DOI 10.1007/s10589-008-9166-9

**A semismooth Newton method for SOCCPs based** **on a one-parametric class of SOC complementarity** **functions**

**Shaohua Pan· Jein-Shan Chen**

Received: 29 March 2007 / Revised: 29 October 2007 / Published online: 7 February 2008

© Springer Science+Business Media, LLC 2008

**Abstract In this paper, we present a detailed investigation for the properties of a**
one-parametric class of SOC complementarity functions, which include the glob-
ally Lipschitz continuity, strong semismoothness, and the characterization of their
B-subdifferential. Moreover, for the merit functions induced by them for the second-
order cone complementarity problem (SOCCP), we provide a condition for each sta-
tionary point to be a solution of the SOCCP and establish the boundedness of their
*level sets, by exploiting Cartesian P -properties. We also propose a semismooth New-*
ton type method based on the reformulation of the nonsmooth system of equations
involving the class of SOC complementarity functions. The global and superlinear
convergence results are obtained, and among others, the superlinear convergence is
established under strict complementarity. Preliminary numerical results are reported
for DIMACS second-order cone programs, which confirm the favorable theoretical
properties of the method.

**Keywords Second-order cone**· Complementarity · B-subdifferential ·
Semismooth· Newton’s method

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

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

S. Pan

School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, People’s Republic of 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**

*We 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 and G are the mappings from*
R* ^{n}*toR

*which are assumed to be continuously differentiable, and*

^{n}*K is the Cartesian*product of second-order cones (SOCs), also called Lorentz cones [10]. In other words,

*K = K*

^{n}^{1}

*× K*

^{n}^{2}

*× · · · × K*

^{n}

^{m}*,*(2)

*where m, n*1

*, . . . , n*

_{m}*≥ 1, n*1

*+ n*2

*+ · · · + n*

*m*

*= n, and*

*K*^{n}^{i}*:= {(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 real
numbersR_{+}. We refer to (1)–(2) as the second-order cone complementarity problem
*(SOCCP). In the sequel, corresponding to the Cartesian structure ofK, we write*
*x= (x*1*, . . . , x*_{m}*)with x**i* ∈ R^{n}^{i}*for any x* ∈ R^{n}*, and F* *= (F*1*, . . . , F*_{m}*)* *and G*=
*(G*_{1}*, . . . , G*_{m}*)with F**i**, G** _{i}*: R

*→ R*

^{n}

^{n}*.*

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

*F (ζ )∈ K, ζ ∈ K, F (ζ ), ζ = 0,* (3)
which is a natural extension of the nonlinear complementarity problem (NCP) where
*K = K*^{1}*×· · ·×K*^{1}. Another important special case corresponds to the Karush-Kuhn-
Tucker (KKT) conditions of the convex second-order cone program (SOCP):

*min g(x)*

*s.t.* *Ax= b, x ∈ K,* (4)

*where A*∈ R^{m}^{×n}*has full row rank, b*∈ R^{m}*and g*: R* ^{n}*→ R is a convex twice con-
tinuously differentiable function. From [6], the KKT conditions for (4), which are
sufficient but not necessary for optimality, can be written in the form of (1) and (2)
with

*F (ζ ):= d +(I −A*^{T}*(AA*^{T}*)*^{−1}*A)ζ,* *G(ζ ):= ∇g(F (ζ ))−A*^{T}*(AA*^{T}*)*^{−1}*Aζ,* (5)
*where d*∈ R^{n}*is any vector satisfying Ax= b. For large problems with a sparse A,*
(5) has an advantage that the main cost of evaluating the Jacobian*∇F and ∇G lies*
*in inverting AA** ^{T}*, which can be done efficiently via sparse Cholesky factorization.

There have been various methods proposed for solving SOCPs and SOCCPs, which include interior-point methods [1–3, 18, 19, 23, 26], non-interior smooth- ing Newton methods [7,13], smoothing-regularization methods [15], merit function methods [6] and semismooth Newton methods [16]. Among others, the last four kinds

of methods are all based on an SOC complementarity function or a smooth merit function induced by it.

*Given a mapping φ*: R* ^{l}*× R

*→ R*

^{l}

^{l}*, we call φ an SOC complementarity function*associated with the cone

*K*

^{l}*if for any (x, y)*∈ R

*× R*

^{l}*,*

^{l}*φ (x, y)= 0 ⇐⇒ x ∈ K*^{l}*,* *y∈ K*^{l}*,* *x, y = 0.* (6)
*Clearly, when l*= 1, an SOC complementarity function reduces to an NCP function,
which plays an important role in the solution of NCPs; see [24] and references therein.

