DOI 10.1007/s00245-008-9054-9

**A Damped Gauss-Newton Method**

**for the Second-Order Cone Complementarity Problem**

**Shaohua Pan· Jein-Shan Chen**

Published online: 6 August 2008

© Springer Science+Business Media, LLC 2008

**Abstract We investigate some properties related to the generalized Newton method**
for the Fischer-Burmeister (FB) function over second-order cones, which allows us
to reformulate the second-order cone complementarity problem (SOCCP) as a semi-
smooth system of equations. Specifically, we characterize the B-subdifferential of
the FB function at a general point and study the condition for every element of the B-
subdifferential at a solution being nonsingular. In addition, for the induced FB merit
function, we establish its coerciveness and provide a weaker condition than Chen
and Tseng (Math. Program. 104:293–327,2005) for each stationary point to be a
*solution, under suitable Cartesian P -properties of the involved mapping. By this, a*
damped Gauss-Newton method is proposed, and the global and superlinear conver-
gence results are obtained. Numerical results are reported for the second-order cone
programs from the DIMACS library, which verify the good theoretical properties of
the method.

**Keywords Second-order cones**· Complementarity · Fischer-Burmeister function ·
B-subdifferential· Generalized Newton method

S. Pan’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.

J.-S. Chen is member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. J.-S. Chen’s work is partially supported by National Science Council of Taiwan.

S. Pan (

^{)}

School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China e-mail:shhpan@scut.edu.cn

J.-S. Chen

Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan e-mail:jschen@math.ntnu.edu.tw

**1 Introduction**

*Consider the following conic complementarity problem of finding ζ*∈ R* ^{n}*such that

*F (ζ )∈ K,*

*G(ζ )∈ K,*

*F (ζ ), G(ζ ) = 0,*(1) where

*·, · represents the Euclidean inner product, F, G : R*

*→ R*

^{n}*are the map- ping assumed to be continuously differentiable throughout this paper, and*

^{m}*K is the*Cartesian product of second-order cones (SOCs), or called Lorentz cones. In other words,

*K = K*^{n}^{1}*× K*^{n}^{2}*× · · · × K*^{n}^{q}*,* (2)
*where q, n*1*, . . . , n*_{q}*≥ 1, n*1*+ · · · + n**q**= m and*

*K*^{n}* ^{i}*:=

*x= (x*1*, x*_{2}*)*∈ R × R^{n}^{i}^{−1}*| x*1*≥ x*2

with* · denoting the Euclidean norm and K*^{1} denoting the set of nonnegative re-
alsR_{+}. We will refer to (1)–(2) as the second-order cone complementarity problem
*(SOCCP). Corresponding to the Cartesian structure ofK, in the rest of this paper, we*
*always write F= (F*1*, . . . , F**q**)and G= (G*1*, . . . , G**q**)with F**i**, G**i* : R* ^{n}*→ R

^{n}*.*

^{i}*An important special case of the SOCCP corresponds to n= m and G(ζ ) = ζ for*
*all ζ*∈ R* ^{n}*. Then (1) and (2) reduce to

*F (ζ )∈ K, ζ ∈ K,* *F (ζ ), ζ = 0,* (3)

which is a natural extension of the nonlinear complementarity problem (NCP) over
the nonnegative orthant coneR^{n}_{+}. Another special case corresponds to the Karush-
Kuhn-Tucker (KKT) conditions for the convex second-order cone program (CSOCP):

*min g(x)*

s.t. *Ax= b, x ∈ K,* (4)

*where A*∈ R^{p}^{×m}*has full row rank, b*∈ R^{p}*and g*: R* ^{m}*→ R is a twice continu-
ously differentiable convex function. From [6], the KKT conditions of (4), which are
sufficient but not necessary for optimality, can be rewritten in the form of (1) with

*F (ζ ):= ˆx +(I −A*^{T}*(AA*^{T}*)*^{−1}*A)ζ,* *G(ζ ):= ∇g(F (ζ ))−A*^{T}*(AA*^{T}*)*^{−1}*Aζ,* (5)
where *ˆx ∈ R*^{m}*is any vector satisfying Ax= b. When g is a linear function, (4) re-*
duces to the standard second-order cone program which has extensive applications in
engineering design, finance, control, and robust optimization; see [1,14] and refer-
ences therein.

There have been many methods proposed for solving SOCPs and SOCCPs. They include the interior-point methods [1,2,14,16,24,26], the non-interior smoothing Newton methods [7, 11], the smoothing-regularization method [13], and the merit function approach [6]. Among others, the last three kinds of methods are all based

*on an SOC complementarity function. Specifically, a mapping φ*: R* ^{l}*× R

*→ R*

^{l}*is*

^{l}*called an SOC complementarity function associated with the coneK*

^{l}*(l≥ 1) if*

*φ (x, y)= 0 ⇐⇒ x ∈ K*^{l}*,* *y∈ K*^{l}*,* *x, y = 0.* (6)
*A popular choice of φ is the vector-valued Fischer-Burmeister (FB) function, de-*
fined by

*φ (x, y):= (x*^{2}*+ y*^{2}*)*^{1/2}*− (x + y) ∀x, y ∈ R** ^{l}* (7)

*where x*

^{2}

*= x ◦ x denotes the Jordan product of x and itself, x*

*denotes a vector*

^{1/2}*such that (x*

^{1/2}*)*

^{2}

*= x, and x + y means the usual componentwise addition of vectors.*

*From the next section, we see that φ in (7) is well-defined for all (x, y)*∈ R* ^{l}*×R

*. The function was shown in [11] to satisfy the equivalence (6), and therefore its squared norm*

^{l}*ψ (x, y)*:=1

2*φ(x, y)*^{2} (8)

*is a merit function for the SOCCP, i.e., ψ(x, y)= 0 if and only if x ∈ K*^{l}*, y* *∈ K** ^{l}*
and

*x, y = 0. The functions φ and ψ were studied in the literature [6,*21], where ψ was shown to be continuously differentiable everywhere by Chen and Tseng [6] and

*φ*was proved to be strongly semismooth by Sun and Sun [21].

