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

Shaohua Pan ^{1}

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China

Jein-Shan Chen ^{2}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan

March 26, 2007 (revised on July 30, 2007) (2nd revised on October 28, 2007)

Abstract. In this paper, we present a detailed investigation for the properties of a one-
parametric class of SOC complementarity functions, which include the globally Lipschitz
continuity, strong semismoothness, and the characterization of the B-subdifferential at
a general point. Moreover, for the merit functions induced by them for the second-order
cone complementarity problem (SOCCP), we provide a condition for each stationary
point being a solution of the SOCCP and establish the boundedness of their level sets,
*by exploiting Cartesian P -properties. We also propose a semismooth Newton 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.

Key words. Second-order cone, complementarity, semismooth, B-subdifferential, New- ton’s method.

1The author’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. E-mail:shhpan@scut.edu.cn.

2Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.

The author’s work is partially supported by National Science Council of Taiwan. E-mail:

jschen@math.ntnu.edu.tw.

## 1 Introduction

*We consider the following conic complementarity problem of finding ζ ∈ IR** ^{n}* such that

*F (ζ) ∈ K,*

*G(ζ) ∈ K,*

*hF (ζ), G(ζ)i = 0,*(1)

*where h·, ·i denotes the Euclidean inner product, F and G are the mappings from IR*

*to IR*

^{n}

^{n}*which are assumed to be continuously differentiable, and K is the Cartesian product*of second-order cones (SOCs), also called Lorentz cones [8]. 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}*) ∈ IR × IR*^{n}^{i}^{−1}*| x*_{1} *≥ kx*_{2}*k*ª
*,*

*with k · k denoting the Euclidean norm and K*^{1} denoting the set of nonnegative reals IR_{+}.
*We will refer to (1)–(2) as the second-order cone complementarity problem (SOCCP). In*
*addition, we write F = (F*1*, . . . , F**m**) and G = (G*1*, . . . , G**m**) with F**i**, G**i* : IR^{n}*→ IR*^{n}* ^{i}*.

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

*F (ζ) ∈ K,* *ζ ∈ K,* *hF (ζ), ζi = 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 ∈ IR*^{m×n}*has full row rank, b ∈ IR*^{m}*and g : IR*^{n}*→ IR is a twice continuously*
differentiable convex function. From [7], the KKT conditions for (4), which are sufficient
but not necessary for optimality, can be written in the form of (1) 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 ∈ IR*

^{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*

*, which can be done efficiently via sparse Cholesky factorization.*

^{T}There have been various methods proposed for solving SOCPs and SOCCPs. They include interior-point methods [1, 2, 17, 18, 24], non-interior smoothing Newton methods [4, 9], the smoothing-regularization method [13], the merit function method [7] and the semismooth Newton method [15]. 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 : IR*^{l}*× IR*^{l}*→ IR*^{l}*(l ≥ 1), we call φ an SOC complementarity function*
*associated with the cone K*^{l}*if for any (x, y) ∈ IR*^{l}*× IR** ^{l}*,

*φ(x, y) = 0 ⇐⇒ x ∈ K*^{l}*,* *y ∈ K*^{l}*,* *hx, yi = 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 [22] and references therein.

*A popular choice of φ is the Fischer-Burmeister (FB) function [10, 11], defined by*
*φ*_{FB}*(x, y) := (x*^{2}*+ y*^{2})^{1/2}*− (x + y),* (7)
*where x*^{2} *means x ◦ x with “◦” denoting the Jordan product, and x + y denotes the usual*
*componentwise addition of vectors. More specifically, for any x = (x*_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2}*) ∈*
*IR × IR*^{l−1}*, we define their Jordan product associated with K** ^{l}* as

*x ◦ y := (hx, yi, 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}*∈ IR*^{l}*. It is known that x*^{2} *∈ K*^{l}*for all x ∈ IR** ^{l}*.

