case σ = 1, the nonlinear function in (27) is always
smooth.

Note that the equations (27) are penalized equations corresponding to the SOCLCP
(1) because the penalty term αΦ^{−}(µ, x)^{σ} penalizes the ‘negative part’ of x when µ → 0^{+}.
By Lemma 3.2 and from equations (27), it is easy to see that Ax_{µ,α} − b ∈ K (noting
αΦ^{−}(µ, x_{µ,α})^{σ} ∈ K). Our goal is to show that the solution sequence {x_{µ,α}} converges to
the solution of SOCLCP (1) when α → +∞ and µ → 0^{+}. In order to achieve this, we
need to make the assumption for matrix A as below.

Assumption 4.1. The matrix A is positive definite, but not necessarily symmetric, i.e., there exists a constant a0 > 0, such that

y^{T}Ay ≥ a_{0}kyk^{2}, ∀y ∈ IR^{n}. (29)
This assumption just implies that ¯A = (A + A^{T})/2 is symmetric positive definite with
a_{0} = λ_{min}( ¯A) since y^{T}Ay = y^{T}Ay. Here is an example of A. Let¯

A = 2 −1 3 1

,

it is easy to see that matrix A is positive definite satisfying (29), but not symmetric.

Under Assumption 4.1, the SOCLCP (1) has a unique solution and the LOPEs (27) also has a unique solution, see for more details in [17, 21].

Proposition 4.1. For any α ≥ 1, σ ∈ (0, 1] and sufficiently small µ, the solution of the
LOPEs (27) is bounded, i.e., there exists a positive constant M , independent of x_{µ,α}, µ, α
and σ, such that kx_{µ,α}k ≤ M .

Proof. By multiplying x_{µ,α} on both sides of (27), we observe that
x^{T}_{µ,α}Axµ,α = x^{T}_{µ,α}b + αx^{T}_{µ,α}Φ^{−}(µ, xµ,α)^{σ} =

r

X

i=1

(xµ,α)^{T}_{i} bi+ α(xµ,α)^{T}_{i} Φ^{−}_{i} (µ, (xµ,α)i)^{σ}
(30)
by (26),(28) and denoting b = (b_{1}, . . . , b_{r}) ∈ IR^{n}^{1}× · · · × IR^{n}^{r}. For any (x_{µ,α})_{i}, i = 1, . . . , r,
to proceed, we consider three cases to evaluate the term

Ξ_{i} := (x_{µ,α})^{T}_{i} b_{i} + α(x_{µ,α})^{T}_{i} Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} ≤ kx_{µ,α}k (kbk + 1) . (31)

Case 1: (x_{µ,α})_{i} ∈ K^{n}^{i}. From Cauchy-Schwarz inequality, spectral decomposition of
(x_{µ,α})_{i}, and the fact that the norm of the piece component is less than that of the whole
vector, we have

Ξi ≤ k(xµ,α)ik kbik + αkΦ^{−}_{i} (µ, (xµ,α)i)^{σ}k

≤ kx_{µ,α}k kbk + αkφ^{−}(µ, λ_{1})^{σ}u^{(1)}+ φ^{−}(µ, λ_{2})^{σ}u^{(2)}k

≤ kx_{µ,α}k kbk +√

2αφ^{−}(µ, 0)^{σ} ,

(32)

where the second inequality holds by the definition of Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} as in (22), and
the last inequality holds by the triangle inequality, the nonnegativity of φ^{−}(µ, 0)^{σ} from
(10) and the monotone decreasing of φ^{−}(µ, t) about t since 0 ≤ λ_{1} ≤ λ_{2} in this case.

Now, applying Lemma3.1, we have lim_{µ→0}^{+}φ^{−}(µ, 0)^{σ} = 0. This means, for any penalty
parameter α, there exists a positive real number ν, such that √

2αφ^{−}(µ, 0)^{σ} ≤ 1 for all
µ ∈ (0, ν]. Therefore, from (32), we obtain the conclusion (31).

Case 2: (x_{µ,α})_{i} ∈ −K^{n}^{i}. In light of Lemma 3.2, we know Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} ∈ K^{n}^{i}, and
hence

(x_{µ,α})^{T}_{i} Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} ≤ 0.

Thus, we have Ξ_{i} ≤ (x_{µ,α})^{T}_{i} b_{i} ≤ k(x_{µ,α})_{i}kkb_{i}k ≤ kx_{µ,α}k (kbk + 1), which says conclusion
(31) holds.

