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∈ Rn 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 RntoRnwhich are assumed to be continuously differentiable, andK is the Cartesian product of second-order cones (SOCs), also called Lorentz cones [10]. In other words, K = Kn1× Kn2× · · · × Knm, (2) where m, n1, . . . , nm≥ 1, n1+ n2+ · · · + nm= n, and
Kni:= {(x1, x2)∈ R × Rni−1| x1≥ x2},
with· denoting the Euclidean norm and K1denoting 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= (x1, . . . , xm)with xi ∈ Rni for any x ∈ Rn, and F = (F1, . . . , Fm) and G= (G1, . . . , Gm)with Fi, Gi: Rn→ Rni.
An important special case of the SOCCP corresponds to G(ζ )= ζ for all ζ ∈ Rn. 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 = K1×· · ·×K1. 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∈ Rm×nhas full row rank, b∈ Rmand g: Rn→ 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 −AT(AAT)−1A)ζ, G(ζ ):= ∇g(F (ζ ))−AT(AAT)−1Aζ, (5) where d∈ Rnis 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 AAT, 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: Rl× Rl→ Rl, we call φ an SOC complementarity function associated with the coneKlif for any (x, y)∈ Rl× Rl,
φ (x, y)= 0 ⇐⇒ x ∈ Kl, y∈ Kl, 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 φFB(x, y):= (x2+ y2)1/2− (x + y). (7) More specifically, for any x= (x1, x2), y= (y1, y2)∈ R × Rl−1, we define their Jor- dan product associated withKlas
x◦ y := (x, y, y1x2+ x1y2). (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 ∈ Rl. We write x2to mean x◦ x and write x+ y to mean the usual componentwise addition of vectors. It is known that x2∈ Kl for all x∈ Rl. Moreover, if x∈ Kl, then there exists a unique vector inKl, denoted by x1/2, such that (x1/2)2= x1/2◦ x1/2= x. Thus, φFBin (7) is well-defined for all (x, y)∈ Rl× Rland mapsRl× RltoRl. The function φFBwas proved in [13]
to satisfy the equivalence (6), and therefore its squared norm, denoted by ψFB(x, y):=1
2φFB(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 φNR: Rl× Rl→ Rl given by
φNR(x, y):= x − [x − y]+,
where[·]+means the minimum Euclidean distance projection ontoKl. 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
ψNR(x, y):=1
2φNR(x, y)2,
but, compared to ψFB, it has a remarkable drawback, i.e. the non-differentiability.
In this paper, we consider a one-parametric class of vector-valued functions φτ(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, φτ reduces to the FB function φFB in (7) while it becomes a multiple of the natural residual function φNR 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 ):=
⎛
⎜⎜
⎜⎜
⎜⎜
⎝
φτ(F1(ζ ), G1(ζ )) ...
φτ(Fi(ζ ), Gi(ζ )) ...
φτ(Fm(ζ ), Gm(ζ ))
⎞
⎟⎟
⎟⎟
⎟⎟
⎠
= 0, (10)
which induces a natural merit function τ : Rn→ R+given by
τ(ζ )=1
2τ(ζ )2=
m i=1
ψτ(Fi(ζ ), Gi(ζ )), (11)
with ψτ being the natural merit function associated with φτ, i.e.,
ψτ(x, y)=1
2φτ(x, y)2. (12)
In [4], we studied the continuous differentiability of ψτ 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 φτ, 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 Rn1 × · · · × Rnm is identified with Rn1+···+nm. For a differentiable mapping F : Rn→ Rm,∇F (x) denotes the transpose of the Jacobian F (x). For a symmet- ric matrix A∈ Rn×n, we write A O (respectively, A O) to mean A is positive semidefinite (respectively, positive definite). Given a finite number of square matrices Q1, . . . , Qn, we denote the block diagonal matrix with these matrices as block diag- onals by diag(Q1, . . . , Qn)or by diag(Qi, i= 1, . . . , n). If J and B are index sets such thatJ , B ⊆ {1, 2, . . . , m}, we denote PJ Bby the block matrix consisting of the sub-matrices Pj k∈ Rnj×nk of P with j∈ J , k ∈ B, and by xBa vector consisting of sub-vectors xi∈ Rni 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 ofKl. It is known thatKlis a closed convex self-dual cone with nonempty interior given by
int(Kl):= {x = (x1, x2)∈ R × Rl−1| x1>x2}
and the boundary given by
bd(Kl):= {x = (x1, x2)∈ R × Rl−1| x1= x2}.