*A popular choice of φ is the Fischer-Burmeister (FB) function [11,*12], defined by
*φ*FB*(x, y):= (x*^{2}*+ y*^{2}*)*^{1/2}*− (x + y).* (7)
*More specifically, for any x= (x*1*, x*_{2}*), y= (y*1*, y*_{2}*)*∈ R × R^{l}^{−1}*, we define their Jor-*
*dan product associated withK** ^{l}*as

*x◦ y := (x, y, y*1*x*2*+ x*1*y*2*).* (8)
The Jordan product “◦”, unlike scalar or matrix multiplication, is not associative,
which is the main source on complication in the analysis of SOCCPs. The identity
*element under this product is e:= (1, 0, . . . , 0)** ^{T}* ∈ R

^{l}*. We write x*

^{2}

*to mean x◦ x and*

*write x+ y to mean the usual componentwise addition of vectors. It is known that*

*x*

^{2}

*∈ K*

^{l}*for all x*∈ R

^{l}*. Moreover, if x∈ K*

*, then there exists a unique vector in*

^{l}*K*

*,*

^{l}*denoted by x*

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

^{1/2}*)*

^{2}

*= x*

^{1/2}*◦ x*

^{1/2}*= x. Thus, φ*FBin (7) is well-defined

*for all (x, y)*∈ R

*× R*

^{l}*and mapsR*

^{l}*× R*

^{l}*toR*

^{l}

^{l}*. The function φ*FBwas proved in [13]

to satisfy the equivalence (6), and therefore its squared norm, denoted by
*ψ*FB*(x, y)*:=1

2*φ*FB*(x, y)*^{2}*,*

is a merit function for the SOCCP. The merit function is shown to be continuously differentiable by Chen and Tseng [6], and a merit function approach was proposed by use of it.

*Another popular choice of φ is the natural residual function φ*NR: R* ^{l}*× R

*→ R*

^{l}*given by*

^{l}*φ*_{NR}*(x, y):= x − [x − y]*_{+}*,*

where[·]+means the minimum Euclidean distance projection onto*K** ^{l}*. The function
was studied in [13,15] which is involved in smoothing methods for the SOCCP,
recently it was used to develop a semismooth Newton method for nonlinear SOCPs
by Kanzow and Fukushima [16]. We note that φNRinduces a natural residual merit
function

*ψ*NR*(x, y)*:=1

2*φ*NR*(x, y)*^{2}*,*

*but, compared to ψ*FB, it has a remarkable drawback, i.e. the non-differentiability.

In this paper, we consider a one-parametric class of vector-valued functions
*φ*_{τ}*(x, y):= [(x − y)*^{2}*+ τ(x ◦ y)]*^{1/2}*− (x + y)* (9)

*with τ being any but fixed parameter in (0, 4). The class of functions is a natural ex-*
tension of the family of NCP functions proposed by Kanzow and Kleinmichel [17],
and has been shown in [4] to satisfy the characterization (6). It is not hard to see that
*as τ= 2, φ**τ* *reduces to the FB function φ*FB in (7) while it becomes a multiple of
*the natural residual function φ*NR *as τ* → 0^{+}. With the class of SOC complemen-
tarity functions, clearly, the SOCCP can be reformulated as a nonsmooth system of
equations

*τ**(ζ )*:=

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

*φ*_{τ}*(F*1*(ζ ), G*1*(ζ ))*
*...*

*φ**τ**(F**i**(ζ ), G**i**(ζ ))*
*...*

*φ*_{τ}*(F*_{m}*(ζ ), G*_{m}*(ζ ))*

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

*= 0,* (10)

*which induces a natural merit function **τ* : R* ^{n}*→ R

_{+}given by

*τ**(ζ )*=1

2*τ**(ζ )*^{2}=

*m*
*i*=1

*ψ**τ**(F**i**(ζ ), G**i**(ζ )),* (11)

*with ψ**τ* *being the natural merit function associated with φ**τ*, i.e.,

*ψ*_{τ}*(x, y)*=1

2*φ**τ**(x, y)*^{2}*.* (12)

In [4], we studied the continuous differentiability of ψ*τ* and showed that each sta-
*tionary point of **τ* is a solution of the SOCCP if *∇F and −∇G are column*
*monotone. In this paper, we concentrate on the properties of φ**τ*, including the glob-
ally Lipschitz continuity, the strong semismoothness, and the characterization of the
B-subdifferential. Particularly, we provide a weaker condition than [4] for each sta-
*tionary point of **τ* to be a solution of the SOCCP and establish the boundedness of
*the level sets of **τ**, by using Cartesian P -properties. We also propose a semismooth*
Newton method based on the system (10), and obtain the corresponding global and
the superlinear convergence results. Among others, the superlinear convergence is
established under strict complementarity.

*Throughout this paper, I represents an identity matrix of suitable dimension,*
and R^{n}^{1} × · · · × R^{n}* ^{m}* is identified with R

^{n}^{1}

^{+···+n}*. For a differentiable mapping*

^{m}*F*: R

*→ R*

^{n}*,*

^{m}*∇F (x) denotes the transpose of the Jacobian F*

^{}

*(x). For a symmet-*

*ric matrix A*∈ R

^{n}

^{×n}*, we write A O (respectively, A O) to mean A is positive*semidefinite (respectively, positive definite). Given a finite number of square matrices

*Q*

_{1}

*, . . . , Q*

*, we denote the block diagonal matrix with these matrices as block diag-*

_{n}*onals by diag(Q*1

*, . . . , Q*

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

*i*

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

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

*by the block matrix consisting of the*

_{J B}*sub-matrices P*

*j k*∈ R

^{n}

^{j}

^{×n}

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

*a vector consisting of*

_{B}*sub-vectors x*

*i*∈ R

^{n}

^{i}*with i∈ B.*

**2 Preliminaries**

In this section, we recall some background materials and preliminary results that will
be used in the subsequent sections. We begin with the interior and the boundary of*K** ^{l}*.
It is known that

