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 ∈ IRn, such that
x ∈ K, Ax − b ∈ K, xT(Ax − b) = 0, (1) where A is an n × n matrix, b is a vector in IRn, and K is the Cartesian product of second-order cones (SOCs), also called Lorentz cones [7,18]. In other words,
K := Kn1 × · · · × Knr (2)
with r, n1, . . . , nr≥ 1, n1+ · · · + nr = n and
Kni :=(x1, x2) ∈ IR × IRni−1| kx2k ≤ x1 , i = 1, . . . , r,
where k · k denotes the Euclidean norm and (x1, x2) := (x1, xT2)T. Note that K1 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 l1 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 = Kn, i.e.,
x ∈ Kn, Ax − b ∈ Kn, xT(Ax − b) = 0, (3) where A ∈ IRn×n and b ∈ IRn. 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− = [λ1(x)]1/k− u(1)x + [λ2(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 Kn by int(Kn). For any x, y in IRn, we write x Kn y if x − y ∈ Kn and write x Kn y if x − y ∈ int(Kn). In other words, we have x Kn 0 if and only if x ∈ Kn, and x Kn 0 if and only if x ∈ int(Kn).
We usually denote (x, y) := (xT, yT)T for the concatenation of two column vectors x, y for simplicity. The notation k · kp denotes the usual lp-norm on IRn 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 = Kn 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 = (x1, x2) ∈ IR × IRn−1, y = (y1, y2) ∈ IR × IRn−1, their Jordan product [7,18] is defined as
x ◦ y := (hx, yi , y1x2+ x1y2).
We write x + y to mean the usual componentwise addition of vectors and x2 to mean x ◦ x. The identity element under this product is e = (1, 0, . . . , 0)T ∈ IRn. It is known that x2 ∈ Kn for all x ∈ IRn. Moreover, if x ∈ Kn, then there is a unique vector in Kn, denoted by x12, such that (x12)2 = x12◦ x12 = x. For any x ∈ IRn, we define x0 = e if x 6= 0.
For any integer k ≥ 1, we recursively define the powers of element as xk = x ◦ xk−1, and define x−k = (xk)−1 if x ∈ int(Kn). 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 xp is definite, and xm+n = xm ◦ xn for all positive integer m and n. Note that Kn 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 × IRn−1, the spectral decomposition of x with respect to SOC is given by
x = λ1(x)u(1)x + λ2(x)u(2)x , (5) where for i = 1, 2,
λi(x) = x1+ (−1)ikx2k, u(i)x = ( 1
2(1, (−1)i xkx2
2k) if kx2k 6= 0,
1
2(1, (−1)iw) if kx2k = 0, (6) with w ∈ IRn−1 being any unit vector. The two scalars λ1(x) and λ2(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 ∈ IRn 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 = (x1, x2) ∈ IR × IRn−1 with the spectral values λ1(x), λ2(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 k2 = 1/2 for i = 1, 2.
(b) λ1(x), λ2(x) are nonnegative (positive) if and only if x ∈ Kn (x ∈ int(Kn)).
(c) For any x ∈ IRn, x Kn 0 if and only if hx, yi ≥ 0 for all y Kn 0.
The spectral decomposition (5)-(6) and the Proposition 2.1 indicate that xk can be described as xk = λk1(x)u(1)x + λk2(x)u(2)x . For any x ∈ IRn, let [x]+ denote the projection of x onto Kn, and [x]− be the projection of −x onto the dual cone (Kn)∗ of Kn, where the dual cone (Kn)∗ is defined by (Kn)∗ := {y ∈ IRn| hx, yi ≥ 0, ∀x ∈ Kn}. In fact, by Proposition 2.1, the dual cone of Kn being itself, i.e., (Kn)∗ = Kn. Due to the special structure of Kn, the explicit formula of projection of x = (x1, x2) ∈ IR × IRn−1 onto Kn is obtained in [14,17, 19] as below
[x]+=
x if x ∈ Kn, 0 if x ∈ −Kn, u otherwise,
where u =
" x1+kx2k
2
x1+kx2k 2
x2
kx2k
# .
Similarly, the expression of [x]− can be written out as
[x]− =
0 if x ∈ Kn,
−x if x ∈ −Kn, v otherwise,
where v =
"
−x1−kx2 2k
x1−kx2k 2
x2
kx2k
# .
It is easy to verify that x = [x]+− [x]− and
[x]+= [λ1(x)]+u(1)x + [λ2(x)]+u(2)x , [x]−= [λ1(x)]−u(1)x + [λ2(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]− ∈ Kn and [x]+◦ [x]− = 0.
