# The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP

Authors: Xiaoni Chi; Zhongping Wan; Zijun Hao
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.


---

## Body

## 1. Introduction

The second-order cone complementarity problem (SOCCP) [1] is to find (x,y,ζ)∈Rn×Rn×Rl such that
(1)x∈K,y∈K,xTy=0,F(x,y,ζ)=0,
where F:Rn×Rn×Rl→Rn×Rl is a continuously differentiable mapping, K⊂Rn is the Cartesian product of second-order cones (SOC), that is, K=Kn1×Kn2×⋯×Knm with n=n1+n2+⋯+nm and the ni-dimensional SOC defined by
(2)Kni={xi=(x1i;x2i)∈R×Rni-1:x1i-∥x2i∥≥0}.Without loss of generality, we may assume thatm=1 and K=Kn in the following analysis, since our analysis can be easily extended to the general case.The SOCCP contains a wide class of problems, such as nonlinear complementarity problems [2], second-order cone programming [1, 3, 4], and has a variety of engineering and management applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics [5, 6].Recently, great attention has been paid to smoothing methods, partially due to their superior theoretical and numerical performances [7–10]. Smoothing methods usually reformulate the SOCCP as a system of equations by using smoothing functions of SOC complementarity functions [10, 11]. The smoothing parameter involved in smoothing functions may be treated as a variable [9] or a parameter with an appropriate parameter control [7]. In the latter case, the Jacobian consistency plays a key role for achieving a rapid convergence of Newton the methods or the Newton-like methods. Hayashi et al. [7] propose a combined smoothing and regularized method for monotone SOCCP and show its global and quadratic convergence based on the Jacobian consistency of the smoothing natural residual function. Ogasawara and Narushima [12] show the Jacobian consistency of a smoothed Fischer-Burmeister (FB) function. Chen et al. [13] present a smoothing function of a generalized FB function in the context of nonlinear complementarity programming and study some of its favorable properties, including the Jacobian consistency property. Based on the results, they [13] propose a smoothing algorithm for the mixed complementarity problem, which is shown to possess global convergence and local superlinear (or quadratic) convergence.In this paper, we are concerned with the one-parametric class of SOC complementarity functionsϕτ:Rn→Rn defined by [14]
(3)ϕτ(x,y):=x+y-x2+y2+(τ-2)(x∘y),
where τ∈(0,4) is an arbitrary but fixed parameter. When τ=2,  ϕτ reduces to the vector-valued Fischer-Burmeister function given by
(4)ϕFB(x,y):=x+y-x2+y2,
and as τ→0, it becomes a multiple of the following vector-valued residual function:
(5)ϕNR(x,y):=x-ΠKn(x-y),
where ΠKn(·) denotes the metric projection on the second-order cone Kn. Thus, the one-parametric class of vector-valued functions (3) cover two popular second-order cone complementarity functions.In this paper, we aim to show the Jacobian consistency of smoothing functions of the one-parametric class of SOC complementarity functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of the smoothing functions, which will help to adjust a parameter appropriately in smoothing methods.The organization of this paper is as follows. In Section2, we review some preliminaries including the Euclidean Jordan algebra associated with SOC and subdifferentials. In Section 3, we derive the computable formula for the Jacobian of the one-parametric class of smoothing functions in the SOCCP. In Section 4, we prove the Jacobian consistency of the one-parametric class of smoothing functions and estimate the distance between the gradient of the smoothing functions and the subgradient of the one-parametric class of the SOC complementarity functions. In Section 5, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions and then present an alternative way to prove the Jacobian consistency of the one-parametric class of smoothing functions. Finally, we close this paper with some conclusions in Section 6.In what follows, we denote the nonnegative orthant ofR by R+. We use the symbol ∥·∥ to denote the Euclidean norm defined by ∥x∥:=xTx for a vector x or the corresponding induced matrix norm. For simplicity, we often use x=(x1;x2) for the column vector x=(x1,x2T)T. For the SOC Kn,  intKn and bdKn mean the topological interior and the boundary of Kn, respectively. For a given set S⊂Rm×n,  convS denotes the convex hull of S in Rm×n, and dist(X,S) denotes inf{∥X-Y∥:Y∈S} for a matrix X∈Rm×n.


