for the Second-Order Cone Complementarity Problem

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∈ Rⁿsuch that F (ζ )∈ K, G(ζ )∈ K, F (ζ ), G(ζ ) = 0, (1) where·, · represents the Euclidean inner product, F, G : Rⁿ→ R^m are the map- ping assumed to be continuously differentiable throughout this paper, andK is the Cartesian product of second-order cones (SOCs), or called Lorentz cones. In other words,

K = Kⁿ¹× Kⁿ²× · · · × K^nq, (2) where q, n1, . . . , n_q≥ 1, n1+ · · · + nq= m and

Kⁿⁱ:=

x= (x1, x₂)∈ R × Rⁿⁱ⁻¹| x1≥ x2

with ·  denoting the Euclidean norm and K¹ 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= (F1, . . . , Fq)and G= (G1, . . . , Gq)with Fi, Gi : Rⁿ→ Rⁿⁱ.

An important special case of the SOCCP corresponds to n= m and G(ζ ) = ζ for all ζ∈ Rⁿ. 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 coneRⁿ₊. 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)⁻¹A)ζ, G(ζ ):= ∇g(F (ζ ))−A^T(AA^T)⁻¹Aζ, (5) where ˆx ∈ R^mis 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^l→ R^l is 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):= (x²+ y²)^1/2− (x + y) ∀x, y ∈ R^l (7) where x²= x ◦ x denotes the Jordan product of x and itself, x^1/2 denotes a vector 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^l. The function was shown in [11] to satisfy the equivalence (6), and therefore its squared norm

ψ (x, y):=1

2φ(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 andx, 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 φwas 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 (F₁(ζ ), G₁(ζ )) ... φ (Fi(ζ ), Gi(ζ ))

... φ (Fq(ζ ), Gq(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 : Rⁿ→ R^min (9) is semismooth. Furthermore, from Proposition 2 of [6], its squared norm induces a smooth merit function, given by

(ζ ):=1

2(ζ )²= q

i=1

ψ (Fi(ζ ), Gi(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,Rⁿ denotes the space of n-dimensional real column vectors, andRⁿ¹ × · · · × R^nq is identified withR^n1+···+nq. Thus, (x1, . . . , x_q)∈ Rⁿ¹× · · · × R^nq is viewed as a col- umn vector inR^n1+···+nq. For any differentiable mapping F : Rⁿ→ R^m, the nota- tion∇F (x) ∈ R^n×mdenotes 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 Q1, . . . , Qq, we denote the block diagonal matrix with these matrices as block diagonals by diag(Q1, . . . , Q_q)or by diag(Qi, i= 1, . . . , q). If J and B are index sets such thatJ , B ⊆ {1, 2, . . . , q}, we denote by P_{J B}the block matrix consisting of the sub- matrices Pj k∈ R^nj×nk of P with j∈ J , k ∈ B, and denote by x_Ba vector consisting of sub-vectors xi∈ Rⁿⁱ 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 ofK^l (l >1). It is known thatK^lis a closed convex self-dual cone with nonempty interior given by

int(K^l):=

x= (x1, x2)∈ R × R^l−1| x1>x2 and the boundary given by

bd(K^l):=

x= (x1, x2)∈ R × R^l−1| x1= x2 .

For any x= (x1, x2), y= (y1, y2)∈ R × R^l−1, we define their Jordan product [9] as x◦ y :=

x, y, x1y₂+ y1x₂ .

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= (x1, x₂)∈ R × R^l−1, define the matrix Lxby

L_x:=

x₁ x₂^T x₂ x₁I

 ,

which can be viewed as a linear mapping fromR^ltoR^lwith the following properties.

Property 2.1

(a) Lxy= x ◦ y and Lx+y= Lx+ Lyfor any x, y∈ R^l. (b) x∈ K^l⇐⇒ Lx O and x ∈ int(K^l)⇐⇒ Lx O.

(c) Lxis invertible whenever x∈ int(K^l)with the inverse L⁻¹_x given by

L⁻¹_x = 1 det(x)

x₁ −x^T₂

−x2 det(x)

x1 I+^x2_x^x₁^2T



, (11)

where det(x):= x₁²− x2²denotes the determinant of x.

In the following, we recall from [9,11] that each x= (x1, x₂)∈ R × R^l−1admits a spectral factorization, associated withK^l, of the form

x= λ1(x)· u⁽¹⁾_x + λ2(x)· u⁽²⁾_x ,

where λ1(x), λ2(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)= x1+ (−1)ⁱx2, u⁽ⁱ⁾_x =1

2(1, (−1)ⁱ¯x2), i= 1, 2,

with ¯x2= x2/x2 if x2= 0 and otherwise ¯x2being any vector in R^l−1 satisfying

 ¯x2 = 1. If x2= 0, the factorization is unique. The spectral factorizations of x, x² and x^1/2have various interesting properties, and some of them are summarized as follows.