*K*

*is a closed convex self-dual cone with nonempty interior given by*

^{l}*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 each x= (x*1*, x*_{2}*)*∈ R × R^{l}^{−1}*, the determinant and the trace of x are defined by*
*det(x):= x*_{1}^{2}*− x*2^{2}*,* *tr(x):= 2x*1*.*

*In general, det(x◦ y) = det(x) det(y) unless x*2*= αy*2 *for some α*∈ R. A vector
*x*∈ R^{l}*is said to be invertible if det(x)= 0, and its inverse is denoted by x*^{−1}. Given
*a vector x= (x*1*, x*_{2}*)*∈ R × R^{l}^{−1}, we often use the following symmetry matrix

*L** _{x}*:=

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

*,* (13)

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

^{l}*. It is easy to verify L*

*x*

*y*=

*x◦ y and L*

*x*

*+y*

*= L*

*x*

*+ L*

*y*

*for any x, y*∈ R

^{l}*. Furthermore, x∈ K*

*if and only if*

^{l}*L*

_{x}*O, and x ∈ int(K*

^{l}*)if and only if L*

*x*

*O. Then L*

*x*is invertible with

*L*^{−1}* _{x}* = 1

*det(x)*

*x*_{1} *−x*_{2}^{T}

*−x*2 *det(x)*
*x*_{1} *I*+_{x}^{1}

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

*.* (14)

We recall from [13] that each x*= (x*1*, x*2*)*∈ R × R^{l}^{−1}admits a spectral factoriza-
tion, associated with*K** ^{l}*, of the form

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

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

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

2*(1, (−1)*^{i}*¯x*2*)* (15)
with *¯x*2*= x*2*/x*2* if x*2*= 0, and otherwise ¯x*2 being any vector inR^{l}^{−1}satisfying

* ¯x*2* = 1. If x*2= 0, then the factorization is unique. The spectral decompositions
*of x, x*^{2} *and x** ^{1/2}*have some basic properties as below, whose proofs can be found
in [13].

* Property 2.1 For any x= (x*1

*, x*

_{2}

*)*∈ R × R

^{l}^{−1}

*with the spectral values λ*1

*(x), λ*

_{2}

*(x)*

*and spectral vectors u*

^{(1)}*x*

*, u*

^{(2)}

_{x}*given as above, we have that*

*(a) x∈ K*^{l}*if and only if λ*1*(x)≥ 0, and x ∈ int(K*^{l}*)if and only if λ*1*(x) >*0.

*(b) x*^{2}*= λ*^{2}_{1}*(x)u*^{(1)}_{x}*+ λ*^{2}_{2}*(x)u*^{(2)}_{x}*∈ K** ^{l}*.

*(c) x*

*=√*

^{1/2}*λ*1*(x) u*^{(1)}* _{x}* +√

*λ*2*(x) u*^{(2)}_{x}*∈ K*^{l}*if x∈ K** ^{l}*.

*(d) det(x)= λ*1*(x)λ*2*(x),* *tr(x)= λ*1*(x)+ λ*2*(x)andx*^{2}*= [λ*^{2}_{1}*(x)+ λ*^{2}_{2}*(x)]/2.*

For the sake of notation, throughout the rest of this paper, we always let
*w= (w*1*, w*2*)= w(x, y) := (x − y)*^{2}*+ τ(x ◦ y),*

*z= (z*1*, z*2*)= z(x, y) := [(x − y)*^{2}*+ τ(x ◦ y)]** ^{1/2}* (16)

*for any x= (x*1

*, x*

_{2}

*), y= (y*1

*, y*

_{2}

*)*∈ R × R

^{l}^{−1}. It is easy to compute

*w*_{1}*= x*^{2}*+ y*^{2}*+ (τ − 2)x*^{T}*y,*

*w*2*= 2(x*1*x*2*+ y*1*y*2*)+ (τ − 2)(x*1*y*2*+ y*1*x*2*).*

*Moreover, w∈ K*^{l}*and z∈ K** ^{l}*hold by considering that

*w= x*

^{2}

*+ y*

^{2}

*+ (τ − 2)(x ◦ y)*

=

*x*+*τ*− 2
2 *y*

2

+*τ (4− τ)*
4 *y*^{2}=

*y*+*τ*− 2
2 *x*

2

+*τ (4− τ)*

4 *x*^{2}*.* (17)
In what follows, we present several important technical lemmas. Since their proofs
can be found in [4], we here omit them for simplicity.

**Lemma 2.1 [4, Lemma 3.4] For any x***= (x*1*, x*2*), y= (y*1*, y*2*)*∈ R × R^{l}^{−1}*and τ*∈
*(0, 4), let w= (w*1*, w*2*)be defined as in (16). Ifw*2* = 0, then*

*x*1+*τ* − 2
2 *y*1

*+ (−1)*^{i}

*x*2+*τ* − 2
2 *y*2

*T* *w*_{2}

*w*2

2

≤

*x*2+*τ* − 2
2 *y*2

*+ (−1)*^{i}

*x*1+*τ* − 2
2 *y*1

*w*_{2}

*w*2

^{2}

*≤ λ**i**(w)* *for i= 1, 2.*

*Furthermore, these relations also hold when interchanging x and y.*

**Lemma 2.2 [4, Lemma 3.2] For any x***= (x*1*, x*_{2}*), y= (y*1*, y*_{2}*)*∈ R × R^{l}^{−1}*and τ*∈
*(0, 4), let w= (w*1*, w*2*)be given as in (16). If w /∈ int(K*^{l}*), then*