Case 3: (xµ,α)i ∈ K/ ^{n}^{i} ∪ −K^{n}^{i}. In this case, we know that λ1 < 0 < λ2 and [(xµ,α)i]+ =
λ_{2}u^{(2)}. From the definition of Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} as in (22), Proposition 2.1, we have

(x_{µ,α})^{T}_{i} Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} = (λ_{1}u^{(1)}+ λ_{2}u^{(2)})^{T} φ^{−}(µ, λ_{1})^{σ}u^{(1)}+ φ^{−}(µ, λ_{2})^{σ}u^{(2)} .

= ^{1}_{2} (λ_{1}φ^{−}(µ, λ_{1})^{σ} + λ_{2}φ^{−}(µ, λ_{2})^{σ})

≤

√ 2 2 (

√ 2

2 λ2)φ^{−}(µ, λ2)^{σ}

≤

√ 2

2 kx_{µ,α}kφ^{−}(µ, λ_{2})^{σ},

(33)

where the first inequality holds due to λ1φ^{−}(µ, λ1)^{σ} < 0 < λ2φ^{−}(µ, λ2)^{σ}, and the second
inequality holds due to

√2

2 λ_{2} = k[(x_{µ,α})_{i}]_{+}k ≤ k(x_{µ,α})_{i}k ≤ kx_{µ,α}k. Substituting (33) from
Ξ_{i} and using Cauchy-Schwarz inequality, we obtain

Ξ_{i} ≤ k(x_{µ,α})_{i}kkb_{i}k +

√2

2 αkx_{µ,α}kφ^{−}(µ, λ_{2})^{σ}

≤ kx_{µ,α}kkbk +

√2

2 αkx_{µ,α}kφ^{−}(µ, λ_{2})^{σ}

≤ kxµ,αk kbk +

√ 2

2 αφ^{−}(µ, 0)^{σ}

,

(34)

where the third inequality holds by the monotone decreasing of φ^{−}(µ, t) about t. Similar
to case 1, for any penalty parameter α, there exists a positive real number ν, such that_{√}

2

2 αφ^{−}(µ, 0)^{σ} ≤ 1 for all µ ∈ (0, ν]. Hence, we reach the conclusion (31) by (34).

From above three cases, the conclusion (31) holds, which shows an evaluation of Ξ_{i}.
Thus, from (30) and Assumption 4.1, there exists a constants a_{0} > 0 such that

a_{0}kx_{µ,α}k^{2} ≤ x^{T}_{µ,α}Ax_{µ,α} =

r

X

i=1

Ξ_{i} ≤ rkx_{µ,α}k (kbk + 1) .

This implies kx_{µ,α}k · (a_{0}kx_{µ,α}k − r (kbk + 1)) ≤ 0, and hence kx_{µ,α}k ≤ _{a}^{r}

0 (kbk + 1) . By
taking M = _{a}^{r}

0 (kbk + 1), the proof is completed. 2

It is well-known that the affine function g(x) := Ax − b is continuous function and
by Proposition 4.1, kg(x_{µ,α})k is bounded for any α ≥ 1, σ ∈ (0, 1] and sufficiently small
µ. We are able to establish an upper bound for kΦ^{−}(µ, x_{µ,α})k in next proposition. The
upper bound is also applicable for k[x_{µ,α}]−k (see Remark 4.1), which plays an impor-
tant role in the convergence analysis. The detailed proof is based on the definition of
Φ^{−}_{i} (µ, (x_{µ,α})_{i}) stated as in (18) and uses the same techniques as in [21, Proposition 3.2]

by left multiplying Φ^{−}_{i} (µ, (x_{µ,α})_{i}) on both sides of the i-th block of (27):

(Ax_{µ,α})_{i}− αΦ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} = b_{i}.

Therefore, we omit it and only present the result, i.e., there exists a positive constant Ci,
independent of x_{µ,α}, µ and α, such that

kΦ^{−}_{i} (µ, (x_{µ,α})_{i})k ≤ C_{i}

α^{1/σ} (35)

holds for any α ≥ 1, σ ∈ (0, 1] and sufficiently small µ. By the definition of Φ^{−}(µ, xµ,α)
as shown in (24) and setting C = C_{1}+ · · · + C_{r}, we obtain the following proposition.