Property 2.2 For any x= (x1, x₂)∈ R × R^l−1, let λ1(x), λ₂(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 ≤ λ1(x)≤ λ2(x)and x∈ int(K^l) ⇐⇒ 0 < λ1(x)≤ λ2(x).

(b) x²= [λ1(x)]²u⁽¹⁾_x + [λ2(x)]²u⁽²⁾_x ∈ K^l for any x∈ R^l. (c) If x∈ K^l, then x^1/2=√

λ₁(x) u⁽¹⁾_x +√

λ₂(x) u⁽²⁾_x ∈ K^l.

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

Given a mapping H: Rⁿ→ R^m, if H is locally Lipschitz continuous, then the set

∂_BH (z):=

V ∈ R^m×n| ∃{z^k} ⊆ DH: z^k→ z, H (z^k)→ 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 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: Rⁿ→ R^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(h²);

H is called (strongly) semismooth if it is (strongly) semismooth everywhere. Here, o(h) means a function α : Rⁿ → R^m satisfying limh→0α(h)/h = 0, while O(h²)denotes a function α: Rⁿ→ R^msatisfyingα(h) ≤ Ch²for allh ≤ 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×mis said to have

(a) the Cartesian P -property if for any 0= x = (x1, . . . , x_q)∈ R^m with xi ∈ Rⁿⁱ, there exists an index ν∈ {1, 2, . . . , q} such that xν, (Mx)_ν > 0;

(b) the Cartesian P0-property if for any 0= x = (x1, . . . , xq)∈ R^m with xi ∈ Rⁿⁱ, there exists an index ν∈ {1, 2, . . . , q} 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, . . . , F_q)with Fi: Rⁿ→ Rⁿⁱ, F is said to (a) have the uniform Cartesian P -property if for any x= (x1, . . . , x_q), y= (y1, . . . ,

y_q)∈ R^m, there is an index ν∈ {1, 2, . . . , q} 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, . . . , x_q), y= (y1, . . . , y_q)∈ R^mand x= y, there exists an index ν ∈ {1, 2, . . . , q} such that

x_ν= yν and xν− yν, F_ν(x)− Fν(y) ≥ 0.

From the above definitions, if a continuously differentiable mapping F: Rⁿ→ Rⁿ has the uniform Cartesian P -property (Cartesian P0-property), then ∇F (x) at any x∈ Rⁿenjoys the Cartesian P -property (Cartesian P0-property). In addition, we may see that, when n1= · · · = nq= 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= (x1, x2), y= (y1, y2)∈ R × R^l−1, we define w, z: R^l× R^l→ R^lby w= (w1, w2)= (w1(x, y), w2(x, y))= w(x, y) := x²+ y²,

z= (z1, z₂)= (z1(x, y), z₂(x, y))= z(x, y) := (x²+ y²)^1/2.

(12)

Clearly, w∈ K^l with w1= x² + y² and w2= 2(x1x₂ + y1y₂). Let ¯w2= w₂/w2 if w2= 0, and otherwise ¯w2be any vector inR^l−1satisfying ¯w2 = 1.

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

√λ2(w)+√ λ1(w)

2 ,

√λ2(w)−√ λ1(w)

2 ¯w2



∈ 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^l. For this purpose, we need several important technical lemmas.

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 x²+ y²∈int(K^l). Moreover,∇xz(x, y)=LxL⁻¹_z and∇yz(x, y)= LyL⁻¹_z , where L⁻¹_z = (1/√

w1)Iif w2= 0, and otherwise L⁻¹_z =

 b c¯w^T₂ c¯w2 aI+ (b − a) ¯w2¯w₂^T



(13) 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)

 .

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= (x1, x2), y= (y1, y2)∈ R × R^l−1 with w= x²+ y²∈ bd(K^l), we have

x²₁= x2², y₁²= y2², x1y1= x2^Ty2, x1y2= y1x2. If, in addition, w2= 0, then w²= 2w₁²= 2w2²= 4(x₁²+ y₁²)= 0 and

x₁¯w2= x2, x₂^T ¯w2= x1, y₁¯w2= y2, y₂^T ¯w2= y1.