Putting these analyses into a single SOC Kni, i = 1, . . . , r in (2), we can extend them to the general case K = Kn1 × · · · × Knr. More specifically, for any x = (x1, . . . , xr) ∈ IRn1 × · · · × IRnr, y = (y1, . . . , yr) ∈ IRn1 × · · · × IRnr, 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]+:= ([x1]+, . . . , [xr]+), [x]−:= ([x1]−, . . . , [xr]−), (7) where [xi]+, [xi]− for i = 1, . . . , r respectively denote the projection of xi onto the single SOC Kni and the projection of −xi onto (Kni)∗.
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) := 1µd
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) = 1µd
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) −D2µ ≤ φ+(µ, t) − [t]+ ≤ D1µ, where D1 =
Z 0
−∞
|t|d(t)dt and D2 = 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) −D2µ ≤ φ−(µ, t) − [t]− ≤ D1µ, where D1 =
Z +∞
0
|t|d(t)dt and D2 = 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 limµ→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 + eµt
, (11)
φ−2(µ, t) =
0 if t ≥ µ2,
1
2µ −t + µ22
if − µ2 < t < µ2,
−t if t ≤ −µ2,
(12)
φ−3(µ, t) = p4µ2+ t2− t
2 , (13)
φ−4(µ, t) =
0 if t > 0,
t2
2µ if − µ ≤ t ≤ 0,
−t − µ2 if t < −µ,
(14)
where the corresponding kernel functions are d1(t) = et
(1 + et)2,
d2(t) = 1 if − 12 ≤ t ≤ 12, 0 otherwise, d3(t) = 2
(t2+ 4)32,
d4(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 maxt∈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 limµ→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 = (x1, . . . , xr) ∈ IRn1 × · · · × IRnr, we are ready to show how to construct a smoothing function for vectors [x]+ and [x]− associated with K = Kn1× · · · × Knr. We start by constructing a smoothing function for vectors [xi]+, [xi]− on a single SOC Kni, i = 1, . . . , r since [x]+ and [x]− are shown as (7).
First, given smoothing functions φ+, φ− in (9),(10) and xi ∈ IRni, i = 1, . . . , r, we define vector-valued function Φ+i , Φ−i : IR++× IRni → IRni, i = 1, . . . , r as
Φ+i (µ, xi) := φ+(µ, λ1(xi)) u(1)x
i + φ+(µ, λ2(xi)) 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, xi) := φ−(µ, λ1(xi)) u(1)x
i + φ−(µ, λ2(xi)) u(2)x
i, (18)
where µ ∈ IR++is a parameter, λ1(xi), λ2(xi) are the spectral values, and u(1)xi , u(2)xi are the spectral vectors of xi. Consequently, Φ+i (µ, xi), Φ−i (µ, xi) are also smooth on IR++× IRni [8]. Moreover, it is easy to assert that
lim
µ→0+Φ+i (µ, xi) = [λ1(xi)]+u(1)xi + [λ2(xi)]+u(2)xi = [xi]+, (19) lim
µ→0+
Φ−i (µ, xi) = [λ1(xi)]−u(1)x
i + [λ2(xi)]−u(2)x
i = [xi]−, (20)
which means each function Φ+i (µ, xi), Φ−i (µ, xi) serves as a smoothing function of [xi]+, [xi]−
associated with single SOC Kni, i = 1, . . . , r, respectively. Due to Lemma 3.1, Remark 3.1 and from definition of Φ+i (µ, xi), Φ−i (µ, xi) in (17), (18), it is not difficult to verify that for any σ > 0, the below two functions