*x*_{1}^{2}*= 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; (18)
*x*_{1}^{2}*+ y*_{1}^{2}*+ (τ − 2)x*1*y*_{1}*= x*1*x*_{2}*+ y*1*y*_{2}*+ (τ − 2)x*1*y*_{2}

*= x*2^{2}*+ y*2^{2}*+ (τ − 2)x*^{T}_{2}*y*_{2}*.* (19)
*If, in addition, (x, y)= (0, 0), then w*2* = 0, and moreover,*

*x*_{2}^{T}*w*_{2}

*w*2*= x*1*,* *x*1

*w*_{2}

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

*w*2 *= y*1*,* *y*1

*w*_{2}

*w*2*= y*2*.* (20)

**Lemma 2.3 [4, Proposition 3.2] For any x***= (x*1*, x*_{2}*), y= (y*1*, y*_{2}*)*∈ R × R^{l}^{−1}*, let*
*z(x, y)be defined by (16). Then z(x, y) is continuously differentiable at a point (x, y)*
*if and only if (x− y)*^{2}*+ τ(x ◦ y) ∈ int(K*^{l}*), and furthermore,*

∇*x**z(x, y)= L*_{x}_{+}* ^{τ}*−2

2 *y**L*^{−1}_{z}*,* ∇*y**z(x, y)= L*_{y}_{+}* ^{τ}*−2
2

*x*

*L*

^{−1}

_{z}*,*

*where*

*L*^{−1}* _{z}* =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎛

⎝ *b* *c* ^{w}

*T*

*w*22

*c*_{w}^{w}^{2}

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

2^{2}

⎞

*⎠ if w*2= 0;

*(1/*√

*w*1*)I* *if w*2*= 0,*

(21)

*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)*

*.*

(22)

To close this section, we recall some definitions that will be used in the subsequent
*sections. Given a mapping H*: R* ^{n}*→ R

^{m}*, if H is locally Lipschitz continuous, 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 H at z in the sense of Clarke [8]. For the concepts of*(strongly) semismooth functions, please refer to [21,22] for details. We next present

*definitions of Cartesian P -properties for a matrix M*∈ R

^{n}*, which are in fact special cases of those introduced by Chen and Qi [5] for a linear transformation.*

^{×n}* Definition 2.1 A matrix M*∈ R

^{n}*is said to have*

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

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

*x**ν**, (Mx)**ν** > 0;*

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

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

*x*_{ν}*= 0 and x**ν**, (Mx)*_{ν}* ≥ 0.*

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*

_{m}*)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*_{m}*), y* =
*(y*1*, . . . , y*_{m}*)*∈ R^{n}*, there exists an index ν∈ {1, 2, . . . , m} 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**m**), y= (y*1*, . . . , y**m**)*∈
R^{n}*, there exists an index ν∈ {1, 2, . . . , m} such that*

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

*If a continuously differentiable mapping F has the Cartesian P -properties, then*
the matrix*∇F (x) at any x ∈ R*^{n}*enjoys the corresponding Cartesian P -properties.*

**3 Properties of the functions φ****τ****and ****τ**

*This section is devoted to investigating the favorable properties of φ**τ*, which include
the globally Lipschitz continuity, the strong semismoothness and the characterization
of the B-subdifferential at any point. Based on these results, we also present some
*properties of the operator **τ* related to the generalized Newton method.

*From the definition of φ**τ* *and z(x, y) given as in (9) and (16), respectively, we*
have

*φ**τ**(x, y)= z(x, y) − (x + y) = z − (x + y)* (23)
*for any x= (x*1*, x*_{2}*), y= (y*1*, y*_{2}*)*∈ R × R^{l}^{−1}*. Recall that the vectors w= (w*1*, w*_{2}*)*
*and z= (z*1*, z*_{2}*)*in (16) satisfy w, z*∈ K** ^{l}*, and hence, from Property2.1(b) and (c),

*z*=

√*λ*2*(w)*+√
*λ*1*(w)*

2 *,*

√*λ*2*(w)*−√
*λ*1*(w)*

2 *¯w*2

*,* (24)

where *¯w*2= _{w}^{w}^{2}_{2}_{} *if w*2*= 0, and otherwise ¯w*2 is any vector in R^{l}^{−1} satisfying

* ¯w*2 = 1. The following proposition states some favorable properties possessed
*by φ**τ*.

**Proposition 3.1 The function φ***τ* *defined as in (9) has the following properties.*

*(a) φ**τ* *is continuously differentiable at a point (x, y)*∈ R* ^{l}* × R

^{l}*if and only if*

*(x− y)*

^{2}

*+ τ(x ◦ y) ∈ int(K*

^{l}*). Moreover,*

∇*x**φ*_{τ}*(x, y)= L*_{x}_{+}* ^{τ}*−2

2 *y**L*^{−1}_{z}*− I,* ∇*y**φ*_{τ}*(x, y)= L*_{y}_{+}* ^{τ}*−2

2 *x**L*^{−1}_{z}*− I.*

*(b) φ**τ* *is globally Lipschitz continuous with the Lipschitz constant independent of τ .*
*(c) φ**τ* *is strongly semismooth at any (x, y)*∈ R* ^{l}*× R

*.*

^{l}*(d) ψ**τ* *defined by (12) is continuously differentiable everywhere.*

*Proof (a) The proof directly follows from Lemma*2.3and (23).

