**second-order cone linear complementarity problems**

**Zijun Hao**^{1}**· Chieu Thanh Nguyen**^{2}**· Jein-Shan Chen**^{2}

Received: 24 March 2020 / Accepted: 15 November 2021 / Published online: 3 December 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 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 specific 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 mono- tonically decreases to zero, we show that the solution sequence 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 approach. The performance profile of four spe- cific smoothing approximations is presented, and the generalization of two approximations are also investigated.

**Keywords Second-order cone**· Linear complementarity problem · Lower order penalty
approach· Exponential convergence rate

**Mathematics Subject Classification 90C25**· 90C30 · 90C33

The author’s work is supported by the National Natural Science Foundation of China Nos. 11661002, 11871383), the Natural Science Fund of Ningxia (No. 2020AAC03236), the First-class Disciplines Foundation of Ningxia (No. NXYLXK2017B09). 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 Zijun Haozijunhao@126.com Chieu Thanh Nguyen thanhchieu90@gmail.com

1 School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China 2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan

**1 Introduction**

This paper targets the following second-order cone linear complementarity problem
*(SOCLCP), 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

*is the Cartesian product of second-order cones (SOCs), also called Lorentz cones [7,18]. In other words,*

_{K}*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}*| x*2* ≤ x*1

*, i = 1, . . . , r,*

where* · denotes the Euclidean norm and (x*1*, x*2*) := (x*1*, x*_{2}^{T}*)** ^{T}*. Note that

*K*

^{1}denotes the set of nonnegative real numbers IR

_{+}. The SOCLCP, as an extension of the linear complemen- tarity 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 SOCLCPs (1)–(2), including the interior-point method [1,28,32], the smoothing Newton method [14,19,24], the smoothing-regularization method [23], the semismooth Newton 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 substantially in recent
years, the fact remains that there still have many complementarity problems require effi-
cient and accurate numerical methods. The penalty methods are well-known for solving
constrained optimization problems which possess many nice properties. More specifically,
*the l*1exact penalty function method and lower order penalty function method are known as the
approaches which hold many nice properties and attracts 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 assumptions, 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, asα → +∞, 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

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 corresponding algorithm is
constructed and numerical results are also reported to examine the feasibility of the proposed
method. The performance profile of those specific smoothing approximations is presented,
and the generalization of two approximations are also investigated.

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

For simplicity, we denote the interior of single SOC*K** ^{n}*by int(

*K*

^{n}*). For any x, y in IR*

*, we*

^{n}*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 ·

*p*

*denotes the usual l*

*-norm on IR*

_{p}

^{n}*for any p*≥ 1. In particular, it is Euclidean norm

*· 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 := (x, y , 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*

*, then there is a unique vector in*

^{n}*K*

^{n}*, denoted by x*

^{1}

^{2}, such that

*(x*

^{1}

^{2}

*)*

^{2}

*= x*

^{2}

^{1}

*◦ x*

^{1}

^{2}

*= x. For any x ∈ IR*

^{n}*, we define x*

^{0}

*= e if x = 0. For any*

*integer k*

*≥ 1, we recursively define the powers of element as x*

^{k}*= x ◦ x*

*, and define*

^{k−1}*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*

*is*

^{p}*definite, and x*

^{m+n}*= x*

^{m}*◦ x*

^{n}*for all positive integer m and n. Note thatK*

^{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 [5–8,10–12,18,19,33]. 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}*,* (5)

*where for i= 1, 2,*

*λ**i**(x) = x*1*+ (−1)*^{i}*x*2*, u*^{(i)}* _{x}* =

1

2*(1, (−1)*^{i x}_{x}^{2}_{2}_{}*) if x*2*
= 0,*

1

2*(1, (−1)*^{i}*w) if x*2* = 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 x*

_{2}= 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 valuesλ*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* *,u*^{(i)}*x* ^{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 ifx, y ≥ 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