Φ+i (µ, xi)σ := φ+(µ, λ1(xi))σu(1)x
i + φ+(µ, λ2(xi))σu(2)x
i , (21)
Φ−i (µ, xi)σ := φ−(µ, λ1(xi))σu(1)x
i + φ−(µ, λ2(xi))σu(2)x
i (22)
are continuous functions approximate to [xi]σ+ and [xi]σ−, respectively. In other words, lim
µ→0+
Φ+i (µ, xi)σ = [λ1(xi)]σ+u(1)x
i + [λ2(xi)]σ+u(2)x
i = [xi]σ+, lim
µ→0+Φ−i (µ, xi)σ = [λ1(xi)]σ−u(1)xi + [λ2(xi)]σ−u(2)xi = [xi]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++× IRn→ IRn as
Φ+(µ, x) := Φ+1(µ, x1), . . . , Φ+r(µ, xr) , (23) Φ−(µ, x) := Φ−1(µ, x1), . . . , Φ−r(µ, xr) , (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 = Kn1 × · · · × Knr, respectively. At the same time, from (21), (22), Φ+(µ, x)σ := Φ+1(µ, x1)σ, . . . , Φ+r(µ, xr)σ , (25) Φ−(µ, x)σ := Φ−1(µ, x1)σ, . . . , Φ−r(µ, xr)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 xi ∈ IRni, i = 1, . . . , r, since φ+(µ, λk(xi)) ≥ 0, φ−(µ, λk(xi)) ≥ 0 for k = 1, 2 from (9), (10), we have Φ+i (µ, xi), Φ−i (µ, xi) ∈ Kni 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, xµ,α means the solution of (27), and corre- sponding to the structure of (2), we denote
xµ,α = ((xµ,α)1, . . . , (xµ,α)r) ∈ IRn1 × · · · × IRnr. (28) For simplicity and without causing confusion, we always denote the spectral values and spectral vectors of (xµ,α)i, i = 1, . . . , r as λk := λk((xµ,α)i), u(k) := u(k)(x
µ,α)i for k = 1, 2.
Accordingly, [λk]− := [λk((xµ,α)i)]− and φ−(µ, λk) := φ−(µ, λk((xµ,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 Axµ,α − b ∈ K (noting αΦ−(µ, xµ,α)σ ∈ K). Our goal is to show that the solution sequence {xµ,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
yTAy ≥ a0kyk2, ∀y ∈ IRn. (29) This assumption just implies that ¯A = (A + AT)/2 is symmetric positive definite with a0 = λmin( ¯A) since yTAy = yTAy. 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 xµ,α, µ, α and σ, such that kxµ,αk ≤ M .
Proof. By multiplying xµ,α on both sides of (27), we observe that xTµ,αAxµ,α = xTµ,αb + αxTµ,αΦ−(µ, xµ,α)σ =
r
X
i=1
(xµ,α)Ti bi+ α(xµ,α)Ti Φ−i (µ, (xµ,α)i)σ (30) by (26),(28) and denoting b = (b1, . . . , br) ∈ IRn1× · · · × IRnr. For any (xµ,α)i, i = 1, . . . , r, to proceed, we consider three cases to evaluate the term
Ξi := (xµ,α)Ti bi + α(xµ,α)Ti Φ−i (µ, (xµ,α)i)σ ≤ kxµ,αk (kbk + 1) . (31)
Case 1: (xµ,α)i ∈ Kni. From Cauchy-Schwarz inequality, spectral decomposition of (xµ,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)σk
≤ kxµ,αk kbk + αkφ−(µ, λ1)σu(1)+ φ−(µ, λ2)σu(2)k
≤ kxµ,αk kbk +√
2αφ−(µ, 0)σ ,
(32)
where the second inequality holds by the definition of Φ−i (µ, (xµ,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 ≤ λ1 ≤ λ2 in this case.
Now, applying Lemma3.1, we have limµ→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: (xµ,α)i ∈ −Kni. In light of Lemma 3.2, we know Φ−i (µ, (xµ,α)i)σ ∈ Kni, and hence
(xµ,α)Ti Φ−i (µ, (xµ,α)i)σ ≤ 0.