*(b) It suffices to prove that z(x, y) is globally Lipschitz continuous by (23). Let*
*ˆz = (ˆz*1*,ˆz*2*)= ˆz(x, y,
) := [(x − y)*^{2}*+ τ(x ◦ y) +
e]** ^{1/2}* (25)

*for any > 0 and x= (x*1

*, x*2

*), y= (y*1

*, y*2

*)*∈ R × R

*. Then, applying Lemma A.1 inAppendixand the Mean-Value Theorem, we have*

^{l−1}*z(x, y) − z(a, b) =* lim

*
*→0^{+}*ˆz(x, y,
) − lim*

*
*→0^{+}*ˆz(a, b,
)*

≤ lim

*
*→0^{+}*ˆz(x, y,
) − ˆz(a, y,
) + ˆz(a, y,
) − ˆz(a, b,
)*

≤ lim

*
*→0^{+}

1 0

∇*x**ˆz(a + t(x − a), y,
)(x − a)dt*

+ lim

*
*→0^{+}

_{1}

0

∇*y**ˆz(a, b + t(y − b),
)(y − b)dt*

≤√

*2C(x, y) − (a, b)*

*for any (x, y), (a, b)*∈ R* ^{l}*× R

^{l}*, where C > 0 is a constant independent of τ .*

*(c) From the definition of φ*

*τ*

*and φ*FB, it is not hard to check that

*φ**τ**(x, y)= φ*FB

*x*+*τ*− 2
2 *y,*

√*τ (4− τ)*

2 *y*

+1

2*(τ*− 4 +

*τ (4− τ))y.*

*Note that φ*FBis strongly semismooth everywhere by Corollary 3.3 of [25], and the
*functions x*+ ^{τ}^{−2}_{2} *y*, ^{1}_{2}√

*τ (4− τ)y and* ^{1}_{2}*(τ* − 4 +√

*τ (4− τ))y are also strongly*
*semismooth at any (x, y)*∈ R* ^{l}*× R

^{l}*. Therefore, φ*

*τ*is a strongly semismooth func- tion since by [12, Theorem 19] the composition of strongly semismooth functions is strongly semismooth.

(d) The proof can be found in Proposition 3.3 of the literature [4].
Proposition3.1(c) indicates that, when a smoothing or nonsmooth Newton method
is employed to solve the system (10), a fast convergence rate (at least superlinear) can
be expected. To develop a semismooth Newton method for the SOCCP, we need to
*characterize the B-subdifferential ∂**B**φ*_{τ}*(x, y)at a general point (x, y). The discussion*
*of B-subdifferential for φ*FBwas given in [20]. Here, we generalize it to φ*τ* for any
*τ∈ (0, 4). The detailed derivation process is included in*Appendixfor completeness.

* Proposition 3.2 Given a general point x= (x*1

*, x*2

*), y= (y*1

*, y*2

*)*∈ R × R

^{l}^{−1}

*, each*

*element in ∂*

*B*

*φ*

_{τ}*(x, y)is of the form V*

*= [V*

*x*

*− I V*

*y*

*− I] with V*

*x*

*and V*

*y*

*having the*

*following representation:*

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

2 *y**and V**y**= L*^{−1}_{z}*L*_{y}_{+}*τ*−2
2 *x*.
*(b) If (x− y)*^{2}*+ τ(x ◦ y) ∈ bd(K*^{l}*)and (x, y)= (0, 0), then*

*V** _{x}*∈

1 2√

*2w*1

1 *¯w*^{T}_{2}

*¯w*2 *4I− 3 ¯w*2*¯w*^{T}_{2}

*L** _{x}*+

*τ*− 2 2

*L*

_{y}+1

2 1

*− ¯w*2

*u*^{T}

*,*

(26)
*V** _{y}*∈

1 2√

*2w*1

1 *¯w*^{T}_{2}

*¯w*2 *4I− 3 ¯w*2*¯w*^{T}_{2}

*L** _{y}*+

*τ*− 2 2

*L*

_{x}+1

2 1

*− ¯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, where ¯w*2=_{w}^{w}^{2}_{2}_{}.

*(c) If (x, y)= (0, 0), then V**x* *∈ {L*_{ˆu}*}, V**y* *∈ {L*_{ˆv}*} for some ˆu = ( ˆu*1*,ˆu*2*),* *ˆv =*
*(ˆv*1*,ˆv*2*)*∈ R × R^{l}^{−1}*satisfying ˆu, ˆv ≤ 1 and ˆu*1*ˆv*2*+ ˆv*1*ˆu*2*= 0, 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*_{2}^{T}*)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*_{2}^{T}*)ω*2 *(I− ¯w*2*¯w*^{T}_{2}*)ω*1

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

*η*2* ≤ 1, and s = (s*1*, s*_{2}*), ω= (ω*1*, ω*_{2}*)*∈ R × R^{l}^{−1}*such thats*^{2}*+ ω*^{2}≤ 1.

*In what follows, we focus on the properties of the operator **τ* defined in (10). We
*start with the semismoothness of **τ**. Since **τ* 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 [12, Theorem 19], we
obtain the following conclusion as an immediate consequence of Proposition3.1(c).