*Moreover, if x ∈ K*

^{l}*, then there exists a unique vector in K*

^{l}*, denoted by x*

*, such that*

^{1/2}*(x*

*)*

^{1/2}^{2}

*= x*

^{1/2}*◦ x*

^{1/2}*= x. Thus, φ*

_{FB}

*in (7) is well-defined for all (x, y) ∈ IR*

^{l}*× IR*

*. The*

^{l}*function φ*

_{FB}was proved in [9] to satisfy the equivalence (6), and its squared norm

*ψ*_{FB}*(x, y) :=* 1

2*kφ*_{FB}*(x, y)k*^{2}*,*

has been shown to be continuously differentiable everywhere by Chen and Tseng [7].

*Another popular choice of φ is the residual function φ*_{NR}: IR^{l}*× IR*^{l}*→ IR** ^{l}* given by

*φ*

_{NR}

*(x, y) := x − [x − y]*

_{+}

*,*

*where [ · ]*_{+} *means the minimum Euclidean distance projection onto K** ^{l}*. The function
was studied in [9, 13] which is involved in smoothing methods for the SOCCP, and
recently it was used to develop a semismooth Newton method for nonlinear SOCPs by

*Kanzow and Fukushima [15]. The function φ*

_{NR}also induces a merit function

*ψ*_{NR}*(x, y) :=* 1

2*kφ*_{NR}*(x, y)k*^{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 an arbitrary fixed parameter from (0, 4). The class of functions is a natural*
extension of the family of NCP functions proposed by Kanzow and Kleinmichel [14], and
has been shown to satisfy the characterization (6) in [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 complementarity functions, the
SOCCP can be reformulated as a nonsmooth system of equations

Φ_{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 Ψ* _{τ}* : IR

^{n}*→ IR*

_{+}given by

Ψ_{τ}*(ζ) =* 1

2*kΦ*_{τ}*(ζ)k*^{2} =
X*m*

*i=1*

*ψ*_{τ}*(F*_{i}*(ζ), G*_{i}*(ζ),* (11)

with

*ψ**τ**(x, y) =* 1

2*kφ**τ**(x, y)k*^{2}*.* (12)

*In [6], we studied the continuous differentiability of ψ** _{τ}* and proved that each stationary
point of Ψ

*τ*

*is a solution of the SOCCP if ∇F and −∇G are column monotone. This*

*paper focuses on other properties of φ*

*, including the globally Lipschitz continuity, the strong semismoothness, and the characterization of the B-subdifferential. Particularly, we provide a weaker condition than [6] for each stationary point of Ψ*

_{τ}*to be a solution of the SOCCP and establish the boundedness of the level sets of Ψ*

_{τ}*τ*, by using Cartesian

*P -properties. We also propose a semismooth Newton method based on (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*
IR^{n}^{1}*×· · ·×IR*^{n}* ^{m}*is identified with IR

^{n}^{1}

^{+···+n}

^{m}*. For a differentiable mapping F : IR*

^{n}*→ IR*

*,*

^{m}*∇F (x) denotes the transpose of the Jacobian F*^{0}*(x). For a symmetric matrix A ∈ IR** ^{n×n}*,

*we write A º O (respectively, A Â O) to mean A is positive semidefinite (respectively,*

*positive definite). Given a finite number of square matrices Q*

_{1}

*, . . . , Q*

*, we denote the*

_{m}*block diagonal matrix with these matrices as block diagonals by diag(Q*

_{1}

*, . . . , Q*

*) or by*

_{m}*diag(Q*

_{i}*, i = 1, . . . , m). If J and B are index sets such that J , B ⊆ {1, 2, . . . , m}, we*

*denote P*

*J B*

*by the block matrix consisting of the submatrices P*

*jk*

*∈ IR*

^{n}

^{j}

^{×n}

^{k}*of P with*

*j ∈ J , k ∈ B, and by x*

_{B}*a vector consisting of subvectors x*

_{i}*∈ IR*

^{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 begin with the interior and the boundary of K*^{l}*(l ≥ 1).*

*It is known that K** ^{l}* is a closed convex self-dual cone with nonempty interior given by

*int(K*

*) := ©*

^{l}*x = (x*_{1}*, x*_{2}*) ∈ IR × IR*^{l−1}*| x*_{1} *> kx*_{2}*k*ª
and the boundary given by