In view of the characterization in (6), clearly, the SOCCP can be reformulated as the following nonsmooth system of equations:

*(ζ )*:=

⎛

⎜⎜

⎜⎜

⎜⎝

*φ (F*_{1}*(ζ ), G*_{1}*(ζ ))*
*...*
*φ (F**i**(ζ ), G**i**(ζ ))*

*...*
*φ (F**q**(ζ ), G**q**(ζ ))*

⎞

⎟⎟

⎟⎟

⎟⎠= 0 (9)

*where φ is defined as in (7) with a suitable dimension l. By Corollary 3.3 of [21], it is*
*not hard to show that the operator *: R* ^{n}*→ R

*in (9) is semismooth. Furthermore, from Proposition 2 of [6], its squared norm induces a smooth merit function, given by*

^{m}*(ζ )*:=1

2*(ζ )*^{2}=
*q*

*i*=1

*ψ (F**i**(ζ ), G**i**(ζ )).* (10)
*In this paper, we mainly characterize the B-subdifferential of φ at a general point*
*and present an estimate for the B-subdifferential of . By this, a condition is given*
*to guarantee every element of the B-subdifferential of at a solution to be nonsin-*
gular, which plays an important role in the local convergence analysis of nonsmooth
Newton methods for the SOCCP. In addition, two important results are also presented
*for the merit function (ζ ). One of them shows that each stationary point of is a*
solution of the SOCCP under a weaker condition than the one used by [6], and the

*other establishes the coerciveness of for the SOCCP (3) under the uniform Carte-*
*sian P -property of F . Based on these results, we finally propose a damped Gauss-*
Newton method by applying the generalized Newton method [19,20] for the system
(9), and analyze its global and superlinear (quadratic) convergence. Numerical re-
sults are reported for the SOCPs from the DIMACS library [18], which verify the
good theoretical properties of the method.

*Throughout this paper, I represents an identity matrix of suitable dimension,*R^{n}*denotes the space of n-dimensional real column vectors, and*R^{n}^{1} × · · · × R^{n}* ^{q}* is
identified withR

^{n}^{1}

^{+···+n}

^{q}*. Thus, (x*1

*, . . . , x*

_{q}*)*∈ R

^{n}^{1}× · · · × R

^{n}*is viewed as a col- umn vector inR*

^{q}

^{n}^{1}

^{+···+n}

^{q}*. For any differentiable mapping F*: R

*→ R*

^{n}*, the nota- tion*

^{m}*∇F (x) ∈ R*

^{n}

^{×m}*denotes the transpose of the Jacobian F*

^{ }

*(x). For a symmetric*

*matrix A, we write A O (respectively, A O) if A is positive definite (respec-*

*tively, positive semidefinite). Given a finite number of square matrices Q*1

*, . . . , Q*

*q*, we denote the block diagonal matrix with these matrices as block diagonals by

*diag(Q*1

*, . . . , Q*

_{q}*)or by diag(Q*

*i*

*, i= 1, . . . , q). If J and B are index sets such*that

*J , B ⊆ {1, 2, . . . , q}, we denote by P*

*the block matrix consisting of the sub-*

_{J B}*matrices P*

*j k*∈ R

^{n}

^{j}

^{×n}

^{k}*of P with j∈ J , k ∈ B, and denote by x*

*a vector consisting*

_{B}*of sub-vectors x*

*i*∈ R

^{n}

^{i}*with i∈ B.*

**2 Preliminaries**

This section recalls some background materials and preliminary results that will be
used in the subsequent sections. We start with the interior and the boundary of*K*^{l}*(l >1). It is known thatK** ^{l}*is a closed convex self-dual cone with nonempty interior
given by

*int(K*^{l}*)*:=

*x= (x*1*, x*2*)*∈ R × R^{l}^{−1}*| x*1*>x*2
and the boundary given by

*bd(K*^{l}*)*:=

*x= (x*1*, x*2*)*∈ R × R^{l}^{−1}*| x*1*= x*2
*.*

*For any x= (x*1*, x*2*), y= (y*1*, y*2*)*∈ R × R^{l}^{−1}*, we define their Jordan product [9] as*
*x◦ y :=*

*x, y, x*1*y*_{2}*+ y*1*x*_{2}
*.*

The Jordan product “◦”, unlike scalar or matrix multiplication, is not associative,
which is the main source on complication in the analysis of SOCCP. The identity
*element under this product is e:= (1, 0, . . . , 0)** ^{T}* ∈ R

^{l}*. For each x= (x*1

*, x*

_{2}

*)*∈ R × R

^{l}^{−1}

*, define the matrix L*

*x*by

*L** _{x}*:=

*x*_{1} *x*_{2}^{T}*x*_{2} *x*_{1}*I*

*,*

which can be viewed as a linear mapping fromR* ^{l}*toR

*with the following properties.*

^{l}**Property 2.1**

*(a) L**x**y= x ◦ y and L**x**+y**= L**x**+ L**y**for any x, y*∈ R* ^{l}*.