*, let*

^{n}*[x]*

_{+}denote the projec-

*tion of x ontoK*

*, and*

^{n}*[x]*−be the projection of

*−x onto the dual cone (K*

^{n}*)*

^{∗}of

*K*

*, where the dual cone*

^{n}*(K*

^{n}*)*

^{∗}is defined by

*(K*

^{n}*)*

^{∗}

*:= {y ∈ IR*

^{n}*| x, y ≥ 0, ∀x ∈K*

*}. In fact, by Proposition2.1, the dual cone of*

^{n}*K*

*being itself, i.e.,*

^{n}*(K*

^{n}*)*

^{∗}=

*K*

*. Due to the special structure of*

^{n}*K*

^{n}*, the explicit formula of projection of x*

*= (x*1

*, x*2

*) ∈ IR × IR*

*onto*

^{n−1}*K*

*is obtained in [14,17,19] as below*

^{n}*[x]*+=

⎧⎨

⎩

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

*where u*=

_{x}_{1}_{+x}_{2}_{}
2

*x*1*+x*2
2

*x*2

*x*2

*.*

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}^{−x}_{2} ^{2}^{}

*x*_{1}*−x*2
2

*x*2

*x*2

*.*

*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 := (x*1*◦ y*1*, . . . , x**r* *◦ y**r**).*

Let*[x]*+,*[x]*−*respectively denote the projection of x ontoK*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}*)*

^{∗}.

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*
_{+∞}

−∞ *d(t)dt = 1.* (8)

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

*μ*

, where*μ is a positive parameter. If*
_{+∞}

−∞ *|t| d(t)dt <*

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

_{+∞}

−∞ *(t − s)*_{+}*ˆs(μ, s)ds =*
_{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) =* _{μ}^{1}*d*
*t*

*μ*

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

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

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

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

_{0}

−∞*|t|d(t)dt and D*2= max _{+∞}

−∞ *td(t)dt, 0*

*.*

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

From Proposition3.1(b), we have

*μ→0*lim^{+}*φ*^{+}*(μ, t) = [t]*+

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

*φ*^{−}*(μ, t) =*
_{−t}

−∞*(−t − s)ˆs(μ, −s)ds =*
_{+∞}

*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 functionφ*^{−}*(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}=

_{+∞}

0

*|t|d(t)dt and D*2= max

_{+∞}

−∞ *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*Proposition3.1and3.2, we know that

*φ*

^{+}

*(μ, t) defined by (9) andφ*

^{−}

*(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, letφ^{+}*(μ, t), φ*^{−}*(μ, t) be the smooth-*
*ing functions of[t]*_{+}*, [t]*_{−}*, defined by (9) and (10) respectively. Then, for anyσ > 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 smoothing
functions for*[t]*−as well:

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

*,* (11)

*φ*^{−}_{2}*(μ, t) =*

⎧⎪

⎨

⎪⎩

0 if t≥ _{2}^{¯}*,*

2μ1

*−t +*^{μ}_{2}_{2}

if−_{2}^{¯} *< t <* _{2}^{¯}*,*

*−t* if t≤ −_{2}^{¯}*,*

(12)

*φ*^{−}_{3}*(μ, t) =*

4*μ*^{2}*+ t*^{2}*− t*

2 *,* (13)

*φ*^{−}_{4}*(μ, t) =*

⎧⎪

⎨

⎪⎩

0 if t*> 0,*

*t*^{2}

2μ if*− ¯ ≤ t ≤ 0,*

*−t −*^{μ}_{2} if t*< −¯,*

(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*.*

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 Fig.*1.