**Proposition 3.3 The operator ***τ*: R* ^{n}*→ R

^{n}*defined as in (10) is semismooth. More-*

*over, it is strongly semismooth if F*

^{}

*and G*

^{}

*are locally Lipschitz continuous.*

*To characterize the B-subdifferential of **τ*, in the rest of this paper, we let
*F*_{i}*(ζ )= (F**i*1*(ζ ), F**i2**(ζ )),* *G*_{i}*(ζ )= (G**i1**(ζ ), G**i2**(ζ ))*∈ R × R^{n}^{i}^{−1}
*and w**i*: R* ^{n}*→ R

^{n}

^{i}*and z*

*i*: R

*→ R*

^{n}

^{n}

^{i}*for i= 1, 2, . . . , m be given as follows:*

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

*z*_{i}*= (z**i1**(ζ ), z*_{i2}*(ζ ))= z(F**i**(ζ ), G*_{i}*(ζ )).* (27)
**Proposition 3.4 Let ***τ*: R* ^{n}*→ R

^{n}*be defined as in (10). Then, for any ζ*∈ R

*,*

^{n}*∂*_{B}_{τ}*(ζ )*^{T}*⊆ ∇F (ζ )(A(ζ ) − I) + ∇G(ζ )(B(ζ ) − I),* (28)
*where A(ζ ) and B(ζ ) are possibly multivalued n× n block diagonal matrices whose*
*ith blocks A**i**(ζ )and B**i**(ζ )for i= 1, 2, . . . , m have the following representation:*

*(a) If (F**i**(ζ )− G**i**(ζ ))*^{2}*+ τ(F**i**(ζ )◦ G**i**(ζ ))∈ int(K*^{n}^{i}*), then*
*A*_{i}*(ζ )= L*_{F}_{i}_{+}*τ−2*

2 *G**i**L*^{−1}_{z}

*i* *and* *B*_{i}*(ζ )= L*_{G}_{i}_{+}*τ−2*

2 *F**i**L*^{−1}_{z}

*i* *.*

*(b) If (F**i**(ζ )− G**i**(ζ ))*^{2}*+ τ(F**i**(ζ )◦ G**i**(ζ ))∈ bd(K*^{n}^{i}*)and (F**i**(ζ ), G*_{i}*(ζ ))= (0, 0),*
*then*

*A*_{i}*(ζ )*∈

1

2√
*2w**i1*

*L*_{F}* _{i}*+

*τ*− 2 2

*L*

_{G}

_{i} 1 *¯w*_{i2}^{T}

*¯w**i2* *4I− 3 ¯w**i2**¯w*_{i2}^{T}

+1

2*u*_{i}*(1,− ¯w*_{i}* ^{T}*2

*)*

*B*_{i}*(ζ )*∈

1

2√
*2w**i1*

*L*_{G}* _{i}*+

*τ*− 2 2

*L*

_{F}

_{i} 1 *¯w*_{i2}^{T}

*¯w**i2* *4I− 3 ¯w**i2**¯w*_{i2}^{T}

+1

2*v*_{i}*(1,− ¯w*^{T}*i2**)*

*for some u**i**= (u**i1**, u*_{i}_{2}*), v**i**= (v**i1**, v*_{i2}*)*∈ R × R^{n}^{i}^{−1}*satisfying|u**i1**| ≤ u**i2* ≤ 1
*and|v**i*1*| ≤ v**i2** ≤ 1, where ¯w**i2*=_{w}^{w}^{i2}

*i2*.
*(c) If (F**i**(ζ ), G**i**(ζ ))= (0, 0), then*

*A*_{i}*(ζ )∈ {L*_{ˆu}_{1}} ∪

1

2*ξ*_{i}*(1,* *¯w*^{T}_{i2}*)*+1

2*u*_{i}*(1,− ¯w*^{T}_{i2}*)*+

0 *2s*_{i2}^{T}*(I− ¯w**i2**¯w*_{i2}^{T}*)*
0 *2s**i1**(I− ¯w**i2**¯w*_{i2}^{T}*)*

*B*_{i}*(ζ )∈ {L*_{ˆv}_{1}} ∪

1

2*η*_{i}*(1,* *¯w*^{T}*i2**)*+1

2*v*_{i}*(1,− ¯w**i2*^{T}*)*+

0 *2ω*^{T}_{i2}*(I− ¯w**i*2*¯w*^{T}_{i2}*)*
0 *2ω**i1**(I− ¯w**i*2*¯w*^{T}_{i2}*)*

*for some* *ˆu**i* *= ( ˆu**i1**,ˆu**i2**),* *ˆv**i**= (ˆv**i1**,ˆv**i2**)*∈ R × R^{n}^{i}^{−1} *satisfying ˆu**i**, ˆv**i* ≤ 1
*and* *ˆu**i*1*ˆv**i2**+ ˆv**i1**ˆu**i2**= 0, some u**i* *= (u**i1**, u*_{i}_{2}*), v**i* *= (v**i1**, v*_{i2}*), ξ**i* *= (ξ**i1**, ξ*_{i2}*),*
*η*_{i}*= (η**i1**, η*_{i2}*)*∈ R × R^{n}^{i}^{−1}*with|u**i1**| ≤ u**i2** ≤ 1, |v**i1**| ≤ v**i2** ≤ 1, |ξ**i1*| ≤

*ξ**i*2* ≤ 1 and |η**i1**| ≤ η**i*2* ≤ 1, ¯ω**i2*∈ R^{n}^{i}^{−1} *satisfying* * ¯ω**i2** = 1, and s**i* =
*(s**i1**, s**i*2*), ω**i**= (ω**i*1*, ω**i2**)*∈ R × R^{n}^{i}^{−1}*such thats**i*^{2}*+ ω**i*^{2}≤ 1.