*(b) x∈ K*

^{l}*⇐⇒ L*

*x*

*O and x ∈ int(K*

^{l}*)⇐⇒ L*

*x*

*O.*

*(c) L**x**is invertible whenever x∈ int(K*^{l}*)with the inverse L*^{−1}_{x}*given by*

*L*^{−1}* _{x}* = 1

*det(x)*

*x*_{1} *−x*^{T}_{2}

*−x*2 *det(x)*

*x*1 *I*+^{x}^{2}_{x}^{x}_{1}^{2}^{T}

*,* (11)

*where det(x):= x*_{1}^{2}*− x*2^{2}*denotes the determinant of x.*

In the following, we recall from [9,11] that each x*= (x*1*, x*_{2}*)*∈ R × R* ^{l−1}*admits
a spectral factorization, associated with

*K*

*, of the form*

^{l}*x= λ*1*(x)· u*^{(1)}_{x}*+ λ*2*(x)· u*^{(2)}_{x}*,*

*where λ*1*(x), λ*2*(x)and u*^{(1)}*x* *, u*^{(2)}*x* are the spectral values and the associated spectral
*vectors of x, respectively, defined by*

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

2*(1, (−1)*^{i}*¯x*2*),* *i= 1, 2,*

with *¯x*2*= x*2*/x*2* if x*2*= 0 and otherwise ¯x*2being any vector in R^{l}^{−1} satisfying

* ¯x*2* = 1. If x*2*= 0, the factorization is unique. The spectral factorizations of x, x*^{2}
*and x** ^{1/2}*have various interesting properties, and some of them are summarized as
follows.

* Property 2.2 For any x= (x*1

*, x*

_{2}

*)*∈ R × R

^{l}^{−1}

*, let λ*1

*(x), λ*

_{2}

*(x)and u*

^{(1)}*x*

*, u*

^{(2)}

_{x}*be the*

*spectral values and the associated spectral vectors. Then, the following results hold.*

*(a) x∈ K*^{l}*⇐⇒ 0 ≤ λ*1*(x)≤ λ*2*(x)and x∈ int(K*^{l}*)* *⇐⇒ 0 < λ*1*(x)≤ λ*2*(x).*

*(b) x*^{2}*= [λ*1*(x)*]^{2}*u*^{(1)}_{x}*+ [λ*2*(x)*]^{2}*u*^{(2)}_{x}*∈ K*^{l}*for any x*∈ R* ^{l}*.

*(c) If x∈ K*

^{l}*, then x*

*=√*

^{1/2}*λ*_{1}*(x) u*^{(1)}* _{x}* +√

*λ*_{2}*(x) u*^{(2)}_{x}*∈ K** ^{l}*.

Now we recall the concepts of the B-subdifferential and (strong) semismoothness.

*Given a mapping H*: R* ^{n}*→ R

^{m}*, if H is locally Lipschitz continuous, then the set*

*∂*_{B}*H (z)*:=

*V* ∈ R^{m}^{×n}*| ∃{z*^{k}*} ⊆ D**H**: z*^{k}*→ z, H*^{
}*(z*^{k}*)→ V*

*is nonempty and is called the B-subdifferential of H at z, where D**H*⊆ R* ^{n}*denotes the

*set of points at which H is differentiable. The convex hull ∂H (z):= conv∂*

*B*

*H (z)*is the generalized Jacobian of Clarke [4]. Semismoothness was originally introduced by Mifflin [15] for functionals. Smooth functions, convex functionals, and piecewise lin- ear functions are examples of semismooth functions. Later, Qi and Sun [20] extended

*the definition of semismooth functions to a mapping H*: R

*→ R*

^{n}

^{m}*. H is called semi-*

*smooth at x if H is directionally differentiable at x and for all V*

*∈ ∂H(x + h) and*

*h*→ 0,

*V h− H*^{
}*(x; h) = o(h);*

*H* *is called strongly semismooth at x if H is semismooth at x and for all V* ∈

*∂H (x+ h) and h → 0,*

*V h− H*^{
}*(x; h) = O(h*^{2}*)*;

*H* *is called (strongly) semismooth if it is (strongly) semismooth everywhere. Here,*
*o(h) means a function α : R** ^{n}* → R

*satisfying lim*

^{m}*h*→0

*α(h)/h = 0, while*

*O(h*

^{2}

*)denotes a function α*: R

*→ R*

^{n}*satisfying*

^{m}*α(h) ≤ Ch*

^{2}for all

*h ≤ δ*

*and some C > 0, δ > 0.*

*Next, we present the definitions of Cartesian P -properties for a matrix M*∈ R^{m}* ^{×m}*,
which are special cases of those introduced by Chen and Qi [5] for a linear transfor-
mation.

* Definition 2.1 A matrix M*∈ R

^{m}*is said to have*

^{×m}*(a) the Cartesian P -property if for any 0= x = (x*1*, . . . , x*_{q}*)*∈ R^{m}*with x**i* ∈ R^{n}* ^{i}*,

*there exists an index ν∈ {1, 2, . . . , q} such that x*

*ν*

*, (Mx)*

_{ν}*> 0;*

*(b) the Cartesian P*0-property if for any 0*= x = (x*1*, . . . , x**q**)*∈ R^{m}*with x**i* ∈ R^{n}* ^{i}*,

*there exists an index ν∈ {1, 2, . . . , q} such that x*

*ν*

*= 0 and x*

*ν*

*, (Mx)*

*≥ 0.*

_{ν}Some nonlinear generalizations of these concepts in the setting of*K are defined*
as follows.