## 2. Preliminaries

In this section, we recall some concepts and results, which include the Euclidean Jordan algebra [3, 15] associated with the SOC Kn and subdifferentials [16].First, we recall the Euclidean Jordan algebra associated with the SOC and some useful definitions. The Euclidean Jordan algebra for the SOCKn is the algebra defined by
(6)x∘y=(xTy;x1y2+y1x2),∀x,y∈Rn,
with e=(1,0,…,0)∈Rn being its unit element. Given an element x=(x1;x2)∈R×Rn-1, we define
(7)L(x)=(x1x2Tx2x1I),
where I represents the (n-1)×(n-1) identity matrix. It is easy to verify that x∘y=L(x)y for any y∈Rn. Moreover, L(x) is symmetric positive definite (and hence invertible) if and only if x∈intKn.Now, we give the spectral factorization of vectors inRn associated with the SOC Kn. Let x=(x1;x2)∈R×Rn-1. Then x can be decomposed as
(8)x=λ1u(1)+λ2u(2),
where λ1,  λ2, and u(1),  u(2) are the spectral values and the associated spectral vectors of x given by
(9)λi=x1+(-1)i∥x2∥,u(i)={12(1,(-1)ix2∥x2∥)ifx2≠0,12(1,(-1)iω)otherwise,
for i=1,2, with any ω∈Rn-1 such that ∥ω∥=1. By the spectral factorization, a scalar function can be extended to a function for the SOC. For any x∈Rn, we define
(10)x2=λ12u(1)+λ22u(2).
Since both eigenvalues of any x∈Kn are nonnegative, we define
(11)x=λ1u(1)+λ2u(2).For anyx=(x1;x2)∈R×Rn-1, we define [12] x′=(x1;-x2). Obviously, x′′=x,  (x+y)′=x′+y′, and (cx)′=cx′ for any c∈R. Moreover, x∘x′=x12-∥x2∥2=0 if x∈bdKn.LetG:Rm→Rn be a locally Lipschitzian function. Then, G is differentiable almost everywhere by Rademacher’s theorem [17]. The Bouligand- (B-) subdifferential and the Clarke subdifferential of G at z are defined by
(12)∂GB(z):={limz^→z∇G(z^):z^∈DG},∂G(z)=conv∂GB(z),
respectively, where DG denotes the set of points at which G is differentiable. It is obvious that ∂G(z)={∇G(z)} if G is continuously differentiable at z.By using the concepts of subdifferentials, we give the definition of the Jacobian consistency, which was first introduced by Chen et al. [16], which is a concept relating the generalized Jacobian of a nonsmooth function with the Jacobian of a smoothing function [7].Definition 1 (see [16]).
LetG:Rm→Rn be a locally Lipschitzian function. Let Gε:Rm→Rn be a continuously differentiable function for any ε>0 such that limε↓0Gε(z)=G(z) for any z∈Rm. We say that Gε satisfies the Jacobian consistency property if for any z∈Rm,  limε↓0dist(∇Gε(z),∂G(z))=0.It should be noted that the “inf” appearing in the definition ofdist(∇Gε(z),∂G(z)) can be replaced by “min,” since the set ∂G(z) is compact at all z∈Rm [17].