From Fig.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)

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

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*
max*t**∈(−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*

*φ*_{1}^{−}*(μ, t) − [t]*_{−}_{∞}*= (ln 2)μ,*

*φ*_{2}^{−}*(μ, t) − [t]*−∞*= μ/8,*

*φ*_{3}^{−}*(μ, t) − [t]*−∞*= μ.*

On the other hand, it is obvious that max_{t∈IR}*φ*4^{−}*(μ, t) − [t]*−* = μ/*2, which says

*φ*^{−}_{4}*(μ, t) − [t]*−∞*= μ/2.*

In summary, we have

*φ*^{−}_{3}*(μ, t) − [t]*−∞*> φ*_{1}^{−}*(μ, t) − [t]*−∞*> φ*_{4}^{−}*(μ, t) − [t]*−∞*> φ*^{−}_{2}*(μ, t) − [t]*−∞*.*
(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 Fig.2.

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

**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 differ-*
entiable. For example,*φ*^{−}_{1}*(μ, t)** ^{σ}*,

*φ*

_{3}

^{−}

*(μ, t)*

*are continuously differentiable, but*

^{σ}*φ*

_{2}

^{−}

*(μ, t)*

*,*

^{σ}*φ*

_{4}

^{−}

*(μ, t)*

*are not continuously differentiable for*

^{σ}*σ = 1/2 since the non-differentiable points*

*are t= μ/2 and t = 0 respectively. Their geometric views are depicted in Fig.*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

^{+}

_{i}*,*

^{−}

*: IR++× IR*

_{i}

^{n}*→ IR*

^{i}

^{n}

^{i}*, i = 1, . . . , r as*

_{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)

_{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)}*x**i* *, u*^{(2)}_{x}* _{i}* are the

*spectral vectors of x*

*.*

_{i}Consequently,^{+}_{i}*(μ, x**i**), *^{−}_{i}*(μ, x**i**) are also smooth on IR*++× IR^{n}* ^{i}* [8]. Moreover, it is
easy to assert that

*μ→0*lim^{+}^{+}_{i}*(μ, x**i**) = [λ*1*(x**i**)]*+*u*^{(1)}_{x}

*i* *+ [λ*2*(x**i**)]*+*u*^{(2)}_{x}

*i* *= [x**i*]+*,* (19)

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

*μ→0*lim^{+}^{−}_{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 ^{+}_{i}*(μ, x**i**), *^{−}_{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, Remark3.1and from definition of_{i}^{+}*(μ, x**i**), *^{−}_{i}*(μ, x**i**) in (17), (18), it is not difficult*
to verify that for any*σ > 0, the below two functions*

^{+}_{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)

^{−}_{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,

*μ→0*lim^{+}^{+}_{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}_{+}*,*

*μ→0*lim^{+}^{−}_{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^{+}*, *^{−}: IR++× IR* ^{n}*→ IR

*as*

^{n}^{+}*(μ, x) :=*

^{+}_{1}*(μ, x*1*), . . . , *^{+}_{r}*(μ, x**r**)*

*,* (23)

^{−}*(μ, x) :=*

^{−}_{1}*(μ, x*1*), . . . , *^{−}_{r}*(μ, x**r**)*

*,* (24)

where^{+}_{i}*(μ, x**i**), *^{−}_{i}*(μ, x**i**), i = 1, . . . , r are defined by (17), (18), respectively. Therefore,*
from (19), (20) and (7),^{+}*(μ, x), *^{−}*(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),

^{+}*(μ, x)** ^{σ}* :=

^{+}_{1}*(μ, x*1*)*^{σ}*, . . . , *^{+}_{r}*(μ, x**r**)*^{σ}

*,* (25)

^{−}*(μ, x)** ^{σ}* :=

^{−}_{1}*(μ, x*1*)*^{σ}*, . . . , *^{−}_{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**^{+}*(μ, x) and *^{−}*(μ, x) are defined by (23), (24), respectively, and*

^{+}*(μ, x)*^{σ}*and*^{−}*(μ, x)*^{σ}*are defined for anyσ > 0 as in (25), (26), respectively. Then,*
*the following results hold.*

*(a) Both*^{+}*(μ, x) and *^{−}*(μ, x) belong toK,*
*(b) Both*^{+}*(μ, x)*^{σ}*and*^{−}*(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*^{+}_{i}*(μ, x**i**), *_{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, λ**k**(x**i**))*^{σ}*≥ 0, φ*^{−}*(μ, λ**k**(x**i**))** ^{σ}* ≥ 0,

*k= 1, 2. Applying (*25) and (26), the desired result follows.