* Definition 2.2 Given a mapping F= (F*1

*, . . . , F*

_{q}*)with F*

*i*: R

*→ R*

^{n}

^{n}

^{i}*, F is said to*

*(a) have the uniform Cartesian P -property if for any x= (x*1

*, . . . , x*

_{q}*), y= (y*1

*, . . . ,*

*y*_{q}*)*∈ R^{m}*, there is an index ν∈ {1, 2, . . . , q} and a positive constant ρ > 0 such*

that

*x**ν**− y**ν**, F**ν**(x)− F**ν**(y)*

*≥ ρx − y*^{2};

*(b) have the Cartesian P*0*-property if for any x= (x*1*, . . . , x*_{q}*), y= (y*1*, . . . , y*_{q}*)*∈
R^{m}*and x= y, there exists an index ν ∈ {1, 2, . . . , q} such that*

*x*_{ν}*= y**ν* and *x**ν**− y**ν**, F*_{ν}*(x)− F**ν**(y) ≥ 0.*

*From the above definitions, if a continuously differentiable mapping F*: R* ^{n}*→ R

^{n}*has the uniform Cartesian P -property (Cartesian P*0-property), then

*∇F (x) at any*

*x*∈ R

^{n}*enjoys the Cartesian P -property (Cartesian P*0-property). In addition, we may

*see that, when n*1

*= · · · = n*

*q*

*= 1, the above concepts reduce to the definitions of P -*

*matrices and P -functions, respectively, for the NCP.*

Finally, we introduce some notations which will be used in the rest of this paper.

*For any x= (x*1*, x*2*), y= (y*1*, y*2*)*∈ R × R^{l}^{−1}*, we define w, z*: R* ^{l}*× R

*→ R*

^{l}*by*

^{l}*w= (w*1

*, w*2

*)= (w*1

*(x, y), w*2

*(x, y))= w(x, y) := x*

^{2}

*+ y*

^{2}

*,*

*z= (z*1*, z*_{2}*)= (z*1*(x, y), z*_{2}*(x, y))= z(x, y) := (x*^{2}*+ y*^{2}*)*^{1/2}*.*

(12)

*Clearly, w∈ K*^{l}*with w*1*= x*^{2} *+ y*^{2} *and w*2*= 2(x*1*x*_{2} *+ y*1*y*_{2}*). Let* *¯w*2=
*w*_{2}*/w*2* if w*2*= 0, and otherwise ¯w*2be any vector inR^{l}^{−1}satisfying* ¯w*2 = 1.

Then, using Property2.1(b) and (c), it is not hard to compute that
*z*=

√*λ*2*(w)*+√
*λ*1*(w)*

2 *,*

√*λ*2*(w)*−√
*λ*1*(w)*

2 *¯w*2

*∈ K*^{l}*.*

**3 B-Subdifferential of the FB Function**

*In this section, we characterize the B-subdifferential of the FB function φ at a general*
*point (x, y)*∈ R* ^{l}*×R

*. For this purpose, we need several important technical lemmas.*

^{l}*The first lemma characterizes the set of the points where z(x, y) is differentiable.*

Since the proof is direct by [3, Proposition 4] and formula (11), we here omit it.

**Lemma 3.1 The function z(x, y) in (12) is continuously differentiable at a point***(x, y)if and only if x*^{2}*+ y*^{2}*∈int(K*^{l}*). Moreover,*∇*x**z(x, y)=L**x**L*^{−1}_{z}*and*∇*y**z(x, y)*=
*L**y**L*^{−1}_{z}*, where L*^{−1}_{z}*= (1/*√

*w*1*)Iif w*2*= 0, and otherwise*
*L*^{−1}* _{z}* =

*b* *c¯w*^{T}_{2}
*c¯w*2 *aI+ (b − a) ¯w*2*¯w*_{2}^{T}

(13)
*with*

*a*= 2

√*λ*2*(w)*+√

*λ*1*(w),* *b*=1
2

1

√*λ*2*(w)*+ 1

√*λ*1*(w)*

*,*

*c*=1
2

1

√*λ*_{2}*(w)*− 1

√*λ*_{1}*(w)*

*.*

The following two lemmas extend the results of Lemmas 2 and 3 of [6], respec- tively. Since the proofs are direct by using the same technique as [6], we here omit them.

* Lemma 3.2 For any x= (x*1

*, x*2

*), y= (y*1

*, y*2

*)*∈ R × R

^{l}^{−1}

*with w= x*

^{2}

*+ y*

^{2}∈

*bd(K*

^{l}*), we have*

*x*^{2}_{1}*= x*2^{2}*,* *y*_{1}^{2}*= y*2^{2}*,* *x*1*y*1*= x*2^{T}*y*2*,* *x*1*y*2*= y*1*x*2*.*
*If, in addition, w*2*= 0, then w*^{2}*= 2w*_{1}^{2}*= 2w*2^{2}*= 4(x*_{1}^{2}*+ y*_{1}^{2}*)= 0 and*

*x*_{1}*¯w*2*= x*2*,* *x*_{2}^{T}*¯w*2*= x*1*,* *y*_{1}*¯w*2*= y*2*,* *y*_{2}^{T}*¯w*2*= y*1*.*

**Lemma 3.3 For any x***= (x*1*, x*_{2}*), y= (y*1*, y*_{2}*)*∈ R × R^{l−1}*with w*2*= 2(x*1*x*_{2}+
*y*_{1}*y*_{2}*)= 0, there holds that*

*x*1*+ (−1)*^{i}*x*_{2}^{T}*¯w*2

2

*≤ x*2*+ (−1)*^{i}*x*1*¯w*2^{2}*≤ λ**i**(w)* *for i= 1, 2.*

Based on Lemmas3.1–3.3, we are now in a position to present the representa-
*tion for the elements of the B-subdifferential ∂**B**φ (x, y)at a general point (x, y)*∈
R* ^{l}*× R

*.*

^{l}* Proposition 3.1 Given a general point (x, y)*∈ R

*× R*

^{l}

^{l}*, each element in ∂*

*B*

*φ (x, y)*

*is given by[V*

*x*

*− I V*

*y*

*− I] with V*

*x*

*and V*

*y*

*having the following representation:*

*(a) If x*^{2}*+ y*^{2}*∈ int(K*^{l}*), then V**x**= L*^{−1}_{z}*L*_{x}*and V**y**= L*^{−1}_{z}*L** _{y}*.