For each x= (x1, x2)∈ R × Rl−1, the determinant and the trace of x are defined by det(x):= x12− x22, tr(x):= 2x1.
In general, det(x◦ y) = det(x) det(y) unless x2= αy2 for some α∈ R. A vector x∈ Rlis said to be invertible if det(x)= 0, and its inverse is denoted by x−1. Given a vector x= (x1, x2)∈ R × Rl−1, we often use the following symmetry matrix
Lx:=
x1 x2T x2 x1I
, (13)
which can be viewed as a linear mapping fromRl toRl. It is easy to verify Lxy= x◦ y and Lx+y= Lx+ Ly for any x, y∈ Rl. Furthermore, x∈ Kl if and only if Lx O, and x ∈ int(Kl)if and only if Lx O. Then Lxis invertible with
L−1x = 1 det(x)
x1 −x2T
−x2 det(x) x1 I+x1
1x2x2T
. (14)
We recall from [13] that each x= (x1, x2)∈ R × Rl−1admits a spectral factoriza- tion, associated withKl, of the form
x= λ1(x)· u(1)x + λ2(x)· u(2)x ,
where λi(x)and u(i)x for i= 1, 2 are the spectral values and the associated spectral vectors of x, respectively, given by
λi(x)= x1+ (−1)ix2, u(i)x =1
2(1, (−1)i¯x2) (15) with ¯x2= x2/x2 if x2= 0, and otherwise ¯x2 being any vector inRl−1satisfying
¯x2 = 1. If x2= 0, then the factorization is unique. The spectral decompositions of x, x2 and x1/2have some basic properties as below, whose proofs can be found in [13].
Property 2.1 For any x= (x1, x2)∈ R × Rl−1with the spectral values λ1(x), λ2(x) and spectral vectors u(1)x , u(2)x given as above, we have that
(a) x∈ Kl if and only if λ1(x)≥ 0, and x ∈ int(Kl)if and only if λ1(x) >0.
(b) x2= λ21(x)u(1)x + λ22(x)u(2)x ∈ Kl. (c) x1/2=√
λ1(x) u(1)x +√
λ2(x) u(2)x ∈ Kl if x∈ Kl.
(d) det(x)= λ1(x)λ2(x), tr(x)= λ1(x)+ λ2(x)andx2= [λ21(x)+ λ22(x)]/2.
For the sake of notation, throughout the rest of this paper, we always let w= (w1, w2)= w(x, y) := (x − y)2+ τ(x ◦ y),
z= (z1, z2)= z(x, y) := [(x − y)2+ τ(x ◦ y)]1/2 (16) for any x= (x1, x2), y= (y1, y2)∈ R × Rl−1. It is easy to compute
w1= x2+ y2+ (τ − 2)xTy,
w2= 2(x1x2+ y1y2)+ (τ − 2)(x1y2+ y1x2).