**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− α*^{−}*(μ, x)*^{σ}*= b,* (27)

where*σ ∈ (0, 1] is a given power parameter, α ≥ 1 is a penalty parameter and *^{−}*(μ, x)** ^{σ}*is
defined as (26). Throughout this section, x

*means the solution of (27), and corresponding to the structure of (2), we denote*

_{μ,α}*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*α*^{−}*(μ, x)*^{σ}*penalizes the ‘negative part’ of x whenμ → 0*^{+}.
By Lemma3.2and from equations (27), it is easy to see that Ax_{μ,α}*− b ∈* *K* (noting
*α*^{−}*(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 a*0*> 0, such that*

*y*^{T}*Ay≥ a*0*y*^{2}*, ∀y ∈ IR*^{n}*.* (29)

3 1

*it is easy to see that matrix A is positive definite satisfying (29), but not symmetric. Under*
Assumption4.1, 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 thatx*_{μ,α}* ≤ M.*

**Proof By multiplying x*** _{μ,α}*on both sides of (27), we observe that

*x*_{μ,α}^{T}*Ax*_{μ,α}*= x*^{T}_{μ,α}*b**+ αx*_{μ,α}^{T}^{−}*(μ, x**μ,α**)** ^{σ}* =

*r*

*i*=1

*(x**μ,α**)**i*^{T}*b*_{i}*+ α(x**μ,α**)*^{T}*i**i*^{−}*(μ, (x**μ,α**)**i**)** ^{σ}*
(30)
by (26),(28) and denoting b

*= (b*1

*, . . . , b*

*r*

*) ∈ IR*

^{n}^{1}×· · ·×IR

^{n}*. For any*

^{r}*(x*

_{μ,α}*)*

*i*

*, i = 1, . . . , r,*to proceed, we consider three cases to evaluate the term

*i* *:= (x**μ,α**)*^{T}_{i}*b*_{i}*+ α(x**μ,α**)*^{T}_{i}^{−}_{i}*(μ, (x**μ,α**)**i**)*^{σ}*≤ x**μ,α** (b + 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

*i* *≤ (x*_{μ,α}*)**i*

*b**i** + α*^{−}_{i}*(μ, (x*_{μ,α}*)**i**)*^{σ}

*≤ x**μ,α*

*b + αφ*^{−}*(μ, λ*1*)*^{σ}*u*^{(1)}*+ φ*^{−}*(μ, λ*2*)*^{σ}*u*^{(2)}

*≤ x**μ,α*

*b +*√

2*αφ*^{−}*(μ, 0)*^{σ}*,*

(32)

where the second inequality holds by the definition of^{−}_{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)*

*from (10) and the monotone decreasing of*

^{σ}*φ*

^{−}

*(μ, t) about t since 0 ≤ λ*1

*≤ λ*2in 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*√

2*αφ*^{−}*(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 Lemma3.2, we know

^{−}

_{i}*(μ, (x*

_{μ,α}*)*

*i*

*)*

*∈*

^{σ}*K*

^{n}*, and hence*

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

Thus, we have*i* *≤ (x*_{μ,α}*)*^{T}_{i}*b**i**≤ (x*_{μ,α}*)**i**b**i** ≤ x*_{μ,α}* (b + 1), which says conclusion*
(31) holds.