*bd(K** ^{l}*) :=©

*x = (x*_{1}*, x*_{2}*) ∈ IR × IR*^{l−1}*| x*_{1} *= kx*_{2}*k*ª
*.*

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

*In general, det(x ◦ y) 6= det(x) det(y) unless x*_{2} *= αy*_{2} *for some α ∈ IR. A vector x ∈ IR*^{l}*is said to be invertible if det(x) 6= 0, and its inverse is denoted by x** ^{−1}*. Given a vector

*x = (x*

_{1}

*, x*

_{2}

*) ∈ IR × IR*

*, we often use the following symmetry matrix*

^{l−1}*L** _{x}*:=

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

¸

*,* (13)

which can be viewed as a linear mapping from IR* ^{l}* to IR

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

_{x}*y = x ◦ y*

*and L*

_{x+y}*= L*

_{x}*+ L*

_{y}*for any x, y ∈ IR*

^{l}*. Furthermore, x ∈ K*

^{l}*if and only if L*

_{x}*º O, and*

*x ∈ int(K*

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

_{x}*Â O. If x ∈ int(K*

^{l}*), then L*

*is invertible with*

_{x}*L*^{−1}* _{x}* = 1

*det(x)*

*x*1 *−x*^{T}_{2}

*−x*_{2} *det(x)*
*x*_{1} *I +* 1

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

* .* (14)

*We recall from [9] that each x = (x*_{1}*, x*_{2}*) ∈ IR × IR** ^{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*_{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}*kx*_{2}*k, u*^{(i)}* _{x}* = 1
2

¡*1, (−1)*^{i}*x*¯_{2}¢

(15)
with ¯*x*_{2} *= x*_{2}*/kx*_{2}*k if x*_{2} *6= 0, and otherwise ¯x*_{2} being any vector in IR* ^{l−1}* such that

*k¯x*2

*k = 1. If x*2

*6= 0, then the factorization is unique. The spectral decomposition of*

*x, x*

^{2}

*and x*

*has some basic properties as below, whose proofs can be found in [9].*

^{1/2}*Property 2.1 For any x = (x*1*, x*2*) ∈ IR × IR*^{l−1}*with the spectral values λ*1*(x), λ*2*(x)*
*and spectral vectors u*^{(1)}*x* *, u*^{(2)}*x* *given as above, the following results hold:*

*(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}* =p

*λ*_{1}*(x) u*^{(1)}*x* +p

*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) and kxk*^{2} *= [λ*^{2}_{1}*(x) + λ*^{2}_{2}*(x)]/2.*

For the sake of notation, throughout the rest of this paper, we always write
*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}*) ∈ IR × IR** ^{l−1}*, and let ¯

*w*

_{2}

*= w*

_{2}

*/kw*

_{2}

*k if w*

_{2}

*6= 0, and*otherwise ¯

*w*

_{2}be any vector in IR

^{l−1}*satisfying k ¯w*

_{2}

*k = 1. We have*

*w*_{1} *= kxk*^{2}*+ kyk*^{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 noting that

*w = x*

^{2}

*+ y*

^{2}

*+ (τ − 2)(x ◦ y) =*

µ

*x +τ − 2*
2 *y*

¶_{2}

+*τ (4 − τ )*
4 *y*^{2}

= µ

*y +τ − 2*
2 *x*

¶_{2}

+*τ (4 − τ )*

4 *x*^{2}*.* (17)
In addition, using Property 2.1 (b) and (c), it is not hard to compute that

*z =*

Ãp*λ*_{2}*(w) +*p
*λ*_{1}*(w)*

2 *,*

p*λ*_{2}*(w) −*p
*λ*_{1}*(w)*

2 *w*¯2

!

*∈ K*^{l}*.* (18)

*The following lemma characterizes the set of points where z(x, y) is (continuously)*
differentiable. Since the proof is direct by the arguments in Case (2) of [6, Proposition
3.2] and formulas (18) and (14), we here omit it.