*(b) If x*

^{2}

*+ y*

^{2}

*∈ bd(K*

^{l}*)and (x, y)= (0, 0), then*

*V** _{x}*∈

1

2√
*2w*1

1 *¯w*_{2}^{T}

*¯w*2 *4I− 3 ¯w*2*¯w*^{T}_{2}

*L** _{x}*+1

2

1

*− ¯w*2

*u*^{T}

*V**y*∈

1

2√
*2w*1

1 *¯w*_{2}^{T}

*¯w*2 *4I− 3 ¯w*2*¯w*^{T}_{2}

*L**y*+1

2

1

*− ¯w*2

*v*^{T}

(14)

*for some u= (u*1*, u*2*), v= (v*1*, v*2*)*∈ R × R^{l}^{−1}*satisfying|u*1*| ≤ u*2* ≤ 1 and*

*|v*1*| ≤ v*2* ≤ 1, where ¯w*2*= w*2*/w*2.

*(c) If (x, y)= (0, 0), then V**x**∈ {L*_{ˆx}*}, V**y**∈ {L*_{ˆy}*} for some ˆx, ˆy with ˆx*^{2}*+ ˆy*^{2}= 1,
*or*

*V** _{x}*∈

1 2

1

*¯w*2

*ξ** ^{T}* +1

2

1

*− ¯w*2

*u** ^{T}*+ 2

0 0

*(I− ¯w*2*¯w*_{2}^{T}*)s*_{2} *(I− ¯w*2*¯w*^{T}_{2}*)s*_{1}

*V**y*∈

1 2

1

*¯w*2

*η** ^{T}* +1

2

1

*− ¯w*2

*v** ^{T}* + 2

0 0

*(I− ¯w*2*¯w*^{T}_{2}*)ω*2 *(I− ¯w*2*¯w*_{2}^{T}*)ω*1

(15)
*for some u= (u*1*, u*_{2}*), v= (v*1*, v*_{2}*), ξ= (ξ*1*, ξ*_{2}*), η= (η*1*, η*_{2}*)*∈ R × R^{l}^{−1} *such*
*that|u*1*| ≤ u*2* ≤ 1, |v*1*| ≤ v*2* ≤ 1, |ξ*1*| ≤ ξ*2* ≤ 1, |η*1*| ≤ η*2* ≤ 1, ¯w*2∈
R^{l}^{−1}*satisfying ¯w*2* = 1, and s = (s*1*, s*_{2}*), ω= (ω*1*, ω*_{2}*)*∈ R × R^{l}^{−1}*satisfying*

*s*^{2}*+ ω*^{2}*≤ 1/2.*

*Proof Let D**φ**denote the set of points where φ is differentiable. Recall that this set is*
characterized by Lemma3.1*since φ(x, y)= z(x, y) − (x + y), and moreover,*

*φ*_{x}^{
}*(x, y)= L*^{−1}*z* *L**x**− I,* *φ*_{y}^{
}*(x, y)= L*^{−1}*z* *L**y**− I ∀(x, y) ∈ D**φ**.*
*(a) In this case, φ is continuously differentiable at (x, y) by Lemma*3.1. Hence,

*∂*_{B}*φ (x, y)consists of a single element, i.e. φ*^{
}*(x, y)= [L*^{−1}_{z}*L*_{x}*− I L*^{−1}_{z}*L*_{y}*− I], and*
the result is clear.

*(b) Assume that (x, y)= (0, 0) satisfies x*^{2}*+ y*^{2}*∈ bd(K*^{l}*). Then w∈ bd(K*^{l}*)*and
*w*1*>*0, which means*w*2* = w*1*>0 and λ*2*(w) > λ*1*(w)*= 0. Observe that, when
*w*2*= 0, the matrix L*^{−1}*z* in (13) can be decomposed as the sum of

*L*_{1}*(w)*:= 1
2√

*λ*1*(w)*

1 *− ¯w*^{T}_{2}

*− ¯w*2 *¯w*2*¯w*^{T}_{2}

(16) and

*L*2*(w)*:= 1
2√

*λ*_{2}*(w)*

1 *¯w*_{2}^{T}

*¯w*2 4√
*λ*2*(w)*

√*λ*2*(w)*+√

*λ*1*(w)**(I− ¯w*2*¯w*^{T}_{2}*)+ ¯w*2*¯w*_{2}^{T}

(17)

with *¯w*2*= w*2*/w*2*. Consequently, φ*_{x}^{
} *and φ*_{y}^{
} can be rewritten as
*φ*_{x}^{
}*(x, y)= (L*1*(w)+ L*2*(w))L*_{x}*− I,*

*φ*_{y}^{
}*(x, y)= (L*1*(w)+ L*2*(w))L*_{y}*− I.* (18)
Let *{(x*^{k}*, y*^{k}*)} ⊆ D**φ* *be an arbitrary sequence converging to (x, y). Let w** ^{k}* =

*(w*

_{1}

^{k}*, w*

_{2}

^{k}*)= w(x*

^{k}*, y*

^{k}*)and z*

^{k}*= z(x*

^{k}*, y*

^{k}*)for each k, where w(x, y) and z(x, y) are*defined by (12). Since w2

*= 0, we without loss of generality assume w*

^{k}_{2}= 0 for

*each k. Let*

*¯w*

^{k}_{2}

*= w*

^{k}_{2}

*/w*

_{2}

^{k}*for each k. From (18), it follows that*

*φ*_{x}^{
}*(x*^{k}*, y*^{k}*)= (L*1*(w*^{k}*)+ L*2*(w*^{k}*))L*_{x}*k**− I,*

*φ*_{y}^{
}*(x*^{k}*, y*^{k}*)= (L*1*(w*^{k}*)+ L*2*(w*^{k}*))L*_{y}*k**− I.* (19)
Since lim*k*→∞*λ*_{1}*(w*^{k}*)= 0, lim**k*→∞*λ*_{2}*(w*^{k}*)= 2w*1*>*0 and lim*k*→∞ *¯w*^{k}_{2}*= ¯w*2, we
have