*Proof Let **τ,i**(ζ )denote the ith subvector of **τ**, i.e. **τ,i**(ζ )= φ**τ**(F**i**(ζ ), G**i**(ζ ))*
*for all i= 1, 2, . . . , m. From Proposition 2.6.2 of [*8], it follows that

*∂**B**τ**(ζ )*^{T}*⊆ ∂**B**τ,1**(ζ )*^{T}*× ∂**B**τ,2**(ζ )*^{T}*× · · · × ∂**B**τ,m**(ζ )*^{T}*,* (29)
*where the latter denotes the set of all matrices whose (n**i*−1*+ 1) to n**i*th columns
*with n*0*= 0 belong to ∂**B*_{τ,i}*(ζ )** ^{T}*. Using the definition of B-subdifferential and the

*continuous differentiability of F and G, it is not difficult to verify that*

*∂*_{B}_{τ,i}*(ζ )*^{T}*= [∇F**i**(ζ )* *∇G**i**(ζ )]∂**B**φ*_{τ}*(F*_{i}*(ζ ), G*_{i}*(ζ ))*^{T}*,* *i= 1, 2, . . . , m. (30)*
Using Proposition3.2and the last two equations, we get the desired result.
* Proposition 3.5 For any ζ*∈ R

^{n}*, let A(ζ ) and B(ζ ) be the multivalued block diago-*

*nal matrices given as in Proposition*3.4

*Then, for any i∈ {1, 2, . . . , m},*

*(A**i**(ζ )− I)**τ,i**(ζ ), (B*_{i}*(ζ )− I)**τ,i**(ζ ) ≥ 0,*

*with equality holding if and only if **τ,i**(ζ )= 0. Particularly, for the index i such that*
*(F*_{i}*(ζ )− G**i**(ζ ))*^{2}*+ τ(F**i**(ζ )· G**i**(ζ )∈ int(K*^{n}^{i}*)), we have*

*(A**i**(ζ )− I)υ**i**, (B*_{i}*(ζ )− I)υ**i** ≥ 0, for any υ**i*∈ R^{n}^{i}*.*
*Proof From Theorem 2.6.6 of [8] and Proposition 3.1(d), we have that*

*∇ψ**τ**(x, y)= ∂**B**φ*_{τ}*(x, y)*^{T}*φ*_{τ}*(x, y).*

*Consequently, for any i= 1, 2, . . . , m, it follows that*

*∇ψ**τ**(F**i**(ζ ), G**i**(ζ ))= ∂**B**φ**τ**(F**i**(ζ ), G**i**(ζ ))*^{T}*φ**τ**(F**i**(ζ ), G**i**(ζ )).*

In addition, from Propositions3.2and3.4, it is not hard to see that
*[A**i**(ζ )*^{T}*− I B**i**(ζ )*^{T}*− I] ∈ ∂**B**φ*_{τ}*(F*_{i}*(ζ ), G*_{i}*(ζ )).*

*Combining with the last two equations yields that for any i= 1, 2, . . . , m,*

∇*x**ψ*_{τ}*(F*_{i}*(ζ ), G*_{i}*(ζ ))= (A**i**(ζ )− I)**τ,i**(ζ ),*

∇*y**ψ*_{τ}*(F*_{i}*(ζ ), G*_{i}*(ζ ))= (B**i**(ζ )− I)**τ,i**(ζ ).*

(31)

Consequently, the first part of conclusions is a direct consequence of Proposition 4.1
of [4]. Notice that for any i*∈ O(ζ ) and υ**i*∈ R^{n}* ^{i}*,

*(A**i**(ζ )− I)υ**i**, (B**i**(ζ )− I)υ**i*

*= (L*_{F}_{i}_{+}*τ−2*

2 *G**i**− L**z**i**)L*^{−1}_{z}

*i* *υ*_{i}*, (L*_{G}

*i*+* ^{τ−2}*2

*F*

*i*

*− L*

*z*

*i*

*)L*

^{−1}

_{z}*i* *υ*_{i}

*= (L*_{G}_{i}_{+}* ^{τ}*−2

2 *F*_{i}*− L**z**i**)(L*_{F}

*i*+^{τ}^{−2}_{2} *G*_{i}*− L**z**i**)L*^{−1}_{z}

*i* *υ*_{i}*, L*^{−1}_{z}

*i* *υ*_{i}*.* (32)

Using the same argument as Case (2) of [4, Proposition 4.1] then yields the second

part.

**4 Nonsingularity conditions**

*In this section, we show that all elements of the B-subdifferential ∂**B**τ**(ζ )*at a solu-
*tion ζ*^{∗}*of the SOCCP are nonsingular if ζ*^{∗}*satisfies strict complementarity, i.e.,*

*F**i**(ζ*^{∗}*)+ G**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*)* *for all i= 1, 2, . . . , m.* (33)
First, we give a technical lemma which states that the multi-valued matrix
*(A*_{i}*(ζ*^{∗}*)− I) + (B**i**(ζ*^{∗}*)− I) is nonsingular if the i-th block component satisfies*
strict complementarity.