## 3. Smoothing Function

In this section, we propose a smoothing function of the one-parametric class of SOC complementarity functions and derive the computable formula for its Jacobian.Since the one-parametric class of SOC complementarity functionsϕτ defined by (3) is nonsmooth, we consider the function ϕτ,ε defined by
(13)ϕτ,ε(x,y):=x+y-x2+y2+(τ-2)(x∘y)+2ε2e,
where the smoothing parameter ε∈R.Definition 2 (see [7]).
For a nondifferentiable functionh:Rm→Rn, one considers a function hε:Rm→Rn with a parameter ε>0 that has the following properties:(i)
hε is differentiable for any ε>0;(ii)
limε↓0hε(x)=h(x) for any x∈Rm.
Such a functionhε is called a smoothing function of h.In the following, we will show that the functionϕτ,ε  given by (13) is a smoothing function of ϕτ. Thus, we can solve a family of smoothing subproblems ϕτ,ε(x,y)=0 for ε>0 and obtain a solution of ϕτ(x,y)=0 by letting ε↓0.For convenience, we give some notations. For anyx=(x1;x2),  y=(y1;y2)∈R×Rn-1, and any ε∈R, we define the mapping zε:R2n→R×Rn-1 by
(14)zε=(z1ε;z2ε)=zε(x,y):=x2+y2+(τ-2)(x∘y)+2ε2e,
and drop the subscript for simplicity for ε=0, and thus,
(15)z=(z1;z2)=z(x,y):=x2+y2+(τ-2)(x∘y).
By direct calculations, we obtain
(16)z1ε=∥x∥2+∥y∥2+(τ-2)xTy+2ε2=2ε2+z1,z2ε=2x1x2+2y1y2+(τ-2)(x1y2+y1x2)=z2,
and therefore zε=(z1ε;z2). Then, the spectral factorization of zε is
(17)zε=λ1(zε)u1(z)+λ2(zε)u2(z),
where λ1(zε),  λ2(zε), and u1(z),  u2(z) are the spectral values and the associated spectral vectors of zε given by
(18)λi(zε)=2ε2+∥x∥2+∥y∥2+(τ-2)xTy+(-1)i∥2x1x2+2y1y2+(τ-2)(x1y2+y1x2)∥,(19)ui(z)=12(1;(-1)iz-2),
for i=1,2. Here,
(20)z-2:=z2∥z2∥=2x1x2+2y1y2+(τ-2)(x1y2+y1x2)∥2x1x2+2y1y2+(τ-2)(x1y2+y1x2)∥,
if z2≠0, and otherwise, z-2 is any vector in Rn-1 such that ∥z-2∥=1.For any(x,y)∈Rn×Rn, it is not difficult to verify that
(21)zε=x2+y2+(τ-2)(x∘y)+2ε2e=(x+τ-22y)2+τ(4-τ)4y2+2ε2e=(y+τ-22x)2+τ(4-τ)4x2+2ε2e∈intKn,
for any ε≠0, and
(22)z=x2+y2+(τ-2)(x∘y)=(x+τ-22y)2+τ(4-τ)4y2=(y+τ-22x)2+τ(4-τ)4x2∈Kn.
Therefore, for any x=(x1;x2),y=(y1;y2)∈R×Rn-1, and any ε∈R, we can also define
(23)wε=(w1ε;w2ε)=wε(x,y):=x2+y2+(τ-2)(x∘y)+2ε2e,
and for ε=0,
(24)w=(w1;w2)=w(x,y):=x2+y2+(τ-2)(x∘y).
The spectral factorization of wε and w is given by, respectively,
(25)wε=λ1(zε)u1(z)+λ2(zε)u2(z),w=λ1(z)u1(z)+λ2(z)u2(z).
By (22), we can partition R2n as R2n=S1∪S2∪{(0,0)}, where