*k*lim→∞*L*2*(w*^{k}*)L*_{x}*k**= C(w)L**x* and lim

*k*→∞*L*2*(w*^{k}*)L*_{y}*k**= C(w)L**y* (20)
where

*C(w)*= 1

2√
*2w*1

1 *¯w*_{2}^{T}

*¯w*2 *4I− 3 ¯w*2*¯w*_{2}^{T}

*.* (21)

*Next we focus on the limit of L*1*(w*^{k}*)L*_{x}*k* *and L*1*(w*^{k}*)L*_{y}*k* *as k*→ ∞. By computing,

*L*1*(w*^{k}*)L*_{x}*k* =1
2

*u*^{k}_{1} *u*^{k}_{2}

*−u*^{k}_{1}*¯w*_{2}^{k}*− ¯w*^{k}_{2}*(u*^{k}_{2}*)*^{T}

*,*

*L*_{1}*(w*^{k}*)L*_{y}*k* =1
2

*v*_{1}^{k}*v*_{2}^{k}

*−v*_{1}^{k}*¯w*_{2}^{k}*− ¯w*^{k}_{2}*(v*_{2}^{k}*)*^{T}

*,*

where

*u*^{k}_{1}=*x*_{1}^{k}*− (x*_{2}^{k}*)*^{T}*¯w*_{2}^{k}

*λ*1*(w*^{k}*)*

*,* *u*^{k}_{2}=*x*_{2}^{k}*− x*_{1}^{k}*¯w*_{2}^{k}

*λ*1*(w*^{k}*)*
*,*

*v*^{k}_{1}=*y*_{1}^{k}*− (y*_{2}^{k}*)*^{T}*¯w*^{k}_{2}

*λ*1*(w*^{k}*)*

*,* *v*^{k}_{2}=*y*_{2}^{k}*− y*_{1}^{k}*¯w*^{k}_{2}

*λ*1*(w*^{k}*)*
*.*

By Lemma3.3,*|u*^{k}_{1}*| ≤ u*^{k}_{2}* ≤ 1 and |v*_{1}^{k}*| ≤ v*_{2}* ^{k}* ≤ 1. So, taking the limit (possibly

*on a subsequence) on L*1

*(w*

^{k}*)L*

_{x}*k*

*and L*1

*(w*

^{k}*)L*

_{y}*k*, respectively, gives

*L*_{1}*(w*^{k}*)L*_{x}*k*→1
2

*u*_{1} *u*_{2}

*−u*1*¯w*2 *− ¯w*2*u*^{T}_{2}

=1 2

1

*− ¯w*2

*u*^{T}

*L*_{1}*(w*^{k}*)L*_{y}*k*→1
2

*v*1 *v*2

*−v*1*¯w*2 *− ¯w*2*v*^{T}_{2}

=1 2

1

*− ¯w*2

*v*^{T}

(22)

*for some u= (u*1*, u*_{2}*), v= (v*1*, v*_{2}*)*∈ R × R^{l}^{−1}satisfying*|u*1*| ≤ u*2* ≤ 1 and |v*1| ≤

*v*2* ≤ 1. In fact, u and v are some accumulation point of the sequences {u** ^{k}*} and

*{v*

*}, respectively. From equations (19)–(22), we immediately obtain*

^{k}*φ*_{x}^{
}*(x*^{k}*, y*^{k}*)→ C(w)L**x*+1
2

1

*− ¯w*2

*u*^{T}*− I,*

*φ*_{y}^{
}*(x*^{k}*, y*^{k}*)→ C(w)L**y*+1
2

1

*− ¯w*2

*v*^{T}*− I.*

*This shows that φ*^{
}*(x*^{k}*, y*^{k}*)→ [V**x**−I V**y**−I] as k → ∞ with V**x**, V** _{y}*satisfying (14).

*(c) Assume that (x, y)= (0, 0). Let {(x*^{k}*, y*^{k}*)} ⊆ D**φ* be an arbitrary sequence
*converging to (x, y). Let w*^{k}*= (w*_{1}^{k}*, w*_{2}^{k}*)= w(x*^{k}*, y*^{k}*)and z*^{k}*= z(x*^{k}*, y*^{k}*)for each k.*

*Since w= 0, we without any loss of generality assume that w*^{k}_{2}*= 0 for all k, or*
*w*^{k}_{2}*= 0 for all k.*

*Case (1): w*^{k}_{2}*= 0 for all k. From Lemma* 3.1, it follows that L^{−1}

*z*^{k}*= (1/*
*w*_{1}^{k}*)I*.
Therefore,

*φ*_{x}^{
}*(x*^{k}*, y*^{k}*)*= 1

*w*_{1}^{k}

*L*_{x}*k**− I and φ*_{y}^{
}*(x*^{k}*, y*^{k}*)*= 1

*w*_{1}^{k}

*L*_{y}*k**− I.*

*Since w*^{k}_{1}*= x*^{k}^{2}*+ y*^{k}^{2}*, every element in φ*^{
}_{x}*(x*^{k}*, y*^{k}*)and φ*_{y}^{
}*(x*^{k}*, y*^{k}*)*is bounded.