**Lemma 4.1 Let ζ**^{∗} *be a solution of the SOCCP, and A(ζ*^{∗}*)and B(ζ*^{∗}*)be the mul-*
*tivalued block diagonal matrices characterized by Proposition* 3.4. Then, for any
*i∈ {1, 2, . . . , m} such that F**i**(ζ*^{∗}*)+ G**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*), we have that **τ,i**(ζ )is con-*
*tinuously differentiable at ζ*^{∗}*and (A**i**(ζ*^{∗}*)− I) + (B**i**(ζ*^{∗}*)− I) is nonsingular.*

*Proof Since ζ*^{∗}*is a solution of the SOCCP, we have for all i= 1, 2, . . . , m*
*F*_{i}*(ζ*^{∗}*)∈ K*^{n}^{i}*,* *G*_{i}*(ζ*^{∗}*)∈ K*^{n}^{i}*,* *F**i**(ζ*^{∗}*), G*_{i}*(ζ*^{∗}*) = 0.*

*It is not hard to verify that F**i**(ζ*^{∗}*)+ G**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*)*if and only if one of the three
cases shown as below holds.

*Case (1) F**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*)and G**i**(ζ*^{∗}*)= 0. Under this case,*

*w**i**(ζ*^{∗}*)= (F**i**(ζ*^{∗}*)− G**i**(ζ*^{∗}*))*^{2}*+ τ(F**i**(ζ*^{∗}*)◦ G**i**(ζ*^{∗}*))= F**i**(ζ*^{∗}*)*^{2}*∈ int(K*^{n}^{i}*).*

By Proposition3.1(a), *τ,i**(ζ )is continuously differentiable at ζ*^{∗}*. Since z**i**(ζ*^{∗}*)*=
*w*_{i}*(ζ*^{∗}*)*^{1/2}*= F**i**(ζ*^{∗}*), from Proposition*3.4(a) it follows that

*A*_{i}*(ζ*^{∗}*)= I and B**i**(ζ*^{∗}*)*=*τ*− 2
2 *I,*

*which implies that (A**i**(ζ*^{∗}*)− I) + (B**i**(ζ*^{∗}*)− I) is nonsingular since 0 < τ < 4.*

*Case (2) F**i**(ζ*^{∗}*)= 0 and G**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*).* *Now, w**i**(ζ*^{∗}*)= G**i**(ζ*^{∗}*)*^{2}*∈ int(K*^{n}^{i}*).*

*So, **τ,i**(ζ )is continuously differentiable at ζ*^{∗}by Proposition3.1(a). Since
*z*_{i}*(ζ*^{∗}*)= w**i**(ζ*^{∗}*)*^{1/2}*= G**i**(ζ*^{∗}*),*

applying Proposition3.4(a) yields that

*A*_{i}*(ζ*^{∗}*)*=*τ*− 2

2 *I* and *B*_{i}*(ζ*^{∗}*)= I,*

*which immediately implies that (A**i**(ζ*^{∗}*)− I) + (B**i**(ζ*^{∗}*)− I) is nonsingular.*

*Case (3) F**i**(ζ*^{∗}*)*∈bd^{+}*(K*^{n}^{i}*)and G**i**(ζ*^{∗}*)*∈bd^{+}*(K*^{n}^{i}*), where bd*^{+}*(K*^{n}^{i}*):=bd(K*^{n}^{i}*)\{0}.*

By Proposition 3.1(a), it suffices to prove w*i**(ζ*^{∗}*)* *∈ int(K*^{n}^{i}*). Suppose that*
*w**i**(ζ*^{∗}*)∈ bd(K*^{n}^{i}*). Then, from (18) in Lemma*2.2, it follows that

*F**i1**(ζ*^{∗}*)G** _{i}*1

*(ζ*

^{∗}

*)= F*

*i*2

*(ζ*

^{∗}

*)*

^{T}*G*

*2*

_{i}*(ζ*

^{∗}

*).*

*Since F**i1**(ζ*^{∗}*)= F**i2**(ζ*^{∗}*) = 0 and G**i1**(ζ*^{∗}*)= G**i2**(ζ*^{∗}*)* = 0, we have

*F**i2**(ζ*^{∗}*) · G**i2**(ζ*^{∗}*) = F**i2**(ζ*^{∗}*)*^{T}*G*_{i}_{2}*(ζ*^{∗}*),*

*which implies that F**i2**(ζ*^{∗}*)= αG**i*2*(ζ*^{∗}*)* *for some constant α > 0. Consequently,*
*F**i**(ζ*^{∗}*)= αG**i**(ζ*^{∗}*). Noting that* *F**i**(ζ*^{∗}*), G**i**(ζ*^{∗}*) = 0, we then get F**i**(ζ*^{∗}*)* =
*G**i**(ζ*^{∗}*)* *= 0. This clearly contradicts the assumptions that F**i**(ζ*^{∗}*)* = 0 and
*G*_{i}*(ζ*^{∗}*)= 0. So, w**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*).*

*From the expression of A**i**(ζ )and B**i**(ζ )*given by Proposition 3.4(a),
*(A*_{i}*(ζ*^{∗}*)− I) + (B**i**(ζ*^{∗}*)− I) = −L**2z*_{i}*(ζ*^{∗}*)*−* ^{τ}*2

*(F*

*i*

*(ζ*

^{∗}

*)*

*+G*

*i*

*(ζ*

^{∗}

*))*

*L*

^{−1}

_{z}*i**(ζ*^{∗}*)**.*