*Lemma 2.1 The function z(x, y) defined by (16) 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}* =

µ *b* *c ¯w*_{2}^{T}

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

¶

*if w*_{2} *6= 0;*

³
*1/√*

*w*_{1}

´

*I* *if w*_{2} *= 0,*

(19)

*with a =* *√* ^{2}

*λ*2*(w)+**√*

*λ*1*(w)**, b =* ^{1}_{2}
µ

*√* 1

*λ*2*(w)* +*√* ^{1}

*λ*1*(w)*

¶

*and c =* ^{1}_{2}
µ

*√* 1

*λ*2*(w)* *−* *√* ^{1}

*λ*1*(w)*

¶
*.*

*Lemma 2.2 [6, Lemma 3.1] For any x = (x*1*, x*2*), y = (y*1*, y*2*) ∈ IR × IR*^{l−1}*, let*
*w = (w*_{1}*, w*_{2}*) be given as in (16). If (x − y)*^{2}*+ τ (x ◦ y) /∈ int(K*^{l}*), then*

*x*^{2}_{1} *= kx*_{2}*k*^{2}*, y*_{1}^{2} *= ky*_{2}*k*^{2}*, x*_{1}*y*_{1} *= x*^{T}_{2}*y*_{2}*, x*_{1}*y*_{2} *= y*_{1}*x*_{2}; (20)
*x*^{2}_{1}*+ y*^{2}_{1}*+ (τ − 2)x*_{1}*y*_{1} *= kx*_{1}*x*_{2}*+ y*_{1}*y*_{2}*+ (τ − 2)x*_{1}*y*_{2}*k*

*= kx*_{2}*k*^{2}*+ ky*_{2}*k*^{2}*+ (τ − 2)x*^{T}_{2}*y*_{2}*.* (21)
*If, in addition, (x, y) 6= (0, 0), then w*2 *6= 0, and moreover,*

*x*^{T}_{2}*w*¯_{2} *= x*_{1}*,* *x*_{1}*w*¯_{2} *= x*_{2}*,* *y*_{2}^{T}*w*¯_{2} *= y*_{1}*,* *y*_{1}*w*¯_{2} *= y*_{2}*.* (22)

*Lemma 2.3 [6, Lemma 3.2] For any x = (x*_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2}*) ∈ IR × IR*^{l−1}*, let w =*
*(w*1*, w*2*) be defined as in (16). If w*2 *6= 0, then for i = 1, 2,*

"µ

*x*1+ *τ − 2*
2 *y*1

¶

*+ (−1)** ^{i}*
µ

*x*2+ *τ − 2*
2 *y*2

¶_{T}

¯
*w*2

#_{2}

*≤*

°°

°° µ

*x*_{2}+ *τ − 2*
2 *y*_{2}

¶

*+ (−1)** ^{i}*
µ

*x*_{1}+ *τ − 2*
2 *y*_{1}

¶

¯
*w*_{2}

°°

°°

2

*≤ λ*_{i}*(w).*

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

To close this section, we recall some concepts that will be used in the sequel. Given
*a mapping H : IR*^{n}*→ IR*^{m}*, if H is locally Lipschitz continuous, the following set*

*∂**B**H(z) :=*©

*V ∈ IR*^{m×n}*| ∃{z*^{k}*} ⊆ D**H* *: z*^{k}*→ z, H*^{0}*(z*^{k}*) → V*ª

*is nonempty and is called the B-subdifferential of H at z, where D*_{H}*⊆ IR** ^{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 [5]. For the concepts of (strongly)*semismooth functions, please refer to [20, 21] for details. We next present definitions of

*Cartesian P -properties for a matrix M ∈ IR*

*, which are in fact special cases of those introduced by Chen and Qi [3] for a linear transformation.*

^{n×n}*Definition 2.1 A matrix M ∈ IR*^{n×n}*is said to have*

*(a) the Cartesian P -property if for any 0 6= x = (x*_{1}*, . . . , x*_{m}*) ∈ IR*^{n}*with x*_{i}*∈ IR*^{n}^{i}*, there*
*exists an index ν ∈ {1, 2, . . . , m} such that hx*_{ν}*, (Mx)*_{ν}*i > 0;*

*(b) the Cartesian P*_{0}*-property if for any 0 6= x = (x*_{1}*, . . . , x*_{m}*) ∈ IR*^{n}*with x*_{i}*∈ IR*^{n}^{i}*,*
*there exists an index ν ∈ {1, 2, . . . , m} such that x*_{ν}*6= 0 and hx*_{ν}*, (Mx)*_{ν}*i ≥ 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* : IR^{n}*→ IR*^{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}*) ∈*