*Taking limit (possibly on a subsequence) on φ*_{x}^{
}*(x*^{k}*, y*^{k}*)and φ*_{y}^{
}*(x*^{k}*, y*^{k}*), we obtain*
*φ*^{
}_{x}*(x*^{k}*, y*^{k}*)* *→ L*_{ˆx}*− I and φ**y*^{
}*(x*^{k}*, y*^{k}*)* *→ L*_{ˆy}*− I*

for some vectors *ˆx, ˆy ∈ R** ^{l}* satisfying

*ˆx*

^{2}

*+ ˆy*

^{2}

*= 1, where ˆx and ˆy are some*accumulation point of the sequences{

^{}

^{x}

^{k}*w*_{1}* ^{k}*} and {

^{}

^{y}

^{k}*w*^{k}_{1}}, respectively. Thus, we prove
*that φ*^{
}*(x*^{k}*, y*^{k}*)→ [V**x**− I V**y**− I] as k → ∞ with V**x**∈ {L*_{ˆx}*} and V**y**∈ {L** _{ˆy}*}.

*Case (2): w*^{k}_{2}*= 0 for all k. Now φ*_{x}^{
}*(x*^{k}*, y*^{k}*)and φ*^{
}_{y}*(x*^{k}*, y*^{k}*)*are given as in (19). Using
the same arguments as part (b) and noting the boundedness of*{ ¯w*^{k}_{2}}, we have

*L*1*(w*^{k}*)L*_{x}*k* → 1
2

1

*− ¯w*2

*u*^{T}*,* *L*1*(w*^{k}*)L*_{y}*k* → 1
2

1

*− ¯w*2

*v** ^{T}* (23)

*for some u= (u*1*, u*2*), v= (v*1*, v*2*)*∈ R × R* ^{l−1}* satisfying

*|u*1

*| ≤ u*2 ≤ 1 and

*|v*1*| ≤ v*2* ≤ 1, and ¯w*2∈ R^{l}^{−1}satisfying * ¯w*2 = 1. We next compute the limit
*of L*2*(w*^{k}*)L*_{x}*k* *and L*2*(w*^{k}*)L*_{y}*k* *as k→ ∞. By the definition of L*2*(w)*in (17),

*L*_{2}*(w*^{k}*)L*_{x}*k*=1
2

*ξ*_{1}^{k}*(ξ*_{2}^{k}*)*^{T}

*ξ*_{1}^{k}*¯w*_{2}^{k}*+ 4(I − ¯w*_{2}^{k}*(¯w*^{k}_{2}*)*^{T}*)s*_{2}^{k}*¯w*^{k}_{2}*(ξ*_{2}^{k}*)*^{T}*+ 4(I − ¯w*_{2}^{k}*(¯w*^{k}_{2}*)*^{T}*)s*_{1}^{k}

*,*

*L*_{2}*(w*^{k}*)L*_{y}*k*=1
2

*η*^{k}_{1} *(η*^{k}_{2}*)*^{T}

*η*^{k}_{1}*¯w*^{k}_{2}*+ 4(I − ¯w*_{2}^{k}*(¯w*^{k}_{2}*)*^{T}*)ω*^{k}_{2} *¯w*_{2}^{k}*(η*^{k}_{2}*)*^{T}*+ 4(I − ¯w*^{k}_{2}*(¯w*_{2}^{k}*)*^{T}*)ω*_{1}^{k}

*,*

where

*ξ*_{1}* ^{k}*=

*x*

_{1}

^{k}*+ (x*

_{2}

^{k}*)*

^{T}*¯w*

^{k}_{2}

*λ*_{2}*(w*^{k}*)* *,* *ξ*_{2}* ^{k}*=

*x*

_{2}

^{k}*+ x*

_{1}

^{k}*¯w*

^{k}_{2}

*λ*

_{2}

*(w*

^{k}*),*

*η*^{k}_{1}=*y*_{1}^{k}*+ (y*_{2}^{k}*)*^{T}*¯w*^{k}_{2}

*λ*_{2}*(w*^{k}*)* *,* *η*_{2}* ^{k}*=

*y*

_{2}

^{k}*+ y*

_{1}

^{k}*¯w*

^{k}_{2}

*λ*_{2}*(w*^{k}*)* *,*

(24)

and

*s*_{1}* ^{k}*=

*x*

_{1}

^{k}*λ*2*(w*^{k}*)*+
*λ*1*(w*^{k}*)*

*,* *s*_{2}* ^{k}*=

*x*

_{2}

^{k}*λ*2*(w*^{k}*)*+
*λ*1*(w*^{k}*)*

*,*

*ω*^{k}_{1}= *y*_{1}^{k}

*λ*2*(w*^{k}*)*+
*λ*1*(w*^{k}*)*

*,* *ω*^{k}_{2}= *y*_{2}^{k}

*λ*2*(w*^{k}*)*+
*λ*1*(w*^{k}*)*

*.*

(25)

By Lemma3.3,*|ξ*_{1}^{k}*| ≤ ξ*_{2}^{k}* ≤ 1 and |η*_{1}^{k}*| ≤ η*^{k}_{2} ≤ 1. In addition,

*s*^{k}^{2}*+ ω*^{k}^{2}= *x*^{k}^{2}*+ y*^{k}^{2}
*2(x*^{k}^{2}*+ y*^{k}^{2}*)*+ 2

*λ*_{1}*(w*^{k}*)*

*λ*_{2}*(w*^{k}*)*≤1
2*.*

*Hence, taking limit (possibly on a subsequence) on L*2*(w*^{k}*)L*_{x}*k* *and L*2*(w*^{k}*)L*_{y}*k*

yields

*L*2*(w*^{k}*)L*_{x}*k*→1
2

*ξ*_{1} *ξ*_{2}^{T}

*ξ*1*¯w*2*+ 4(I − ¯w*2*¯w*_{2}^{T}*)s*2 *¯w*2*ξ*_{2}^{T}*+ 4(I − ¯w*2*¯w*_{2}^{T}*)s*1