Lemma 3.3 For any x = (x1, x₂), y= (y1, y₂)∈ R × R^l−1 with w2= 2(x1x₂+ y₁y₂)= 0, there holds that

x1+ (−1)ⁱx₂^T ¯w2

2

≤ x2+ (−1)ⁱx1¯w2²≤ λi(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 ∂Bφ (x, y)at a general point (x, y)∈ R^l× R^l.

Proposition 3.1 Given a general point (x, y)∈ R^l× R^l, each element in ∂Bφ (x, y) is given by[Vx− I Vy− I] with Vxand Vyhaving the following representation:

(a) If x²+ y²∈ int(K^l), then Vx= L⁻¹_z L_xand Vy= L⁻¹_z L_y. (b) If x²+ y²∈ bd(K^l)and (x, y)= (0, 0), then

V_x∈

 1

2√ 2w1

 1 ¯w₂^T

¯w2 4I− 3 ¯w2¯w^T₂

 L_x+1

2

 1

− ¯w2

 u^T



Vy∈

 1

2√ 2w1

 1 ¯w₂^T

¯w2 4I− 3 ¯w2¯w^T₂

 Ly+1

2

 1

− ¯w2

 v^T

 (14)

for some u= (u1, u2), v= (v1, v2)∈ R × R^l−1satisfying|u1| ≤ u2 ≤ 1 and

|v1| ≤ v2 ≤ 1, where ¯w2= w2/w2.

(c) If (x, y)= (0, 0), then Vx∈ {L_ˆx}, Vy∈ {L_ˆy} for some ˆx, ˆy with  ˆx²+ ˆy²= 1, or

V_x∈

1 2

 1

¯w2

 ξ^T +1

2

 1

− ¯w2

 u^T+ 2

 0 0

(I− ¯w2¯w₂^T)s₂ (I− ¯w2¯w^T₂)s₁



Vy∈

1 2

 1

¯w2

 η^T +1

2

 1

− ¯w2

 v^T + 2

 0 0

(I− ¯w2¯w^T₂)ω2 (I− ¯w2¯w₂^T)ω1



(15) for some u= (u1, u₂), v= (v1, v₂), ξ= (ξ1, ξ₂), η= (η1, η₂)∈ R × R^l−1 such that|u1| ≤ u2 ≤ 1, |v1| ≤ v2 ≤ 1, |ξ1| ≤ ξ2 ≤ 1, |η1| ≤ η2 ≤ 1, ¯w2∈ R^l−1satisfying ¯w2 = 1, and s = (s1, s₂), ω= (ω1, ω₂)∈ R × R^l−1satisfying

s²+ ω²≤ 1/2.

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

φ_x (x, y)= L⁻¹z Lx− I, φ_y (x, y)= L⁻¹z Ly− I ∀(x, y) ∈ Dφ. (a) In this case, φ is continuously differentiable at (x, y) by Lemma3.1. Hence,

∂_Bφ (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 x²+ y²∈ bd(K^l). Then w∈ bd(K^l)and w1>0, which meansw2 = w1>0 and λ2(w) > λ1(w)= 0. Observe that, when w2= 0, the matrix L⁻¹z in (13) can be decomposed as the sum of

L₁(w):= 1 2√

λ1(w)

 1 − ¯w^T₂

− ¯w2 ¯w2¯w^T₂



(16) and

L2(w):= 1 2√

λ₂(w)

 1 ¯w₂^T

¯w2 4√ λ2(w)

√λ2(w)+√

λ1(w)(I− ¯w2¯w^T₂)+ ¯w2¯w₂^T



(17)

with ¯w2= w2/w2. Consequently, φ_x and φ_y can be rewritten as φ_x (x, y)= (L1(w)+ L2(w))L_x− I,

φ_y (x, y)= (L1(w)+ L2(w))L_y− I. (18) Let {(x^k, y^k)} ⊆ Dφ be an arbitrary sequence converging to (x, y). Let w^k = (w₁^k, w₂^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^k₂ = 0 for each k. Let ¯w^k₂= w^k₂/w₂^k for each k. From (18), it follows that

φ_x (x^k, y^k)= (L1(w^k)+ L2(w^k))L_xk− I,

φ_y (x^k, y^k)= (L1(w^k)+ L2(w^k))L_yk− I. (19) Since limk→∞λ₁(w^k)= 0, limk→∞λ₂(w^k)= 2w1>0 and limk→∞ ¯w^k₂= ¯w2, we have

klim→∞L2(w^k)L_xk= C(w)Lx and lim

k→∞L2(w^k)L_yk= C(w)Ly (20) where

C(w)= 1

2√ 2w1

 1 ¯w₂^T

¯w2 4I− 3 ¯w2¯w₂^T



. (21)