Thus, we have Ξi ≤ (xµ,α)Ti bi ≤ k(xµ,α)ikkbik ≤ kxµ,αk (kbk + 1), which says conclusion (31) holds.
Case 3: (xµ,α)i ∈ K/ ni ∪ −Kni. In this case, we know that λ1 < 0 < λ2 and [(xµ,α)i]+ = λ2u(2). From the definition of Φ−i (µ, (xµ,α)i)σ as in (22), Proposition 2.1, we have
(xµ,α)Ti Φ−i (µ, (xµ,α)i)σ = (λ1u(1)+ λ2u(2))T φ−(µ, λ1)σu(1)+ φ−(µ, λ2)σu(2) .
= 12 (λ1φ−(µ, λ1)σ + λ2φ−(µ, λ2)σ)
≤
√ 2 2 (
√ 2
2 λ2)φ−(µ, λ2)σ
≤
√ 2
2 kxµ,αkφ−(µ, λ2)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 λ2 = k[(xµ,α)i]+k ≤ k(xµ,α)ik ≤ kxµ,αk. Substituting (33) from Ξi and using Cauchy-Schwarz inequality, we obtain
Ξi ≤ k(xµ,α)ikkbik +
√2
2 αkxµ,αkφ−(µ, λ2)σ
≤ kxµ,αkkbk +
√2
2 αkxµ,αkφ−(µ, λ2)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 Ξi. Thus, from (30) and Assumption 4.1, there exists a constants a0 > 0 such that
a0kxµ,αk2 ≤ xTµ,αAxµ,α =
r
X
i=1
Ξi ≤ rkxµ,αk (kbk + 1) .
This implies kxµ,αk · (a0kxµ,αk − r (kbk + 1)) ≤ 0, and hence kxµ,αk ≤ ar
0 (kbk + 1) . By taking M = ar
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(xµ,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, xµ,α)k in next proposition. The upper bound is also applicable for k[xµ,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, (xµ,α)i) stated as in (18) and uses the same techniques as in [21, Proposition 3.2]
by left multiplying Φ−i (µ, (xµ,α)i) on both sides of the i-th block of (27):
(Axµ,α)i− αΦ−i (µ, (xµ,α)i)σ = bi.
Therefore, we omit it and only present the result, i.e., there exists a positive constant Ci, independent of xµ,α, µ and α, such that
kΦ−i (µ, (xµ,α)i)k ≤ Ci
α1/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 = C1+ · · · + Cr, 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 xµ,α, µ and α, such that
kΦ−(µ, xµ,α)k ≤ C
α1/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 (xµ,α)i with spectral decomposition (xµ,α)i = λ1u(1) + λ2u(2) and the expres- sion Φ−i (µ, (xµ,α)i) = φ−(µ, λ1)u(1)+ φ−(µ, λ2)u(2), by taking µ → 0+ in φ−(µ, λ1) and φ−(µ, λ2), we obtain k[λ1]−u(1)+ [λ2]−u(2)k ≤ αC1/σi from (35), which yields
k[(xµ,α)i]−k ≤ Ci
α1/σ. (37)
Also, by setting C = C1+ · · · + Cr, we obtain k[(xµ,α)]−k ≤ C
α1/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 xµ,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∗, xµ,α, µ and α, such that
kx∗− xµ,αk ≤ C
α1/σ. (39)
Proof. Follows from (28) and the definition (7), we get xµ,α = [xµ,α]+− [xµ,α]−, where [xµ,α]+ = ([(xµ,α)1]+, . . . , [(xµ,α)r]+), [xµ,α]− = ([(xµ,α)1]−, . . . , [(xµ,α)r]−)