Moreover, w∈ Kland z∈ Klhold by considering that w= x2+ y2+ (τ − 2)(x ◦ y)
=
x+τ− 2 2 y
2
+τ (4− τ) 4 y2=
y+τ− 2 2 x
2
+τ (4− τ)
4 x2. (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= (x1, x2), y= (y1, y2)∈ R × Rl−1and τ∈ (0, 4), let w= (w1, w2)be defined as in (16). Ifw2 = 0, then
x1+τ − 2 2 y1
+ (−1)i
x2+τ − 2 2 y2
T w2
w2
2
≤
x2+τ − 2 2 y2
+ (−1)i
x1+τ − 2 2 y1
w2
w2
2
≤ λi(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= (x1, x2), y= (y1, y2)∈ R × Rl−1and τ∈ (0, 4), let w= (w1, w2)be given as in (16). If w /∈ int(Kl), then
x12= x22, y12= y22, x1y1= x2Ty2, x1y2= y1x2; (18) x12+ y12+ (τ − 2)x1y1= x1x2+ y1y2+ (τ − 2)x1y2
= x22+ y22+ (τ − 2)xT2y2. (19) If, in addition, (x, y)= (0, 0), then w2 = 0, and moreover,
x2T w2
w2= x1, x1
w2
w2 = x2, y2T w2
w2 = y1, y1
w2
w2= y2. (20)
Lemma 2.3 [4, Proposition 3.2] For any x= (x1, x2), y= (y1, y2)∈ R × Rl−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(Kl), and furthermore,
∇xz(x, y)= Lx+τ−2
2 yL−1z , ∇yz(x, y)= Ly+τ−2 2 xL−1z , where
L−1z =
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
⎛
⎝ b c w
T
w22
cww2
2 aI+ (b − a)ww2wT2
22
⎞
⎠ if w2= 0;
(1/√
w1)I if w2= 0,
(21)
with
a= 2
√λ2(w)+√
λ1(w), b=1 2
1
√λ2(w)+ 1
√λ1(w)
,
c=1 2
1
√λ2(w)− 1
√λ1(w)
.
(22)
To close this section, we recall some definitions that will be used in the subsequent sections. Given a mapping H: Rn→ Rm, if H is locally Lipschitz continuous, the set
∂BH (z):= {V ∈ Rm×n| ∃{zk} ⊆ DH: zk→ z, H (zk)→ V }
is nonempty and is called the B-subdifferential of H at z, where DH ⊆ Rndenotes the set of points at which H is differentiable. The convex hull ∂H (z):= conv∂BH (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∈ Rn×n, which are in fact special cases of those introduced by Chen and Qi [5] for a linear transformation.
Definition 2.1 A matrix M∈ Rn×nis said to have
(a) the Cartesian P -property if for any 0= x = (x1, . . . , xm)∈ Rn with xi ∈ Rni, there exists an index ν∈ {1, 2, . . . , m} such that
xν, (Mx)ν > 0;
(b) the Cartesian P0-property if for any 0= x = (x1, . . . , xm)∈ Rn with xi ∈ Rni, there exists an index ν∈ {1, 2, . . . , m} such that
xν= 0 and xν, (Mx)ν ≥ 0.
Some nonlinear generalizations of these concepts in the setting ofK are defined as follows.
Definition 2.2 Given a mapping F= (F1, . . . , Fm)with Fi : Rn→ Rni, F is said to
(a) have the uniform Cartesian P -property if for any x = (x1, . . . , xm), y = (y1, . . . , ym)∈ Rn, there exists an index ν∈ {1, 2, . . . , m} and a positive constant ρ >0 such that
xν− yν, Fν(x)− Fν(y) ≥ ρx − y2;
(b) have the Cartesian P0-property if for any x= (x1, . . . , xm), y= (y1, . . . , ym)∈ Rn, there exists an index ν∈ {1, 2, . . . , m} such that
xν= yν and xν− yν, Fν(x)− Fν(y) ≥ 0.
If a continuously differentiable mapping F has the Cartesian P -properties, then the matrix∇F (x) at any x ∈ Rnenjoys the corresponding Cartesian P -properties.