IR^{n}*, there is an index ν ∈ {1, 2, . . . , m} and a constant ρ > 0 such that*
*hx*_{ν}*− y*_{ν}*, F*_{ν}*(x) − F*_{ν}*(y)i ≥ ρkx − yk*^{2};

*(b) have the Cartesian P*_{0}*-property if for any x = (x*_{1}*, . . . , x*_{m}*), y = (y*_{1}*, . . . , y*_{m}*) ∈ IR*^{n}*and x 6= y, there exists an index ν ∈ {1, 2, . . . , m} such that*

*x*_{ν}*6= y*_{ν}*and hx*_{ν}*− y*_{ν}*, F*_{ν}*(x) − F*_{ν}*(y)i ≥ 0.*

## 3 *Properties of the functions φ*

_{τ}## and Φ

_{τ}*First, we study the favorable properties of φ** _{τ}*, including the globally Lipschitz continuity,
the strong semismoothness and the characterization of the B-subdifferential at any point.

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

*(a) φ*_{τ}*is (continuously) differentiable at (x, y) if and only if w(x, y) ∈ int(K*^{l}*). Also,*

*∇*_{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) ∈ IR*^{l}*× IR*^{l}*.*

*(d) The squared norm of φ*_{τ}*, i.e. ψ*_{τ}*, is continuously differentiable everywhere.*

Proof. (a) The proof directly follows from Lemma 2.1 and the following fact that
*φ*_{τ}*(x, y) = z(x, y) − (x + y).* (23)
*(b) It suffices to prove that z(x, y) is globally Lipschitz continuous by (23). Let*

ˆ

*z = ˆz(x, y, ²) :=* £

*(x − y)*^{2}*+ τ (x ◦ y) + ²e*¤_{1/2}

(24)
*for any ² > 0 and x = (x*_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2}*) ∈ IR × IR** ^{l−1}*. Then, applying Lemma A.1 in
the appendix and the Mean-Value Theorem, we have

°°

*°z(x, y) − z(a, b)*

°°

° =

°°

°° lim* _{²→0}*+

*z(x, y, ²) − lim*ˆ

*²→0*^{+}*z(a, b, ²)*ˆ

°°

°°

*≤* lim

*²→0*^{+}*kˆz(x, y, ²) − ˆz(a, y, ²) + ˆz(a, y, ²) − ˆz(a, b, ²)k*

*≤* lim

*²→0*^{+}

°°

°°
Z _{1}

0

*∇*_{x}*z(a + t(x − a), y, ²)(x − a)dt*ˆ

°°

°°

+ lim

*²→0*^{+}

°°

°°
Z _{1}

0

*∇**y**z(a, b + t(y − b), ²)(y − b)dt*ˆ

°°

°°

*≤* *√*

*2Ck(x, y) − (a, b)k*

*for any (x, y), (a, b) ∈ IR*^{l}*× IR*^{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}
Ã

*x +τ − 2*
2 *y,*

p*τ (4 − τ )*

2 *y*

! +1

2

³

*τ − 4 +*p

*τ (4 − τ )*

´
*y.*

*Notice that φ*_{FB}*is strongly semismooth by [23, Corollary 3.3], and the functions x+*^{τ −2}_{2} *y,*

1 2

p*τ (4 − τ )y and* ^{1}_{2}*(τ − 4 +*p

*τ (4 − τ ))y are also strongly semismooth. Therefore, φ** _{τ}* is
a strongly semismooth function since by [11, Theorem 19] the composition of strongly
semismooth functions is strongly semismooth.

(d) The proof can be found in Proposition 3.3 of [6]. *2*

Proposition 3.1 (c) indicates that, when a smoothing or nonsmooth Newton method is used to solve system (10), a fast convergence rate (at least superlinear) may 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*_{FB} *was given in [19], and we here generalize it to φ*_{τ}*for any τ ∈ (0, 4). The detailed*
derivation process is included in the appendix for completeness.