respectively denotes the projection of xµ,α on K and −xµ,α on K∗. Therefore, the vector x∗− xµ,α can be decomposed as
x∗− xµ,α = x∗− [xµ,α]++ [xµ,α]−= rµ,α+ [xµ,α]−, (40) where
rµ,α = x∗− [xµ,α]+. (41)
Let’s consider the estimation of rµ,α. If rµ,α = 0, the inequality (39) is satisfied due to (38) and (40). Therefore, in the following, we only consider rµ,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∗)TAx∗ ≥ (y − x∗)Tb, ∀y ∈ K. (42) It is known that [xµ,α]+ ∈ K, by (41) and substituting [xµ,α]+ for y in (42) yields
− rµ,αT Ax∗ ≥ −rµ,αT b. (43)
Then, multiplying both sides of (27) by rµ,α yields
rµ,αT Axµ,α− αrTµ,αΦ−(µ, xµ,α)σ = rµ,αT b. (44) Adding up (43) and (44) leads to
rµ,αT A(xµ,α− x∗) − αrµ,αT Φ−(µ, xµ,α)σ ≥ 0. (45) Applying (41) again, we have
rTµ,αΦ−(µ, xµ,α)σ = (x∗− [xµ,α]+)TΦ−(µ, xµ,α)σ. (46) Combining (45) and (46), we achieve rTµ,αA(xµ,α − x∗) ≥ α(x∗ − [xµ,α]+)TΦ−(µ, xµ,α)σ, which says
rTµ,αA(x∗− xµ,α) ≤ α([xµ,α]+− x∗)TΦ−(µ, xµ,α)σ. (47) Now, using (40) and (47) further gives
(x∗− xµ,α− [xµ,α]−)TA(x∗− xµ,α) ≤ α([xµ,α]+− x∗)TΦ−(µ, xµ,α)σ,
which implies
(x∗− xµ,α)TA(x∗− xµ,α) ≤ [xµ,α]T−A(x∗− xµ,α) + α([xµ,α]+− x∗)TΦ−(µ, xµ,α)σ. (48) Follows from (26),(28) and the definition (7), by denoting
Ξi := [(xµ,α)i]T−(A(x∗− xµ,α))i+ α ([(xµ,α)i]+− x∗i)T Φ−i (µ, (xµ,α)i)σ, (49) the inequality (48) is reduced to
(x∗− xµ,α)TA(x∗− xµ,α) ≤
r
X
i=1
Ξi. (50)
To proceed, we discuss three cases of (xµ,α)i to proof the term (49) satisfying Ξi ≤ 1
α1/σ(C0kx∗− xµ,αk + c) , (51) where C0 is a positive constant, independent of xµ,α, µ, α and c ∈ IR++ is undetermined.
Case 1: (xµ,α)i ∈ Kni. Under this case, we see that [(xµ,α)i]− = 0 and [(xµ,α)i]+ = (xµ,α)i. Using (49) and Cauchy-Schwarz inequality, we have
Ξi = α ((xµ,α)i− x∗i)T Φ−i (µ, (xµ,α)i)σ
≤ αk(xµ,α)i− x∗ik · kΦ−i (µ, (xµ,α)i)σk
≤ αkxµ,α− x∗k · kΦ−i (µ, (xµ,α)i)σk
= αkx∗− xµ,αkkφ−(µ, λ1)σu(1)+ φ−(µ, λ2)σu(2)k
≤ kx∗− xµ,αk√
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 ≤ λ1 ≤ λ2 in this case. By Lemma 3.1, we know limµ→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 ≤ α1/σ1 for all µ ∈ (0, ν]. Thus, we achieve the conclusion (51) by setting C0 = 1.
Case 2: (xµ,α)i ∈ −Kni. Under this case, it is clear that [(xµ,α)i]+ = 0, and we have (x∗i)TΦ−i (µ, (xµ,α)i)σ ≥ 0 since Φ−i (µ, (xµ,α)i)σ ∈ Kni and x∗i ∈ Kni. Thus, it follows from (49) and Cauchy-Schwarz inequality that
Ξi ≤ [(xµ,α)i]T−(A(x∗ − xµ,α))i
≤ k[(xµ,α)i]−k · k (A(x∗− xµ,α))ik
≤ k[(xµ,α)i]−k · kA(x∗− xµ,α)k
≤ αC1/σi kAkkx∗− xµ,αk,
(53)