3 Properties of the functions φτ and τ
This section is devoted to investigating the favorable properties of φτ, 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 φτ and z(x, y) given as in (9) and (16), respectively, we have
φτ(x, y)= z(x, y) − (x + y) = z − (x + y) (23) for any x= (x1, x2), y= (y1, y2)∈ R × Rl−1. Recall that the vectors w= (w1, w2) and z= (z1, z2)in (16) satisfy w, z∈ Kl, and hence, from Property2.1(b) and (c),
z=
√λ2(w)+√ λ1(w)
2 ,
√λ2(w)−√ λ1(w)
2 ¯w2
, (24)
where ¯w2= ww22 if w2= 0, and otherwise ¯w2 is any vector in Rl−1 satisfying
¯w2 = 1. The following proposition states some favorable properties possessed by φτ.
Proposition 3.1 The function φτ defined as in (9) has the following properties.
(a) φτ is continuously differentiable at a point (x, y)∈ Rl × Rl if and only if (x− y)2+ τ(x ◦ y) ∈ int(Kl). Moreover,
∇xφτ(x, y)= Lx+τ−2
2 yL−1z − I, ∇yφτ(x, y)= Ly+τ−2
2 xL−1z − I.
(b) φτ is globally Lipschitz continuous with the Lipschitz constant independent of τ . (c) φτ is strongly semismooth at any (x, y)∈ Rl× Rl.
(d) ψτ defined by (12) is continuously differentiable everywhere.
Proof (a) The proof directly follows from Lemma2.3and (23).
(b) It suffices to prove that z(x, y) is globally Lipschitz continuous by (23). Let ˆz = (ˆz1,ˆz2)= ˆz(x, y, ) := [(x − y)2+ τ(x ◦ y) + e]1/2 (25) for any > 0 and x= (x1, x2), y= (y1, y2)∈ R × Rl−1. Then, applying Lemma A.1 inAppendixand the Mean-Value Theorem, we have
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
∇xˆz(a + t(x − a), y, )(x − a)dt
+ lim
→0+
1
0
∇yˆz(a, b + t(y − b), )(y − b)dt
≤√
2C(x, y) − (a, b)
for any (x, y), (a, b)∈ Rl× Rl, where C > 0 is a constant independent of τ . (c) From the definition of φτ and φFB, it is not hard to check that
φτ(x, y)= φFB
x+τ− 2 2 y,
√τ (4− τ)
2 y
+1
2(τ− 4 +
τ (4− τ))y.
Note that φFBis strongly semismooth everywhere by Corollary 3.3 of [25], and the functions x+ τ−22 y, 12√
τ (4− τ)y and 12(τ − 4 +√
τ (4− τ))y are also strongly semismooth at any (x, y)∈ Rl× Rl. Therefore, φτ 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 ∂Bφτ(x, y)at a general point (x, y). The discussion of B-subdifferential for φFBwas given in [20]. Here, we generalize it to φτ for any τ∈ (0, 4). The detailed derivation process is included inAppendixfor completeness.
Proposition 3.2 Given a general point x= (x1, x2), y= (y1, y2)∈ R × Rl−1, each element in ∂Bφτ(x, y)is of the form V = [Vx− I Vy− I] with Vxand Vyhaving the following representation:
(a) If (x− y)2+ τ(x ◦ y) ∈ int(Kl), then Vx= L−1z Lx+τ−2
2 yand Vy= L−1z Ly+τ−2 2 x. (b) If (x− y)2+ τ(x ◦ y) ∈ bd(Kl)and (x, y)= (0, 0), then
Vx∈
1 2√
2w1
1 ¯wT2
¯w2 4I− 3 ¯w2¯wT2
Lx+τ− 2 2 Ly
+1
2 1
− ¯w2
uT
,
(26) Vy∈
1 2√
2w1
1 ¯wT2
¯w2 4I− 3 ¯w2¯wT2
Ly+τ− 2 2 Lx
+1
2 1
− ¯w2
vT
for some u= (u1, u2), v= (v1, v2)∈ R × Rl−1satisfying|u1| ≤ u2 ≤ 1 and
|v1| ≤ v2 ≤ 1, where ¯w2=ww22.