*Proposition 3.2 Given a general point (x, y) ∈ IR × IR*^{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) 6= (0, 0), then*

*V*_{x}*∈*

½ 1

2*√*
*2w*_{1}

µ 1 *w*¯^{T}_{2}

¯

*w*2 *4I − 3 ¯w*2*w*¯^{T}_{2}

¶ µ

*L** _{x}*+

*τ − 2*2

*L*

_{y}¶ +1

2

µ 1

*− ¯w*2

¶
*u*^{T}

¾

*V**y* *∈*

½ 1

2*√*
*2w*1

µ 1 *w*¯_{2}^{T}

¯

*w*_{2} *4I − 3 ¯w*_{2}*w*¯^{T}_{2}

¶ µ

*L**y*+ *τ − 2*
2 *L**x*

¶ +1

2

µ 1

*− ¯w*_{2}

¶
*v*^{T}

¾
(25)
*for some u = (u*_{1}*, u*_{2}*), v = (v*_{1}*, v*_{2}*) ∈ IR × IR*^{l−1}*satisfying |u*_{1}*| ≤ ku*_{2}*k ≤ 1 and*

*|v*_{1}*| ≤ kv*_{2}*k ≤ 1, where ¯w*_{2} = _{kw}^{w}^{2}_{2}_{k}*.*

*(c) If (x, y) = (0, 0), then V**x* *∈ {L**u*ˆ*}, V**y* *∈ {L**v*ˆ*} for some ˆu = (ˆu*1*, ˆu*2*), ˆv = (ˆv*1*, ˆv*2*) ∈*
*IR × IR*^{l−1}*satisfying kˆuk, kˆvk ≤ 1 and ˆu*_{1}ˆ*v*_{2}+ ˆ*v*_{1}*u*ˆ_{2} *= 0, or*

*V**x* *∈*

½1 2

µ 1

¯
*w*_{2}

¶

*ξ** ^{T}* +1
2

µ 1

*− ¯w*_{2}

¶

*u** ^{T}* + 2

µ 0 0

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

¶¾

*V*_{y}*∈*

½1 2

µ 1

¯
*w*_{2}

¶

*η** ^{T}* + 1
2

µ 1

*− ¯w*_{2}

¶

*v** ^{T}* + 2

µ 0 0

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

¶¾ (26)

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

*|u*_{1}*| ≤ ku*_{2}*k ≤ 1, |v*_{1}*| ≤ kv*_{2}*k ≤ 1, |ξ*_{1}*| ≤ kξ*_{2}*k ≤ 1 and |η*_{1}*| ≤ kη*_{2}*k ≤ 1,*

¯

*w*_{2} *∈ IR*^{l−1}*satisfying k ¯w*_{2}*k = 1, and s = (s*_{1}*, s*_{2}*), ω = (ω*_{1}*, ω*_{2}*) ∈ IR × IR*^{l−1}*such*
*that ksk*^{2}*+ kωk*^{2} *≤ 1.*

In what follows, we investigate the properties of the operator Φ* _{τ}* : IR

^{n}*→ IR*

*given by (10). We start with the semismoothness of Φ*

^{n}*. Since Φ*

_{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 [11, Theorem 19], we obtain the following conclusion as an immediate consequence of Proposition 3.1 (c).*

_{τ}*Proposition 3.3 The operator Φ** _{τ}* : IR

^{n}*→ IR*

^{n}*given by (10) is semismooth. Moreover,*

*it is strongly semismooth if F*

^{0}*and G*

^{0}*are locally Lipschitz continuous.*

To characterize the B-subdifferential of Φ_{τ}*, we write F*_{i}*(ζ) = (F*_{i1}*(ζ), F*_{i2}*(ζ)) and*
*G*_{i}*(ζ) = (G*_{i1}*(ζ), G*_{i2}*(ζ)), and denote w*_{i}*and z*_{i}*for i = 1, 2, . . . , m by*

*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)
*For convenience, we sometimes suppress in F*_{i}*(ζ) and G*_{i}*(ζ) the dependence on ζ.*