**Case 3:***(x**μ,α**)**i* */∈K*^{n}* ^{i}* ∪ −

*K*

^{n}*. In this case, we know that*

^{i}*λ*1

*< 0 < λ*2and

*[(x*

*μ,α*

*)*

*i*]+=

*λ*2

*u*

*. From the definition of*

^{(2)}^{−}

_{i}*(μ, (x*

*μ,α*

*)*

*i*

*)*

*as in (22), Proposition2.1, we have*

^{σ}*(x**μ,α**)*_{i}^{T}^{−}_{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}*λ*2*)φ*^{−}*(μ, λ*2*)*^{σ}

≤ ^{√}_{2}^{2}*x**μ,α**φ*^{−}*(μ, λ*2*)*^{σ}*,*

(33)

where the first inequality holds due to*λ*1*φ*^{−}*(μ, λ*1*)*^{σ}*< 0 < λ*2*φ*^{−}*(μ, λ*2*)** ^{σ}*, and the second
inequality holds due to

√2

2 *λ*2*= [(x**μ,α**)**i*]+* ≤ (x**μ,α**)**i** ≤ x**μ,α*. Substituting (33) from

*i*and using Cauchy-Schwarz inequality, we obtain

*i* *≤ (x*_{μ,α}*)**i**b**i* +^{√}_{2}^{2}*αx*_{μ,α}*φ*^{−}*(μ, λ*2*)*^{σ}

*≤ x*_{μ,α}*b +*^{√}_{2}^{2}*αx*_{μ,α}*φ*^{−}*(μ, λ*2*)*^{σ}

*≤ x**μ,α*

*b +*^{√}_{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*i*.
Thus, from (30) and Assumption4.1, there exists a constants a_{0}*> 0 such that*

*a*_{0}*x**μ,α*^{2}*≤ x*_{μ,α}^{T}*Ax** _{μ,α}* =

*r*
*i=1*

*i* *≤ rx**μ,α** (b + 1) .*

This implies*x** _{μ,α}* ·

*a*0*x*_{μ,α}* − r (b + 1)*

*≤ 0, and hence x** _{μ,α}* ≤

_{a}

^{r}0 *(b + 1) .*

*By taking M*= _{a}^{r}_{0}*(b + 1), the proof is completed.*

*It is well-known that the affine function g(x) := Ax − b is continuous function and by*
Proposition4.1,*g(x**μ,α**) is bounded for any α ≥ 1, σ ∈ (0, 1] and sufficiently small μ.*

We are able to establish an upper bound for^{−}*(μ, x**μ,α**) in next proposition. The upper*
bound is also applicable for*[x**μ,α*]− (see Remark4.1), which plays an important role in
the convergence analysis. The detailed proof is based on the definition of^{−}_{i}*(μ, (x*_{μ,α}*)**i**)*
stated as in (18) and uses the same techniques as in [21, Proposition 3.2] by left multiplying

^{−}_{i}*(μ, (x**μ,α**)**i**) on both sides of the ith block of (*27):

*(Ax**μ,α**)**i**− α*^{−}_{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 C** _{i}*,

*independent of x*

_{μ,α}*, μ and α, such that*

^{−}_{i}*(μ, (x**μ,α**)**i**) ≤* *C*_{i}

*α*^{1/σ} (35)

holds for any*α ≥ 1, σ ∈ (0, 1] and sufficiently small μ. By the definition of *^{−}*(μ, x**μ,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*

^{−}*(μ, x*_{μ,α}*) ≤* *C*

*α*^{1}^{/σ}*.* (36)

* Remark 4.1 For any α ≥ 1, σ ∈ (0, 1] and sufficiently small μ, the ith (i = 1, . . . , r)*
component vector

*(x*

_{μ,α}*)*

*i*

*is fixed since x*

*with (28) means the solution of (27). For the fixed*

_{μ,α}*(x*

*μ,α*

*)*

*i*with spectral decomposition

*(x*

*μ,α*

*)*

*i*

*= λ*1

*u*

^{(1)}*+ λ*2

*u*

*and the expression*

^{(2)}^{−}_{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 [λ*1]−

*u*

^{(1)}*+ [λ*2]−

*u*

*≤*

^{(2)}

_{α}*1/σ*

^{C}*from (35),*

^{i}which yields

*[(x*_{μ,α}*)**i*]−* ≤ Cα*^{1/σ}^{i}*.* (37)