Proposition 4.2. For any α ≥ 1, σ ∈ (0, 1] and sufficiently small µ, there exists a
positive constant C, independent of x_{µ,α}, µ and α, such that

kΦ^{−}(µ, xµ,α)k ≤ C

α^{1/σ}. (36)

Remark 4.1. For any α ≥ 1, σ ∈ (0, 1] and sufficiently small µ, the i-th (i = 1, . . . , r)
component vector (xµ,α)i is fixed since xµ,α with (28) means the solution of (27). For
the fixed (x_{µ,α})_{i} with spectral decomposition (x_{µ,α})_{i} = λ_{1}u^{(1)} + λ_{2}u^{(2)} and the expres-
sion Φ^{−}_{i} (µ, (x_{µ,α})_{i}) = φ^{−}(µ, λ_{1})u^{(1)}+ φ^{−}(µ, λ_{2})u^{(2)}, by taking µ → 0^{+} in φ^{−}(µ, λ_{1}) and
φ^{−}(µ, λ_{2}), we obtain k[λ_{1}]−u^{(1)}+ [λ_{2}]−u^{(2)}k ≤ _{α}^{C}1/σ^{i} from (35), which yields

k[(xµ,α)i]−k ≤ C_{i}

α^{1/σ}. (37)

Also, by setting C = C_{1}+ · · · + C_{r}, we obtain
k[(x_{µ,α})]−k ≤ C

α^{1/σ}. (38)

By using Propositions 4.1, 4.2 and Remark 4.1, we are able to obtain the following desired convergence result of SOCLCP (1) is approximated by the LOPEs (27).

Theorem 4.1. For any α ≥ 1, σ ∈ (0, 1] and sufficiently small µ, let x_{µ,α} be the solution
of LOPEs (27), and x^{∗} be the solution of SOCLCP (1). Then, there exists a positive
constant C, independent of x^{∗}, x_{µ,α}, µ and α, such that

kx^{∗}− x_{µ,α}k ≤ C

α^{1/σ}. (39)

Proof. Follows from (28) and the definition (7), we get x_{µ,α} = [x_{µ,α}]_{+}− [x_{µ,α}]−, where
[x_{µ,α}]_{+} = ([(x_{µ,α})_{1}]_{+}, . . . , [(x_{µ,α})_{r}]_{+}), [x_{µ,α}]− = ([(x_{µ,α})_{1}]−, . . . , [(x_{µ,α})_{r}]−)

respectively denotes the projection of x_{µ,α} on K and −x_{µ,α} on K^{∗}. Therefore, the vector
x^{∗}− x_{µ,α} can be decomposed as

x^{∗}− x_{µ,α} = x^{∗}− [x_{µ,α}]_{+}+ [x_{µ,α}]−= r_{µ,α}+ [x_{µ,α}]−, (40)
where

r_{µ,α} = x^{∗}− [x_{µ,α}]_{+}. (41)

Let’s consider the estimation of r_{µ,α}. If r_{µ,α} = 0, the inequality (39) is satisfied due to
(38) and (40). Therefore, in the following, we only consider r_{µ,α} 6= 0. Noting that, the
SOCLCP (1) is equivalent to the variational inequality problem: find x^{∗} ∈ K (see [17,
Proposition 1.1.3]), such that

(y − x^{∗})^{T}Ax^{∗} ≥ (y − x^{∗})^{T}b, ∀y ∈ K. (42)
It is known that [x_{µ,α}]_{+} ∈ K, by (41) and substituting [x_{µ,α}]_{+} for y in (42) yields

− r_{µ,α}^{T} Ax^{∗} ≥ −r_{µ,α}^{T} b. (43)

Then, multiplying both sides of (27) by r_{µ,α} yields

r_{µ,α}^{T} Ax_{µ,α}− αr^{T}_{µ,α}Φ^{−}(µ, x_{µ,α})^{σ} = r_{µ,α}^{T} b. (44)
Adding up (43) and (44) leads to

r_{µ,α}^{T} A(x_{µ,α}− x^{∗}) − αr_{µ,α}^{T} Φ^{−}(µ, x_{µ,α})^{σ} ≥ 0. (45)
Applying (41) again, we have

r^{T}_{µ,α}Φ^{−}(µ, x_{µ,α})^{σ} = (x^{∗}− [x_{µ,α}]_{+})^{T}Φ^{−}(µ, x_{µ,α})^{σ}. (46)
Combining (45) and (46), we achieve r^{T}_{µ,α}A(x_{µ,α} − x^{∗}) ≥ α(x^{∗} − [x_{µ,α}]_{+})^{T}Φ^{−}(µ, x_{µ,α})^{σ},
which says

r^{T}_{µ,α}A(x^{∗}− xµ,α) ≤ α([xµ,α]+− x^{∗})^{T}Φ^{−}(µ, xµ,α)^{σ}. (47)
Now, using (40) and (47) further gives