*Proposition 3.4 Let Φ** _{τ}* : IR

^{n}*→ IR*

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

^{n}*,*

*∂** _{B}*Φ

_{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)) 6= (0, 0) and (F**i**(ζ) − G**i**(ζ))*^{2}*+ τ (F**i**(ζ) ◦ G**i**(ζ)) ∈ bd(K*^{n}^{i}*), then*
*A**i**(ζ) ∈*

½ 1

2*√*
*2w**i1*

µ

*L**F**i* +*τ − 2*
2 *L**G**i*

¶ µ 1 *w*¯_{i2}^{T}

¯

*w*_{i2}*4I − 3 ¯w*_{i2}*w*¯^{T}_{i2}

¶ +1

2*u**i**(1, − ¯w*^{T}* _{i2}*)

¾

*B*_{i}*(ζ) ∈*

½ 1

2*√*
*2w*_{i1}

µ

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

*τ − 2*2

*L*

_{F}

_{i}¶ µ 1 *w*¯^{T}_{i2}

¯

*w**i2* *4I − 3 ¯w**i2**w*¯_{i2}^{T}

¶ + 1

2*v*_{i}*(1, − ¯w*^{T}* _{i2}*)

¾

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

_{kw}

^{w}

^{i2}

_{i2}

_{k}*.*

*(c) If (F**i**(ζ), G**i**(ζ)) = (0, 0), then*
*A*_{i}*(ζ) ∈*

n
*L*_{u}_{ˆ}_{i}

o

*∪*

½1
2*ξ** _{i}*¡

*1, ¯w*^{T}* _{i2}*¢
+ 1

2*u** _{i}*¡

*1, − ¯w*^{T}* _{i2}*¢
+

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

*0 2s*

_{i1}*(I − ¯w*

_{i2}*w*¯

_{i2}*)*

^{T}¶¾

*B*_{i}*(ζ) ∈*
n

*L*_{ˆ}_{v}* _{i}*
o

*∪*

½1
2*η** _{i}*¡

*1, ¯w*_{i2}* ^{T}*¢
+1

2*v** _{i}*¡

*1, − ¯w*^{T}* _{i2}*¢
+

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

*0 2ω*

*i1*

*(I − ¯w*

*i2*

*w*¯

_{i2}*)*

^{T}¶¾

*for some ˆu** _{i}* = (ˆ

*u*

_{i1}*, ˆu*

_{i2}*), ˆv*

*= (ˆ*

_{i}*v*

_{i1}*, ˆv*

_{i2}*) ∈ IR × IR*

^{n}

^{i}

^{−1}*satisfying kˆu*

_{i}*k, kˆv*

_{i}*k ≤ 1*

*and ˆu*

_{i1}*v*ˆ

*+ ˆ*

_{i2}*v*

_{i1}*u*ˆ

_{i2}*= 0, some u*

_{i}*= (u*

_{i1}*, u*

_{i2}*), v*

_{i}*= (v*

_{i1}*, v*

_{i2}*), ξ*

_{i}*= (ξ*

_{i1}*, ξ*

_{i2}*), η*

*=*

_{i}*(η*

*i1*

*, η*

*i2*

*) ∈ IR × IR*

^{n}

^{i}

^{−1}*with |u*

*i1*

*| ≤ ku*

*i2*

*k ≤ 1, |v*

*i1*

*| ≤ kv*

*i2*

*k ≤ 1, |ξ*

*i1*

*| ≤ kξ*

*i2*

*k ≤ 1*

*and |η*

_{i1}*| ≤ kη*

_{i2}*k ≤ 1, ¯w*

_{i2}*∈ IR*

^{n}

^{i}

^{−1}*satisfying k ¯w*

_{i2}*k = 1, and s*

_{i}*= (s*

_{i1}*, s*

_{i2}*), ω*

*=*

_{i}*(ω*

_{i1}*, ω*

_{i2}*) ∈ IR × IR*

^{n}

^{i}

^{−1}*such that ks*

_{i}*k*

^{2}

*+ kω*

_{i}*k*

^{2}

*≤ 1.*

Proof. Let Φ_{τ,i}*(ζ) denote the ith subvector of Φ** _{τ}*, i.e. Φ

_{τ,i}*(ζ) = φ*

_{τ}*(F*

_{i}*(ζ), G*

_{i}*(ζ)) for all*

*i = 1, 2, . . . , m. From Proposition 2.6.2 of [5], it follows that*

*∂** _{B}*Φ

_{τ}*(ζ)*

^{T}*⊆ ∂*

*Φ*

_{B}

_{τ,1}*(ζ)*

^{T}*× ∂*

*Φ*

_{B}

_{τ,2}*(ζ)*

^{T}*× · · · × ∂*

*Φ*

_{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 ∂*

*Φ*

_{B}