(26)S1:={(x,y)∈R2n:z∈intKn}={(x,y)∈R2n:λ2(z)≥λ1(z)>0},S2:={(x,y)∈R2n:z∈bdKn∖{0}}={(x,y)∈R2n:2z1=λ2(z)>λ1(z)=0}.Theorem 3.
For any(x,y)∈Rn×Rn and ε≠0, let ϕτ and ϕτ,ε be, respectively, defined by (3) and (13). Then, the following results hold.(i)
The functionϕτ,ε is continuously differentiable everywhere, and its Jacobian is given by
(27)∇ϕτ,ε(x,y)=(I-L(x+τ-22y)L-1(wε)I-L(y+τ-22x)L-1(wε)),whereL-1(wε)=(1/z1ε)I if z2=0, and otherwise,
(28)L-1(wε)=L1(zε)+L2(zε)=(bεcεz-2Tcεz-2aεI+(bε-aε)z-2z-2T),with(29)L1(zε)=12λ1(zε)(1-z-2T-z-2z-2z-2T),L2(zε)=12λ2(zε)(1z-2Tz-2z-2z-2T)+aε(00T0I-z-2z-2T),(30)aε=2λ1(zε)+λ2(zε),bε=12(1λ1(zε)+1λ2(zε)),cε=12(1λ2(zε)-1λ1(zε)).(ii)
For any(x,y)∈Rn×Rn,  limε↓0ϕτ,ε(x,y)=ϕτ(x,y). Thus, ϕτ,ε is a smoothing function of ϕτ.Proof.
(i) For any(x,y)∈Rn×Rn and any ε≠0, it follows from Corollary  5.4 in [1], the chain rule for differentiation, and (21) that formula (27) holds. Formula (28) is due to Proposition  5.2 and its proof in [1].
(ii) Fix anyx=(x1;x2),  y=(y1;y2)∈R×Rn-1. For any ε>0, it follows from the spectral factorization of z and zε that
(31)ϕτ(x,y)=x+y-(λ1(z)u1(z)+λ2(z)u2(z)),ϕτ,ε(x,y)=x+y-(λ1(zε)u1(z)+λ2(zε)u2(z)),
where
(32)λi(z)=∥x∥2+∥y∥2+(τ-2)xTy--+(-1)i∥2x1x2+2y1y2--------+(τ-2)(x1y2+y1x2)∥,
and λi(zε) and ui(z) are, respectively, given by (18) and (19) for i=1,2. It is obvious that λi(zε)=λi(z)+2ε2 for i=1,2. Then,
(33)limε↓0(λ1(zε)u1(z)+λ2(zε)u2(z))=limε↓0(λ1(z)+2ε2u1(z)+λ2(z)+2ε2u2(z))=λ1(z)u1(z)+λ2(z)u2(z),
and hence, limε↓0ϕτ,ε(x,y)=ϕτ(x,y). Therefore, it follows from (i) and Definition 2 that ϕτ,ε is a smoothing function of ϕτ.Next, we give some properties ofϕτ [14], which will be used in the subsequent analysis.Lemma 4.
For anyx=(x1;x2),  y=(y1;y2)∈R×Rn-1, let z=(z1;z2)  =  x2+y2+  (τ-2)(x∘y)∈bdKn. Then one has
(34)(x+τ-22y)∘(x+τ-22y)′=0,y∘y′=0,(35)(x+τ-22y)∘z′=0,y∘z′=0,(36)(x+τ-22y)∘y′=0,(y+τ-22x)∘x′=0,(37)(y+τ-22x)∘(y+τ-22x)′=0,x∘x′=0,(38)(y+τ-22x)∘z′=0,x∘z′=0,(39)(x+τ-22y)∘(y+τ-22x)′=0,x∘y′=0.
Moreover, the following equivalence holds:
(40)z1=0⟺z2=0⟺z=0⟺x1=y1=0⟺x2=y2=0⟺(x,y)=(0,0).Proof.
From Lemma 3.3 and its proof in [14], it is not difficult to see that relations (34)–(39) hold. The equivalence is also true, since
(41)z=x2+y2+(τ-2)(x∘y)=(x+τ-22y)2+τ(4-τ)4y2=(y+τ-22x)2+τ(4-τ)4x2∈bdKn.
This completes the proof.