Next we focus on the limit of L1(w^k)L_xk and L1(w^k)L_yk as k→ ∞. By computing,

L1(w^k)L_xk =1 2

 u^k₁ u^k₂

−u^k₁¯w₂^k − ¯w^k₂(u^k₂)^T

 ,

L₁(w^k)L_yk =1 2

 v₁^k v₂^k

−v₁^k¯w₂^k − ¯w^k₂(v₂^k)^T

 ,

where

u^k₁=x₁^k− (x₂^k)^T ¯w₂^k

λ1(w^k)

, u^k₂=x₂^k− x₁^k¯w₂^k

λ1(w^k) ,

v^k₁=y₁^k− (y₂^k)^T ¯w^k₂

λ1(w^k)

, v^k₂=y₂^k− y₁^k ¯w^k₂

λ1(w^k) .

By Lemma3.3,|u^k₁| ≤ u^k₂ ≤ 1 and |v₁^k| ≤ v₂^k ≤ 1. So, taking the limit (possibly on a subsequence) on L1(w^k)L_xk and L1(w^k)L_yk, respectively, gives

L₁(w^k)L_xk→1 2

 u₁ u₂

−u1¯w2 − ¯w2u^T₂



=1 2

 1

− ¯w2

 u^T

L₁(w^k)L_yk→1 2

 v1 v2

−v1¯w2 − ¯w2v^T₂



=1 2

 1

− ¯w2

 v^T

(22)

for some u= (u1, u₂), v= (v1, v₂)∈ R × R^l−1satisfying|u1| ≤ u2 ≤ 1 and |v1| ≤

v2 ≤ 1. In fact, u and v are some accumulation point of the sequences {u^k} and {v^k}, respectively. From equations (19)–(22), we immediately obtain

φ_x (x^k, y^k)→ C(w)Lx+1 2

 1

− ¯w2

 u^T − I,

φ_y (x^k, y^k)→ C(w)Ly+1 2

 1

− ¯w2

 v^T − I.

This shows that φ (x^k, y^k)→ [Vx−I Vy−I] as k → ∞ with Vx, V_ysatisfying (14).

(c) Assume that (x, y)= (0, 0). Let {(x^k, y^k)} ⊆ Dφ be an arbitrary sequence converging to (x, y). Let w^k= (w₁^k, w₂^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^k₂= 0 for all k, or w^k₂= 0 for all k.

Case (1): w^k₂= 0 for all k. From Lemma 3.1, it follows that L⁻¹

z^k = (1/ w₁^k)I. Therefore,

φ_x (x^k, y^k)= 1

 w₁^k

L_xk− I and φ_y (x^k, y^k)= 1

 w₁^k

L_yk− I.

Since w^k₁= x^k²+ y^k², 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 φy (x^k, y^k) → L_ˆy− I

for some vectors ˆx, ˆy ∈ R^l satisfying  ˆx²+  ˆy²= 1, where ˆx and ˆy are some accumulation point of the sequences{^xk

w₁^k} and {^yk

w^k₁}, respectively. Thus, we prove that φ (x^k, y^k)→ [Vx− I Vy− I] as k → ∞ with Vx∈ {L_ˆx} and Vy∈ {L_ˆy}.

Case (2): w^k₂= 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^k₂}, we have

L1(w^k)L_xk → 1 2

 1

− ¯w2



u^T, L1(w^k)L_yk → 1 2

 1

− ¯w2



v^T (23)

for some u= (u1, u2), v= (v1, v2)∈ R × R^l−1 satisfying |u1| ≤ u2 ≤ 1 and

|v1| ≤ v2 ≤ 1, and ¯w2∈ R^l−1satisfying  ¯w2 = 1. We next compute the limit of L2(w^k)L_xk and L2(w^k)L_yk as k→ ∞. By the definition of L2(w)in (17),

L₂(w^k)L_xk=1 2

 ξ₁^k (ξ₂^k)^T

ξ₁^k¯w₂^k+ 4(I − ¯w₂^k(¯w^k₂)^T)s₂^k ¯w^k₂(ξ₂^k)^T + 4(I − ¯w₂^k(¯w^k₂)^T)s₁^k

 ,

L₂(w^k)L_yk=1 2

 η^k₁ (η^k₂)^T

η^k₁¯w^k₂+ 4(I − ¯w₂^k(¯w^k₂)^T)ω^k₂ ¯w₂^k(η^k₂)^T + 4(I − ¯w^k₂(¯w₂^k)^T)ω₁^k

 ,

where

ξ₁^k=x₁^k+ (x₂^k)^T ¯w^k₂

λ₂(w^k) , ξ₂^k=x₂^k+ x₁^k ¯w^k₂ λ₂(w^k),

η^k₁=y₁^k+ (y₂^k)^T ¯w^k₂

λ₂(w^k) , η₂^k=y₂^k+ y₁^k ¯w^k₂

λ₂(w^k) ,

(24)

and

s₁^k= x₁^k

λ2(w^k)+ λ1(w^k)

, s₂^k= x₂^k

λ2(w^k)+ λ1(w^k)

,

ω^k₁= y₁^k

λ2(w^k)+ λ1(w^k)

, ω^k₂= y₂^k

λ2(w^k)+ λ1(w^k)

.