_{τ,i}*(ζ)*

*. Using the definition of B-subdifferential and the continuous*

^{T}*differentiability of F and G, it is not difficult to verify that*

*∂**B*Φ*τ,i**(ζ)*^{T}*= [∇F**i**(ζ) ∇G**i**(ζ)]∂**B**φ**τ**(F**i**(ζ), G**i**(ζ))*^{T}*,* *i = 1, . . . , m.* (30)
Using Proposition 3.2 and the last two equations, we readily get the desired result. *2*

*Lemma 3.1 For any ζ ∈ IR*^{n}*, let A(ζ) and B(ζ) be the multivalued block diagonal*
*matrices given as in Proposition 3.4. Then, for any i ∈ {1, 2, . . . , m},*

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

*and the equality holds if and only if Φ*_{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*

*h(A*_{i}*(ζ) − I)υ*_{i}*, (B*_{i}*(ζ) − I)υ*_{i}*i ≥ 0,* *for any υ*_{i}*∈ IR*^{n}^{i}*.*
Proof. From Theorem 2.6.6 of [5] and Proposition 3.1 (d), we have

*∇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 Propositions 3.2 and 3.4, 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)Φ**τ,i**(ζ)*

*∇*_{y}*ψ*_{τ}*(F*_{i}*(ζ), G*_{i}*(ζ)) = (B*_{i}*(ζ) − I)Φ*_{τ,i}*(ζ).* (31)
Consequently, the first part of the conclusions is direct by Proposition 4.1 of [6]. Notice
*that for any i such that (F*_{i}*(ζ) − G*_{i}*(ζ))*^{2}*+ τ (F*_{i}*(ζ) ◦ G*_{i}*(ζ) ∈ int(K*^{n}^{i}*) and any υ*_{i}*∈ IR*^{n}* ^{i}*,

*h(A*_{i}*(ζ) − I)υ*_{i}*, (B*_{i}*(ζ) − I)υ*_{i}*i*

= D³

*L*_{F}_{i}_{+}^{τ −2}

2 *G**i**− L*_{z}_{i}

´
*L*^{−1}_{z}_{i}*υ*_{i}*,*

³

*L*_{G}_{i}_{+}^{τ −2}

2 *F**i**− L*_{z}_{i}

´
*L*^{−1}_{z}_{i}*υ*_{i}

E

= D³

*L*_{G}_{i}_{+}^{τ −2}

2 *F**i**− L*_{z}* _{i}*´³

*L*_{F}_{i}_{+}^{τ −2}

2 *G**i* *− L*_{z}* _{i}*´

*L*^{−1}_{z}_{i}*υ*_{i}*, L*^{−1}_{z}_{i}*υ** _{i}*E

*.* (32)

Therefore, using the same argument as Case (2) of [6, Proposition 4.1], we can obtain
the second part of the conclusions. *2*

## 4 Nonsingularity conditions

*In this section, we show that all elements of the B-subdifferential ∂** _{B}*Φ

_{τ}*(ζ) at a solution*

*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 multivalued matrix (A*_{i}*(ζ*^{∗}*) − I) +*
*(B*_{i}*(ζ*^{∗}*) − I) are nonsingular if the ith block component satisfies strict complementarity.*

*Lemma 4.1 Let ζ*^{∗}*be a solution of the SOCCP, and A(ζ*^{∗}*) and B(ζ*^{∗}*) be the multivalued*
*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 Φ**τ,i**(ζ) is continuously 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}*, hF*_{i}*(ζ*^{∗}*), G*_{i}*(ζ*^{∗}*)i = 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.