## 4. The Jacobian Consistency

In this section, we show the Jacobian consistency of the smoothing functionϕτ,ε, which will play an important role for establishing the rapid convergence of smoothing methods. Moreover, we estimate the distance between the gradient of the smoothing functions and the subgradient of the one-parametric class of the SOC complementarity functions, which will help to adjust a parameter appropriately in smoothing methods.It has been shown in Proposition 3.1 in [14] that the function ϕτ with any τ∈(0,4) satisfies
(42)ϕτ(x,y)=0⟺x∈Kn,y∈Kn,〈x,y〉=0.
Let (x,y,ζ)∈Rn×Rn×Rl and define
(43)Φτ,ε(x,y,ζ):=(F(x,y,ζ)ϕτ,ε(x,y)),Φτ(x,y,ζ):=(F(x,y,ζ)ϕτ(x,y)).
It is easy to see that Φτ,ε(x,y,ζ)=0 is the perturbation of the system of equations Φτ(x,y,ζ)=0. On account of (1), (42), and (43), we have
(44)Φτ(x,y,ζ)=0⟺(x,y,ζ)solves(1).
Since Φτ(x,y,ζ) is typically nonsmooth, we can solve approximately the smooth system Φτ,ε(x,y,ζ)=0 by using Newton’s method at each iteration, and then obtain a solution of Φτ(x,y,ζ)=0 by reducing the parameter ε to zero.First, we show that the functionΦτ,ε(x,y,ζ) satisfies the Jacobian consistency.Lemma 5.
For any(x,y)∈Rn×Rn, one has
(45)Jϕτ0(x,y):=limε→0∇ϕτ,ε(x,y)=(I-L(x+τ-22y)JI-L(y+τ-22x)J),
where
(46)J:={L-1(w)if(x,y)∈S1,122z1(1z-2Tz-24I-3z-2z-2T)if(x,y)∈S2,Oif(x,y)=(0,0).Proof.
By (27) and the symmetry of x and y, it suffices to show that
(47)limε→0L(x+τ-22y)L-1(wε)=L(x+τ-22y)J.Case  (i). If (x,y)∈S1, we have from (18) that
(48)limε→0λi(zε)=limε→0[λi(z)+2ε2]=λi(z)>0,
for i=1,2, and then,
(49)limε→0wε=limε→0[λ1(zε)u1(z)+λ2(zε)u2(z)]=λ1(z)u1(z)+λ2(z)u2(z)=w∈intKn.
Therefore,
(50)limε→0L(x+τ-22y)L-1(wε)=L(x+τ-22y)L-1(w).Case  (ii). If (x,y)∈S2, we obtain (x,y)≠(0,0), and (41) holds, and thus,
(51)∥z2∥=z1=∥x+τ-22y∥2+τ(4-τ)4∥y∥2>0,2z1=λ2(z)>λ1(z)=0.
Then we have from (18) that
(52)λ1(zε)=λ1(z)+2ε2=2ε2>0,λ2(zε)=λ2(z)+2ε2=2z1+2ε2>0.
For any ε≠0, it follows from (28) that L-1(wε)=L1(zε)+L2(zε). We first show that L(x+((τ-2)/2)y)L1(zε)=O for any ε≠0. Let
(53)ω:=(1;z-2)=1∥z2∥(z1;z2)=zz1.
By (35), we have
(54)(x+τ-22y)∘ω′=(x+τ-22y)∘z′z1=0,
and therefore,
(55)L(x+τ-22y)L1(zε)=12λ1(zε)L(x+τ-22y)ω′ω′T=122|ε|(x+τ-22y)∘ω′ω′T=O.
We next show that limε→0L2(zε)=J. In fact, we obtain from (52)
(56)limε→0L2(zε)=limε→0122z1+2ε2(1z-2Tz-2z-2z-2T)+limε→022ε2+2z1+2ε2(00T0I-z-2z-2T),=122z1(1z-2Tz-2z-2z-2T)+22z1(00T0I-z-2z-2T),=122z1(1z-2Tz-24I-3z-2z-2T)=J.
Combining (55) and (56) yields
(57)limε→0L(x+τ-22y)L-1(wε)=limε→0L(x+τ-22y)L2(zε)=L(x+τ-22y)J.Case  (iii). If (x,y)=(0,0), then wε=zε=2|ε|e∈intKn, and
(58)limε→0L(x+τ-22y)L-1(wε)=limε→0O·12|ε|e=O=L(x+τ-22y)J.
This completes the proof.Lemma 6.
For any(x,y)∈Rn×Rn, one has
(59)(I-UxI-Uy)∈∂Bϕτ(x,y),
where