=1 2

1

*¯w*2

*ξ** ^{T}* + 2

0 0

*(I− ¯w*2*¯w*^{T}_{2}*)s*_{2} *(I− ¯w*2*¯w*_{2}^{T}*)s*_{1}

*,*

(26)
*L*2*(w*^{k}*)L*_{y}*k*→1

2

*η*1 *η*^{T}_{2}

*η*1*¯w*2*+ 4(I − ¯w*2*¯w*_{2}^{T}*)ω*2 *¯w*2*η*^{T}_{2} *+ 4(I − ¯w*2*¯w*^{T}_{2}*)ω*1

=1 2

1

*¯w*2

*η** ^{T}* + 2

0 0

*(I− ¯w*2*¯w*^{T}_{2}*)ω*_{2} *(I− ¯w*2*¯w*_{2}^{T}*)ω*_{1}

*for some vectors ξ* *= (ξ*1*, ξ*_{2}*), η= (η*1*, η*_{2}*)*∈ R × R^{l}^{−1} satisfying *|ξ*1*| ≤ ξ*2 ≤
1 and *|η*1*| ≤ η*2* ≤ 1, ¯w*2∈ R^{l}^{−1} satisfying * ¯w*2* = 1, and s = (s*1*, s*_{2}*), ω*=
*(ω*1*, ω*2*)*∈ R × R* ^{l−1}* satisfying

*s*

^{2}

*+ ω*

^{2}

*≤ 1/2. Among others, ξ and η are*some accumulation point of the sequences

*{ξ*

^{k}*} and {η*

^{k}*}, respectively; and s and*

*ω*are some accumulation point of the sequences

*{s*

^{k}*} and {ω*

*}, respectively. From (19), (23) and (26), we obtain*

^{k}*φ*_{x}^{
}*(x*^{k}*, y*^{k}*)*→ 1
2

1

*¯w*2

*ξ** ^{T}* +1

2

1

*− ¯w*2

*u*^{T}

+ 2

0 0

*(I− ¯w*2*¯w*^{T}_{2}*)s*2 *(I− ¯w*2*¯w*^{T}_{2}*)s*1

*− I,*

*φ*_{y}^{
}*(x*^{k}*, y*^{k}*)*→ 1
2

1

*¯w*2

*η** ^{T}* +1

2

1

*− ¯w*2

*v*^{T}

+ 2

0 0

*(I− ¯w*2*¯w*^{T}_{2}*)ω*_{2} *(I− ¯w*2*¯w*^{T}_{2}*)ω*_{1}

*− I.*

*This implies that as k→ ∞, φ*^{
}*(x*^{k}*, y*^{k}*)→ [V**x**− I V**y**− I] with V**x* *and V**y*satis-
fying (15).

Combining with Case (1) then yields the desired result.
*Remark 3.1 When x*^{2}*+ y*^{2}*∈ bd(K*^{l}*)with (x, y)= (0, 0), using Lemma*3.2, we can
*also characterize V**x**and V**y*in Proposition3.1(b) by

*V** _{x}*∈

1

√*2w*1

*x*1 *x*_{2}^{T}*x*2 *2x*1*I*−^{w}_{w}^{2}^{x}_{1}^{T}^{2}

+1

2

1

*−w*2

*w*2

*u*^{T}

*V** _{y}*∈

√1
*2w*1

*y*1 *y*_{2}^{T}*y*_{2} *2y*1*I*−^{w}_{w}^{2}^{y}_{1}^{T}^{2}

+1

2

1

*−w*2

*w*2

*v*^{T}

*for some u= (u*1*, u*2*), v= (v*1*, v*2*)*∈ R × R^{l}^{−1}satisfying*|u*1*| ≤ u*2* ≤ 1 and |v*1| ≤

*v*2 ≤ 1.

**4 Properties of the Operator **

*In this section, we study some properties of related to the generalized Newton*
*method. In particular, we shall present an estimate for the B-subdifferential of and*
*a sufficient condition for all elements of the B-subdifferential of at a solution being*
*nonsingular. For convenience, throughout this section, for any i∈ {1, 2, . . . , q} and*
*ζ*∈ R* ^{n}*, we let

*F**i**(ζ )= (F**i1**(ζ ), F**i2**(ζ )),* *G**i**(ζ )= (G**i1**(ζ ), G**i2**(ζ ))*∈ R × R^{n}^{i}^{−1}*,*
*w*_{i}*(ζ )= (w**i1**(ζ ), w*_{i2}*(ζ ))= w(F**i**(ζ ), G*_{i}*(ζ )),*

*z*_{i}*(ζ )= (z**i1**(ζ ), z**i2**(ζ ))= z(F**i**(ζ ), G*_{i}*(ζ ))*

*where w(x, y) and z(x, y) are the functions defined as in (12).*

*First, since is (strongly) semismooth if and only if all component functions are*
(strongly) semismooth, and since the composite of (strongly) semismooth functions
is (strongly) semismooth by [8, Theorem 19], we have the following proposition as
an immediate consequence of Corollary 3.3 of [21].

* Proposition 4.1 The operator *: R

*→ R*

^{n}

^{m}*defined by (9) is semismooth. Further-*

*more, it is strongly semismooth if F*

^{ }

*and G*

^{ }

*are locally Lipschitz continuous.*

*Let **i* *denote the i-th component of the function . Notice that, for any ζ* ∈ R* ^{n}*,

*∂*_{B}*(ζ )*^{T}*⊆ ∂**B*_{1}*(ζ )*^{T}*× ∂**B*_{2}*(ζ )*^{T}*× · · · × ∂**B*_{q}*(ζ )** ^{T}* (27)