(c) If (x, y)= (0, 0), then Vx ∈ {Lˆu}, Vy ∈ {Lˆv} for some ˆu = ( ˆu1,ˆu2), ˆv = (ˆv1,ˆv2)∈ R × Rl−1satisfying ˆu, ˆv ≤ 1 and ˆu1ˆv2+ ˆv1ˆu2= 0, or
Vx∈
1 2
1
¯w2
ξT +1
2 1
− ¯w2
uT + 2
0 0
(I− ¯w2¯w2T)s2 (I− ¯w2¯w2T)s1
,
Vy∈
1 2
1
¯w2
ηT +1
2 1
− ¯w2
vT + 2
0 0
(I− ¯w2¯w2T)ω2 (I− ¯w2¯wT2)ω1
for some ¯w2 = 1, u = (u1, u2), v= (v1, v2), ξ = (ξ1, ξ2), η= (η1, η2)∈ R × Rl−1 satisfying|u1| ≤ u2 ≤ 1, |v1| ≤ v2 ≤ 1, |ξ1| ≤ ξ2 ≤ 1 and |η1| ≤
η2 ≤ 1, and s = (s1, s2), ω= (ω1, ω2)∈ R × Rl−1such thats2+ ω2≤ 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: Rn→ Rndefined 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 Fi(ζ )= (Fi1(ζ ), Fi2(ζ )), Gi(ζ )= (Gi1(ζ ), Gi2(ζ ))∈ R × Rni−1 and wi: Rn→ Rniand zi: Rn→ Rni for i= 1, 2, . . . , m be given as follows:
wi = (wi1(ζ ), wi2(ζ ))= w(Fi(ζ ), Gi(ζ )),
zi = (zi1(ζ ), zi2(ζ ))= z(Fi(ζ ), Gi(ζ )). (27) Proposition 3.4 Let τ: Rn→ Rnbe defined as in (10). Then, for any ζ∈ Rn,
∂Bτ(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 Ai(ζ )and Bi(ζ )for i= 1, 2, . . . , m have the following representation:
(a) If (Fi(ζ )− Gi(ζ ))2+ τ(Fi(ζ )◦ Gi(ζ ))∈ int(Kni), then Ai(ζ )= LFi+τ−2
2 GiL−1z
i and Bi(ζ )= LGi+τ−2
2 FiL−1z
i .
(b) If (Fi(ζ )− Gi(ζ ))2+ τ(Fi(ζ )◦ Gi(ζ ))∈ bd(Kni)and (Fi(ζ ), Gi(ζ ))= (0, 0), then
Ai(ζ )∈
1
2√ 2wi1
LFi+τ− 2 2 LGi
1 ¯wi2T
¯wi2 4I− 3 ¯wi2¯wi2T
+1
2ui(1,− ¯wiT2)
Bi(ζ )∈
1
2√ 2wi1
LGi+τ− 2 2 LFi
1 ¯wi2T
¯wi2 4I− 3 ¯wi2¯wi2T
+1
2vi(1,− ¯wTi2)
for some ui= (ui1, ui2), vi= (vi1, vi2)∈ R × Rni−1satisfying|ui1| ≤ ui2 ≤ 1 and|vi1| ≤ vi2 ≤ 1, where ¯wi2=wwi2
i2. (c) If (Fi(ζ ), Gi(ζ ))= (0, 0), then
Ai(ζ )∈ {Lˆu1} ∪
1
2ξi(1, ¯wTi2)+1
2ui(1,− ¯wTi2)+
0 2si2T(I− ¯wi2¯wi2T) 0 2si1(I− ¯wi2¯wi2T)
Bi(ζ )∈ {Lˆv1} ∪
1
2ηi(1, ¯wTi2)+1
2vi(1,− ¯wi2T)+
0 2ωTi2(I− ¯wi2¯wTi2) 0 2ωi1(I− ¯wi2¯wTi2)
for some ˆui = ( ˆui1,ˆui2), ˆvi= (ˆvi1,ˆvi2)∈ R × Rni−1 satisfying ˆui, ˆvi ≤ 1 and ˆui1ˆvi2+ ˆvi1ˆui2= 0, some ui = (ui1, ui2), vi = (vi1, vi2), ξi = (ξi1, ξi2), ηi = (ηi1, ηi2)∈ R × Rni−1with|ui1| ≤ ui2 ≤ 1, |vi1| ≤ vi2 ≤ 1, |ξi1| ≤
ξi2 ≤ 1 and |ηi1| ≤ ηi2 ≤ 1, ¯ωi2∈ Rni−1 satisfying ¯ωi2 = 1, and si = (si1, si2), ωi= (ωi1, ωi2)∈ R × Rni−1such thatsi2+ ωi2≤ 1.