(60)Ux=±Z+L(x+τ-22y)J,Uy=±τ-22Z+L(y+τ-22x)J,(61)Z={Oif(x,y)∈S1,12(1-z-2T-z-2z-2z-2T)if(x,y)∈S2,Iif(x,y)=(0,0),J={L-1(w)if(x,y)∈S1,122z1(1z-2Tz-24I-3z-2z-2T)if(x,y)∈S2,Oif(x,y)=(0,0).Proof.
It follows from Proposition  5.2 [1] and the chain rule for differentiation that the one-parametric class of SOC complementarity functions ϕτ is continuously differentiable at any (x,y)∈S1 with
(62)∇ϕτ(x,y)=(I-L(x+τ-22y)L-1(w)I-L(y+τ-22x)L-1(w))∈∂Bϕτ(x,y).
Therefore, it suffices to consider the two cases: (x,y)∈S2 and (x,y)=(0,0).
For any(x,y)∈S2 or (x,y)=(0,0), let (x^,y)=(x+εe,y) with sufficiently small ε≠0, and define
(63)z^=(z^1;z^2):=x^2+y2+(τ-2)(x^∘y),w^=(w^1;w^2):=z^,z~2:=z^2∥z^2∥,λ^i=λi(z^):=z^1+(-1)i∥z^2∥,i=1,2.
Direct calculations yield
(64)z^=(x+εe)2+y2+(τ-2)(x+εe)∘y=x2+y2+(τ-2)(x∘y)+2εx+ε(τ-2)y+ε2e=z+2εx+ε(τ-2)y+ε2e.
Therefore, we obtain
(65)z^1=z1+2εx1+ε(τ-2)y1+ε2,z^2=z2+2εx2+ε(τ-2)y2,(66)λ^i=z1+2εx1+ε(τ-2)y1+ε2+(-1)i×∥z2+2εx2+ε(τ-2)y2∥,i=1,2.
It is obvious that as ε→0, we have (x^,y)→(x,y),  z^→z,w^→w, and λ^i→λi(z) for i=1,2.
By the definition ofB-subdifferential and (3), it suffices to show that
(67)limε→0L(x^+τ-22y)L-1(w^)=Ux,limε→0L(y+τ-22x^)L-1(w^)=Uy,
if ϕτ is differentiable at (x^,y).
Case  (i). If (x,y)=(0,0), it is easy to see that z^=ε2e∈intKn,  w^=|ε|e, and ϕτ is differentiable at (x^,y). Then, we have
(68)limε→±0L(x^+τ-22y)L-1(w^)=limε→±0εI·1|ε|I=limε→±0sgn(ε)I=±I=Ux,limε→±0L(y+τ-22x^)L-1(w^)=limε→±0τ-22εI·1|ε|I=limε→±0τ-22sgn(ε)I=±τ-22I=Uy.Case  (ii). If (x,y)∈S2, we obtain z∈bdKn∖{0}, and thus, from (38) and (39),
(69)∥z^2∥2=∥z2+2εx2+ε(τ-2)y2∥2=∥z2∥2+4ε2∥x2∥2+ε2(τ-2)2∥y2∥2+4εz2Tx2+2ε(τ-2)z2Ty2+4ε2(τ-2)x2Ty2=z12+4ε2x12+ε2(τ-2)2y12+4εz1x1+2ε(τ-2)z1y1+4ε2(τ-2)x1y1=[z1+2εx1+ε(τ-2)y1]2.
Since z1>0 and ε is sufficiently small, we obtain
(70)∥z^2∥=z1+2εx1+ε(τ-2)y1>0.
By (66) and (70), we have
(71)λ^1=z1+2εx1+ε(τ-2)y1+ε2-∥z^2∥=ε2>0,(72)λ^2=z1+2εx1+ε(τ-2)y1+ε2+∥z^2∥=2[z1+2εx1+ε(τ-2)y1]+ε2>0.
Relations (71) and (72) imply that z^∈intKn, and thus, ϕτ is differentiable at (x^,y).
Now we will show that(73)limε→0L(x^+τ-22y)L-1(w^)=Ux.
By Proposition 5.2 in [1], we have L-1(w^)=L1(z^)+L2(z^), where L1(z^) and L2(z^) are given by (29) with z^ and z~2 replacing zε and z-2, respectively. By (70),
(74)ω^:=(1;z~2)=1∥z^2∥(∥z^2∥;z^2)=1∥z^2∥(∥z2+2εx2+ε(τ-2)y2∥;-----∥z2∥z2+2εx2+ε(τ-2)y2)=1∥z^2∥(z1+2εx1+ε(τ-2)y1;-----z2+2εx2+ε(τ-2)y2)=1∥z^2∥[z+2εx+ε(τ-2)y].
According to (35), (39), (71), and (74), we have
(75)(x+τ-22y)∘ω^′=1∥z^2∥(x+τ-22y)∘[z+2εx+ε(τ-2)y]′=1∥z^2∥[(x+τ-22y)∘z′------+2ε(x+τ-22y)∘(x+τ-22y)′]=0,
and thus,
(76)L(x^+τ-22y)L1(z^)=12λ^1(x+εe+τ-22y)∘ω^′ω^′T=12|ε|[(x+τ-22y)∘ω^′ω^′T+εω^′ω^′T]=sgn(ε)2ω^′ω^′T=sgn(ε)2(1-z~2T-z~2z~2z~2T).
It follows from (70), (71), (72), and (74) that λ^1→λ1(z)=0,  λ^2→λ2(z)=2z1, and z~2→z-2 as ε→0. Then, by following the proof of Case (ii) in Lemma 5, we obtain