(25)

By Lemma3.3,|ξ₁^k| ≤ ξ₂^k ≤ 1 and |η₁^k| ≤ η^k₂ ≤ 1. In addition,

s^k²+ ω^k²= x^k²+ y^k² 2(x^k²+ y^k²)+ 2

λ₁(w^k)

λ₂(w^k)≤1 2.

Hence, taking limit (possibly on a subsequence) on L2(w^k)L_xk and L2(w^k)L_yk

yields

L2(w^k)L_xk→1 2

 ξ₁ ξ₂^T

ξ1¯w2+ 4(I − ¯w2¯w₂^T)s2 ¯w2ξ₂^T + 4(I − ¯w2¯w₂^T)s1



=1 2

 1

¯w2

 ξ^T + 2

 0 0

(I− ¯w2¯w^T₂)s₂ (I− ¯w2¯w₂^T)s₁

 ,

(26) L2(w^k)L_yk→1

2

 η1 η^T₂

η1¯w2+ 4(I − ¯w2¯w₂^T)ω2 ¯w2η^T₂ + 4(I − ¯w2¯w^T₂)ω1



=1 2

 1

¯w2

 η^T + 2

 0 0

(I− ¯w2¯w^T₂)ω₂ (I− ¯w2¯w₂^T)ω₁



for some vectors ξ = (ξ1, ξ₂), η= (η1, η₂)∈ R × R^l−1 satisfying |ξ1| ≤ ξ2 ≤ 1 and |η1| ≤ η2 ≤ 1, ¯w2∈ R^l−1 satisfying  ¯w2 = 1, and s = (s1, s₂), ω= (ω1, ω2)∈ R × R^l−1 satisfying s²+ ω²≤ 1/2. Among others, ξ and η are some accumulation point of the sequences {ξ^k} and {η^k}, respectively; and s and ωare some accumulation point of the sequences{s^k} and {ω^k}, respectively. From (19), (23) and (26), we obtain

φ_x (x^k, y^k)→ 1 2

1

¯w2

 ξ^T +1

2

 1

− ¯w2

 u^T

+ 2

 0 0

(I− ¯w2¯w^T₂)s2 (I− ¯w2¯w^T₂)s1



− I,

φ_y (x^k, y^k)→ 1 2

1

¯w2

 η^T +1

2

 1

− ¯w2

 v^T

+ 2

 0 0

(I− ¯w2¯w^T₂)ω₂ (I− ¯w2¯w^T₂)ω₁



− I.

This implies that as k→ ∞, φ (x^k, y^k)→ [Vx− I Vy− I] with Vx and Vysatis- fying (15).

Combining with Case (1) then yields the desired result.  Remark 3.1 When x²+ y²∈ bd(K^l)with (x, y)= (0, 0), using Lemma3.2, we can also characterize Vxand Vyin Proposition3.1(b) by

V_x∈

 1

√2w1

x1 x₂^T x2 2x1I−^w_w^2x₁^T2

 +1

2

 1

−w2

w2

 u^T



V_y∈



√1 2w1

y1 y₂^T y₂ 2y1I−^w_w^2y₁^T2

 +1

2

 1

−w2

w2

 v^T



for some u= (u1, u2), v= (v1, v2)∈ R × R^l−1satisfying|u1| ≤ u2 ≤ 1 and |v1| ≤

v2 ≤ 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∈ Rⁿ, we let

Fi(ζ )= (Fi1(ζ ), Fi2(ζ )), Gi(ζ )= (Gi1(ζ ), Gi2(ζ ))∈ R × Rⁿⁱ⁻¹, w_i(ζ )= (wi1(ζ ), w_i2(ζ ))= w(Fi(ζ ), G_i(ζ )),

z_i(ζ )= (zi1(ζ ), zi2(ζ ))= z(Fi(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 : Rⁿ→ R^mdefined 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 ∈ Rⁿ,

∂_B(ζ )^T ⊆ ∂B₁(ζ )^T × ∂B₂(ζ )^T × · · · × ∂B_q(ζ )^T (27)