Proof Let τ,i(ζ )denote the ith subvector of τ, i.e. τ,i(ζ )= φτ(Fi(ζ ), Gi(ζ )) for all i= 1, 2, . . . , m. From Proposition 2.6.2 of [8], it follows that
∂Bτ(ζ )T ⊆ ∂Bτ,1(ζ )T × ∂Bτ,2(ζ )T × · · · × ∂Bτ,m(ζ )T, (29) where the latter denotes the set of all matrices whose (ni−1+ 1) to nith columns with n0= 0 belong to ∂Bτ,i(ζ )T. Using the definition of B-subdifferential and the continuous differentiability of F and G, it is not difficult to verify that
∂Bτ,i(ζ )T = [∇Fi(ζ ) ∇Gi(ζ )]∂Bφτ(Fi(ζ ), Gi(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∈ Rn, let A(ζ ) and B(ζ ) be the multivalued block diago- nal matrices given as in Proposition3.4Then, for any i∈ {1, 2, . . . , m},
(Ai(ζ )− I)τ,i(ζ ), (Bi(ζ )− I)τ,i(ζ ) ≥ 0,
with equality holding if and only if τ,i(ζ )= 0. Particularly, for the index i such that (Fi(ζ )− Gi(ζ ))2+ τ(Fi(ζ )· Gi(ζ )∈ int(Kni)), we have
(Ai(ζ )− I)υi, (Bi(ζ )− I)υi ≥ 0, for any υi∈ Rni. Proof From Theorem 2.6.6 of [8] and Proposition 3.1(d), we have that
∇ψτ(x, y)= ∂Bφτ(x, y)Tφτ(x, y).
Consequently, for any i= 1, 2, . . . , m, it follows that
∇ψτ(Fi(ζ ), Gi(ζ ))= ∂Bφτ(Fi(ζ ), Gi(ζ ))Tφτ(Fi(ζ ), Gi(ζ )).
In addition, from Propositions3.2and3.4, it is not hard to see that [Ai(ζ )T − I Bi(ζ )T − I] ∈ ∂Bφτ(Fi(ζ ), Gi(ζ )).
Combining with the last two equations yields that for any i= 1, 2, . . . , m,
∇xψτ(Fi(ζ ), Gi(ζ ))= (Ai(ζ )− I)τ,i(ζ ),
∇yψτ(Fi(ζ ), Gi(ζ ))= (Bi(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 υi∈ Rni,
(Ai(ζ )− I)υi, (Bi(ζ )− I)υi
= (LFi+τ−2
2 Gi− Lzi)L−1z
i υi, (LG
i+τ−22 Fi− Lzi)L−1z
i υi
= (LGi+τ−2
2 Fi− Lzi)(LF
i+τ−22 Gi− Lzi)L−1z
i υi, L−1z
i υi. (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 ∂Bτ(ζ )at a solu- tion ζ∗of the SOCCP are nonsingular if ζ∗satisfies strict complementarity, i.e.,
Fi(ζ∗)+ Gi(ζ∗)∈ int(Kni) for all i= 1, 2, . . . , m. (33) First, we give a technical lemma which states that the multi-valued matrix (Ai(ζ∗)− I) + (Bi(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 Fi(ζ∗)+ Gi(ζ∗)∈ int(Kni), we have that τ,i(ζ )is con- tinuously differentiable at ζ∗and (Ai(ζ∗)− I) + (Bi(ζ∗)− I) is nonsingular.