(77)limε→0L2(z^)=122z1(1z-2Tz-24I-3z-2z-2T)=J.
Hence, we have from (76) and (77) that
(78)limε→±0L(x^+τ-22y)L-1(w^)=limε→±0L(x^+τ-22y)L1(z^)+limε→±0L(x^+τ-22y)L2(z^)=limε→±0sgn(ε)2ω^′ω^′T+limε→±0L(x^+τ-22y)J=±Z+L(x+τ-22y)J=Ux.
Next, we will show that(79)limε→0L(y+τ-22x^)L-1(w^)=Uy.
By (38), (39), and (74), we have
(80)(y+τ-22x)∘ω^′=1∥z^2∥(y+τ-22x)∘[z+2εx+ε(τ-2)y]′=1∥z^2∥[(x+τ-22y)′(y+τ-22x)∘z′+2ε(y+τ-22x)-------∘(x+τ-22y)′]=0,
and then,
(81)L(y+τ-22x^)L1(z^)=12λ^1(y+τ-22x+τ-22εe)∘ω^′ω^′T=12|ε|[(y+τ-22x)∘ω^′ω^′T+τ-22εω^′ω^′T]=τ-24sgn(ε)ω^′ω^′T=τ-24sgn(ε)(1-z~2T-z~2z~2z~2T).
Hence, we have from (77) and (81) that
(82)limε→±0L(y+τ-22x^)L-1(w^)=limε→±0L(y+τ-22x^)L1(z^)+limε→±0L(y+τ-22x^)L2(z^)=limε→±0τ-24sgn(ε)ω^′ω^′T+limε→±0L(y+τ-22x^)J=±τ-22Z+L(y+τ-22x)J=Uy.
This completes the proof.Theorem 7.
The functionΦτ,ε defined by (43) with ε>0 satisfies the Jacobian consistency.Proof.
By (43), it suffices to show the Jacobian consistency of the function ϕτ,ε with ε>0. Let
(83)Vi:=(I-UxiI-Uyi),
where
(84)Uxi=(-1)iZ+L(x+τ-22y)J,Uyi=(-1)iτ-22Z+L(y+τ-22x)J,
for i=1,2, and Z and J are given by (61). Define
(85)V:=12(V1+V2)=(I-L(x+τ-22y)JI-L(y+τ-22x)J).
On the one hand, we obtain from Lemma 5 that V=Jϕτ0(x,y)=limε→0∇ϕτ,ε(x,y). On the other hand, we have by Lemma 6 that V1,V2∈∂Bϕτ(x,y), and therefore, V=(1/2)(V1+V2)∈∂ϕτ(x,y). This together with Theorem 3 and Definition 1 implies the Jacobian consistency of ϕτ,ε with ε>0.Now, we are in a position to estimate an upper bound of the parameterε>0 for the predicted accuracy of the distance between the gradient of Φτ,ε and the subgradient of Φτ.Theorem 8.
Letτ∈(0,4),  δ>0 be given, and (x,y,ζ)∈R2n+l be any point. Let θτ(x,y) be any function such that
(86)θτ(x,y)≥∥L(x+τ-22y)JL(y+τ-22x)J∥,
and let ε-:R2n×R+→R+∪{+∞} be defined by
(87)ε-(x,y,δ):={λ1(z)δ2(θτ(x,y)2-λ1(z)δ2)if(x,y)∈S1,δ<θτ(x,y)λ1(z),z1δ2θτ(x,y)(2θτ(x,y)-δ2z1)if(x,y)∈S2,δ<2θτ(x,y)2z1,+∞otherwise.
Then, for any ε∈R such that 0<|ε|≤ε-(x,y,δ), we have
(88)dist(∇Φτ,ε(x,y,ζ),∂Φτ(x,y,ζ))<δ.Proof.
Since it follows from the proof of Theorem7 that Jϕτ0(x,y)=V∈∂ϕτ(x,y) for any (x,y,ζ)∈R2n+l, we obtain
(89)JΦτ0(x,y,ζ):=(Jϕτ0(x,y)∇x,yF(x,y,ζ)O∇ζF(x,y,ζ))∈∂Φτ(x,y,ζ).
Therefore, we have from (27) and (85) that
(90)dist(∇Φτ,ε(x,y,ζ),∂Φτ(x,y,ζ))=min{∥∇Φτ,ε(x,y,ζ)-W∥:W∈∂Φτ(x,y,ζ)}≤∥∇Φτ,ε(x,y,ζ)-JΦτ0(x,y,ζ)∥=∥∇ϕτ,ε(x,y,ζ)-Jϕτ0(x,y,ζ)∥=∥L(x+τ-22y)(L-1(w^)-J)L(y+τ-22x)(L-1(w^)-J)∥.
Then, by following the proof of Proposition 4.1 in [12], we obtain
(91)dist(∇Φτ,ε(x,y,ζ),∂Φτ(x,y,ζ))≤[hτ,0(x,y)-hτ,ε(x,y)]×∥L(x+τ-22y)JL(y+τ-22x)J∥,
where hτ,ε:R2n→R+ is defined as
(92)hτ,ε(x,y):={1λ1(z)+2ε2if(x,y)∈S1,2z1+ε2+|ε|if(x,y)∈S2,0if(x,y)=(0,0).
Hence, similar to the proof of Theorem 4.1 in [12], we have the desired result.