(x^{∗}− x_{µ,α}− [x_{µ,α}]−)^{T}A(x^{∗}− x_{µ,α}) ≤ α([x_{µ,α}]_{+}− x^{∗})^{T}Φ^{−}(µ, x_{µ,α})^{σ},

which implies

(x^{∗}− x_{µ,α})^{T}A(x^{∗}− x_{µ,α}) ≤ [x_{µ,α}]^{T}_{−}A(x^{∗}− x_{µ,α}) + α([x_{µ,α}]_{+}− x^{∗})^{T}Φ^{−}(µ, x_{µ,α})^{σ}. (48)
Follows from (26),(28) and the definition (7), by denoting

Ξ_{i} := [(x_{µ,α})_{i}]^{T}_{−}(A(x^{∗}− x_{µ,α}))_{i}+ α ([(x_{µ,α})_{i}]_{+}− x^{∗}_{i})^{T} Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ}, (49)
the inequality (48) is reduced to

(x^{∗}− x_{µ,α})^{T}A(x^{∗}− x_{µ,α}) ≤

r

X

i=1

Ξ_{i}. (50)

To proceed, we discuss three cases of (x_{µ,α})_{i} to proof the term (49) satisfying
Ξi ≤ 1

α^{1/σ}(C^{0}kx^{∗}− xµ,αk + c) , (51)
where C^{0} is a positive constant, independent of x_{µ,α}, µ, α and c ∈ IR_{++} is undetermined.

Case 1: (x_{µ,α})_{i} ∈ K^{n}^{i}. Under this case, we see that [(x_{µ,α})_{i}]− = 0 and [(x_{µ,α})_{i}]_{+} = (x_{µ,α})_{i}.
Using (49) and Cauchy-Schwarz inequality, we have

Ξ_{i} = α ((x_{µ,α})_{i}− x^{∗}_{i})^{T} Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ}

≤ αk(x_{µ,α})_{i}− x^{∗}_{i}k · kΦ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ}k

≤ αkx_{µ,α}− x^{∗}k · kΦ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ}k

= αkx^{∗}− x_{µ,α}kkφ^{−}(µ, λ_{1})^{σ}u^{(1)}+ φ^{−}(µ, λ_{2})^{σ}u^{(2)}k

≤ kx^{∗}− x_{µ,α}k√

2αφ^{−}(µ, 0)^{σ},

(52)

where the second inequality holds by the fact that the norm of the piece component is
less than that of the whole vector, the second equality holds by the definition as (18) and
the last inequality holds by Proposition2.1, the triangle inequality, the nonnegativity of
φ^{−}(µ, 0)^{σ} from (10) and the monotone decreasing of φ^{−}(µ, t) about t since 0 ≤ λ_{1} ≤ λ_{2}
in this case. By Lemma 3.1, we know lim_{µ→0}^{+}φ^{−}(µ, 0)^{σ} = 0. Therefore, for any α ≥ 1
and σ ∈ (0, 1], there exists a positive real number ν, such that √

2αφ^{−}(µ, 0)^{σ} ≤ _{α}1/σ^{1} for
all µ ∈ (0, ν]. Thus, we achieve the conclusion (51) by setting C^{0} = 1.

Case 2: (x_{µ,α})_{i} ∈ −K^{n}^{i}. Under this case, it is clear that [(x_{µ,α})_{i}]_{+} = 0, and we have
(x^{∗}_{i})^{T}Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} ≥ 0 since Φ^{−}_{i} (µ, (x_{µ,α})_{i})^{σ} ∈ K^{n}^{i} and x^{∗}_{i} ∈ K^{n}^{i}. Thus, it follows from
(49) and Cauchy-Schwarz inequality that

Ξ_{i} ≤ [(x_{µ,α})_{i}]^{T}_{−}(A(x^{∗} − x_{µ,α}))_{i}

≤ k[(x_{µ,α})_{i}]−k · k (A(x^{∗}− x_{µ,α}))_{i}k

≤ k[(x_{µ,α})_{i}]−k · kA(x^{∗}− x_{µ,α})k

≤ _{α}^{C}1/σ^{i} kAkkx^{∗}− xµ,αk,

(53)