Proof Since ζ∗is a solution of the SOCCP, we have for all i= 1, 2, . . . , m Fi(ζ∗)∈ Kni, Gi(ζ∗)∈ Kni, Fi(ζ∗), Gi(ζ∗) = 0.
It is not hard to verify that Fi(ζ∗)+ Gi(ζ∗)∈ int(Kni)if and only if one of the three cases shown as below holds.
Case (1) Fi(ζ∗)∈ int(Kni)and Gi(ζ∗)= 0. Under this case,
wi(ζ∗)= (Fi(ζ∗)− Gi(ζ∗))2+ τ(Fi(ζ∗)◦ Gi(ζ∗))= Fi(ζ∗)2∈ int(Kni).
By Proposition3.1(a), τ,i(ζ )is continuously differentiable at ζ∗. Since zi(ζ∗)= wi(ζ∗)1/2= Fi(ζ∗), from Proposition3.4(a) it follows that
Ai(ζ∗)= I and Bi(ζ∗)=τ− 2 2 I,
which implies that (Ai(ζ∗)− I) + (Bi(ζ∗)− I) is nonsingular since 0 < τ < 4.
Case (2) Fi(ζ∗)= 0 and Gi(ζ∗)∈ int(Kni). Now, wi(ζ∗)= Gi(ζ∗)2∈ int(Kni).
So, τ,i(ζ )is continuously differentiable at ζ∗by Proposition3.1(a). Since zi(ζ∗)= wi(ζ∗)1/2= Gi(ζ∗),
applying Proposition3.4(a) yields that
Ai(ζ∗)=τ− 2
2 I and Bi(ζ∗)= I,
which immediately implies that (Ai(ζ∗)− I) + (Bi(ζ∗)− I) is nonsingular.
Case (3) Fi(ζ∗)∈bd+(Kni)and Gi(ζ∗)∈bd+(Kni), where bd+(Kni):=bd(Kni)\{0}.
By Proposition 3.1(a), it suffices to prove wi(ζ∗) ∈ int(Kni). Suppose that wi(ζ∗)∈ bd(Kni). Then, from (18) in Lemma2.2, it follows that
Fi1(ζ∗)Gi1(ζ∗)= Fi2(ζ∗)TGi2(ζ∗).
Since Fi1(ζ∗)= Fi2(ζ∗) = 0 and Gi1(ζ∗)= Gi2(ζ∗) = 0, we have
Fi2(ζ∗) · Gi2(ζ∗) = Fi2(ζ∗)TGi2(ζ∗),
which implies that Fi2(ζ∗)= αGi2(ζ∗) for some constant α > 0. Consequently, Fi(ζ∗)= αGi(ζ∗). Noting that Fi(ζ∗), Gi(ζ∗) = 0, we then get Fi(ζ∗) = Gi(ζ∗) = 0. This clearly contradicts the assumptions that Fi(ζ∗) = 0 and Gi(ζ∗)= 0. So, wi(ζ∗)∈ int(Kni).
From the expression of Ai(ζ )and Bi(ζ )given by Proposition 3.4(a), (Ai(ζ∗)− I) + (Bi(ζ∗)− I) = −L2zi(ζ∗)−τ2(Fi(ζ∗)+Gi(ζ∗))L−1z
i(ζ∗).