## 5.  An Alternative Proof

In this section, we study the directional derivative andB-subdifferential of the one-parametric class of SOC complementarity functions ϕτ. Based on these results, we present an alternative way to prove the Jacobian consistency of the one-parametric class of smoothing functions ϕτ,ε.By Corollary 3.3 in [18], it is not difficult to see that the function ϕτ given as (3) is directionally differentiable everywhere. However, as far as we know, the expression of its directional derivative is not given in the available literature. In this section, we derive its expression and prove that the B-subdifferential of ϕτ(x,y) at a general point coincides with that of its directional derivative function at the origin.In light of theB-subdifferential of ϕFB(x,y) [4, 10, 19], we obtain the following four results, which can be shown by following the proofs of Proposition 9, Lemma 11, Lemma 12, and Proposition 13 in [4], respectively.Proposition 9.
For any givenx=(x1;x2),  y=(y1;y2)∈R×Rn-1, the directional derivative ϕτ′((x,y);(g,h)) of ϕτ at (x,y) has the following form.(i)
If(x,y)=(0,0), then ϕτ′((x,y);(g,h))=ϕτ(g,h).(ii)
If(x,y)∈S1, then(93)ϕτ′((x,y);(g,h))=[I-L-1(w)L(x+τ-22y)]g+[I-L-1(w)L(y+τ-22x)]h.(iii)
If(x,y)∈S2, then
(94)ϕτ′((x,y);(g,h))=g+h-φ(g,h)2(1-z2)+(x2+((τ-2)/2)y2)Tg2+(y2+((τ-2)/2)x2)Th22x12+y12+(τ-2)x1y1(0z2)-12x12+y12+(τ-2)x1y1×((x+τ-22y)Tg+(y+τ-22x)Th2(x1+τ-22y1)g2+g1(x2+τ-22y2)+2(y1+τ-22x1)h2+h1(y2+τ-22x2)),where g=(g1;g2),  h=(h1;h2)∈R×Rn-1,  z-2 is given as (20), and the function φ:Rn×Rn→R is defined by
(95)φ(g,h)=(θ1(g,h))2+∥θ2(g,h)∥2x12+y12+(τ-2)x1y1,
with
(96)θ1(g,h)=(x1+τ-22y1)g1-(x2+τ-22y2)Tg2+(y1+τ-22x1)h1-(y2+τ-22x2)Th2,θ2(g,h)=(x1+τ-22y1)h2-h1(x2+τ-22y2)+g1(y2+τ-22x2)-(y1+τ-22x1)g2.Lemma 10.
For any given(x,y)∈S2, let φ be defined by (95). Then, the B-subdifferential of the function φ at (0,0) is (97)∂Bφ(0,0)={[(ζ2T(y2+((τ-2)/2)x2)+ζ1(x1+((τ-2)/2)y1)x12+y12+(τ-2)x1y1;-(y1+((τ-2)/2)x1)ζ2-ζ1(x2+((τ-2)/2)y2)x12+y12+(τ-2)x1y1)(-ζ2T(x2+((τ-2)/2)y2)+ζ1(y1+((τ-2)/2)x1)x12+y12+(τ-2)x1y1;(x1+((τ-2)/2)y1)ζ2-ζ1(y2+((τ-2)/2)x2)x12+y12+(τ-2)x1y1)][(-(y1+((τ-2)/2)x1)ζ2-ζ1(x2+y2)x12+y12+(τ-2)x1y1)((x1+y1)ζ2-ζ1(y2+x2)x12+y12+(τ-2)x1y1)]|[(-(y1+((τ-2)/2)x1)ζ2-ζ1(x2+y2)x12+y12+(τ-2)x1y1)((x1+y1)ζ2-ζ1(y2+x2)x12+y12+(τ-2)x1y1)]ζ=(ζ1;ζ2)∈R×Rn-1satisfyingζ12+∥ζ2∥2=1}.Lemma 11.
For any given(x,y)∈Rn×Rn, let ψτ(·,·)≡ϕτ′((x,y);(·,·)). Then,
(98)∂Bϕτ(x,y)=∂Bψτ(0,0).Proposition 12.
For any given(x,y)∈S2, we have
(99)∂Bϕτ(x,y)={(I-X-pu1(z)TI-Y-qu1(z)T)|p=(p1;p2),(ζ2T(y2+((τ-2)/2)x2)+x12+y12+(τ-2)x1y1q=(q1;q2)satisfyingp=(ζ2T(y2+((τ-2)/2)x2)+ζ1(x1+((τ-2)/2)y1)x12+y12+(τ-2)x1y1;-(y1+((τ-2)/2)x1)ζ2-ζ1(x2+((τ-2)/2)y2)x12+y12+(τ-2)x1y1),q=(-ζ2T(x2+((τ-2)/2)y2)+ζ1(y1+((τ-2)/2)x1)x12+y12+(τ-2)x1y1;(x1+(τ-22)y1)ζ2-ζ1(y2+(τ-22)x2)x12+y12+(τ-2)x1y1),(ζ2T(y2+((τ-2)/2)x2)+x12+y12+(τ-2)x1y1forsomeζ=(ζ1;ζ2)suchthatζ12+∥ζ2∥2=1},
where u1(z) and z-2 are defined by (19) and (20), respectively, and (100)X=12x12+y12+(τ-2)x1y1×(x1+τ-22y1x2T+τ-22y2Tx2+τ-22y22(x1+τ-22y1)I-(x2+τ-22y2)z-2T),Y=12x12+y12+(τ-2)x1y1×(y1+τ-22x1y2T+τ-22x2Ty2+τ-22x22(y1+τ-22x1)I-(y2+τ-22x2)z-2T).Theorem 1313.
The functionϕτ,ε defined by (13) with ε>0 satisfies the Jacobian consistency.Proof.
By the proof of Theorem7, it suffices to show that V∈∂ϕτ(x,y), where V is defined by (85). From Proposition 9 and Lemma 11, it is obvious that the result holds if (x,y)∈S1 or (x,y)=(0,0). Now, we prove V∈∂ϕτ(x,y) if (x,y)∈S2. From Proposition 12, it is not difficult to see that
(101)(I-XI-Y)∈∂ϕτ(x,y),
for any given (x,y)∈S2. Direct calculations together with Lemma 4 yield
(102)X=L(x+τ-22y)J,Y=L(y+τ-22x)J,
if (x,y)∈S2. Therefore, it follows from (101) that for any given (x,y)∈S2,
(103)V≡(I-XI-Y)∈∂ϕτ(x,y),
which completes the proof.


## 6. Conclusions

In this paper, we show the Jacobian consistency of the one-parametric class of smoothing functionsϕτ,ε, which will play an important role for establishing the rapid convergence of smoothing methods. Moreover, we estimate the distance between the gradient of the smoothing functions ϕτ,ε and the subgradient of the one-parametric class of the SOC complementarity functions ϕτ, which will help to adjust a parameter appropriately in smoothing methods. The related topics for further study are about smoothing methods for solving the SOCCP via the Jacobian consistency of the one-parametric class of smoothing functions.



---

*Source: 965931-2013-06-10.xml*