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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.

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1 Introduction

We consider the following conic complementarity problem of finding ζ ∈ IRn 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 IRn to IRn 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 = Kn1 × Kn2 × · · · × Knm, (2) where m, n1, . . . , nm ≥ 1, n1+ n2 + · · · + nm = n, and

Kni :=©

(x1, x2) ∈ IR × IRni−1 | x1 ≥ kx2kª ,

with k · k denoting the Euclidean norm and K1 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 = (F1, . . . , Fm) and G = (G1, . . . , Gm) with Fi, Gi : IRn→ IRni.

An important special case of the SOCCP corresponds to G(ζ) = ζ for all ζ ∈ IRn. 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 = 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 ∈ IRm×n has full row rank, b ∈ IRm and g : IRn → 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 − AT(AAT)−1A)ζ, G(ζ) := ∇g(F (ζ)) − AT(AAT)−1Aζ, (5) where d ∈ IRn 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 AAT, which can be done efficiently via sparse Cholesky factorization.

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

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based on an SOC complementarity function or a smooth merit function induced by it.

Given a mapping φ : IRl× IRl → IRl (l ≥ 1), we call φ an SOC complementarity function associated with the cone Kl if for any (x, y) ∈ IRl× IRl,

φ(x, y) = 0 ⇐⇒ x ∈ Kl, y ∈ Kl, 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 φFB(x, y) := (x2+ y2)1/2− (x + y), (7) where x2 means x ◦ x with “◦” denoting the Jordan product, and x + y denotes the usual componentwise addition of vectors. More specifically, for any x = (x1, x2), y = (y1, y2) ∈ IR × IRl−1, we define their Jordan product associated with Kl as

x ◦ y := (hx, yi, 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 ∈ IRl. It is known that x2 ∈ Kl for all x ∈ IRl. Moreover, if x ∈ Kl, then there exists a unique vector in Kl, denoted by x1/2, such that (x1/2)2 = x1/2◦ x1/2 = x. Thus, φFB in (7) is well-defined for all (x, y) ∈ IRl× IRl. The function φFB was proved in [9] to satisfy the equivalence (6), and its squared norm

ψFB(x, y) := 1

2FB(x, y)k2,

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

Another popular choice of φ is the residual function φNR: IRl× IRl → IRl given by φNR(x, y) := x − [x − y]+,

where [ · ]+ means the minimum Euclidean distance projection onto Kl. 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 φNR also induces a merit function

ψNR(x, y) := 1

2NR(x, y)k2,

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)

(4)

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, 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 complementarity functions, the SOCCP can be reformulated as a nonsmooth system of equations

Φτ(ζ) :=







φτ(F1(ζ), G1(ζ)) ...

φτ(Fi(ζ), Gi(ζ)) ...

φτ(Fm(ζ), Gm(ζ))







= 0, (10)

which induces a natural merit function Ψτ : IRn→ IR+ given by

Ψτ(ζ) = 1

2τ(ζ)k2 = Xm

i=1

ψτ(Fi(ζ), Gi(ζ), (11)

with

ψτ(x, y) = 1

2τ(x, y)k2. (12)

In [6], we studied the continuous differentiability of ψτ and proved that each stationary point of Ψτ is a solution of the SOCCP if ∇F and −∇G are column monotone. This paper focuses on other properties of φτ, 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 Ψτ to be a solution of the SOCCP and establish the boundedness of the level sets of Ψτ, 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 IRn1×· · ·×IRnmis identified with IRn1+···+nm. For a differentiable mapping F : IRn → IRm,

∇F (x) denotes the transpose of the Jacobian F0(x). For a symmetric matrix A ∈ IRn×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 Q1, . . . , Qm, we denote the block diagonal matrix with these matrices as block diagonals by diag(Q1, . . . , Qm) or by diag(Qi, i = 1, . . . , m). If J and B are index sets such that J , B ⊆ {1, 2, . . . , m}, we denote PJ B by the block matrix consisting of the submatrices Pjk ∈ IRnj×nk of P with j ∈ J , k ∈ B, and by xB a vector consisting of subvectors xi ∈ IRni with i ∈ B.

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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 Kl (l ≥ 1).

It is known that Kl is a closed convex self-dual cone with nonempty interior given by int(Kl) := ©

x = (x1, x2) ∈ IR × IRl−1 | x1 > kx2kª and the boundary given by

bd(Kl) :=©

x = (x1, x2) ∈ IR × IRl−1 | x1 = kx2kª .

For each x = (x1, x2) ∈ IR × IRl−1, the determinant and the trace of x are defined by det(x) := x21− kx2k2, tr(x) := 2x1.

In general, det(x ◦ y) 6= det(x) det(y) unless x2 = αy2 for some α ∈ IR. A vector x ∈ IRl is said to be invertible if det(x) 6= 0, and its inverse is denoted by x−1. Given a vector x = (x1, x2) ∈ IR × IRl−1, we often use the following symmetry matrix

Lx:=

· x1 xT2 x2 x1I

¸

, (13)

which can be viewed as a linear mapping from IRl to IRl. It is easy to verify Lxy = x ◦ y and Lx+y = Lx+ Ly for any x, y ∈ IRl. Furthermore, x ∈ Kl if and only if Lx º O, and x ∈ int(Kl) if and only if Lx  O. If x ∈ int(Kl), then Lx is invertible with

L−1x = 1 det(x)

x1 −xT2

−x2 det(x) x1 I + 1

x1x2xT2

 . (14)

We recall from [9] that each x = (x1, x2) ∈ IR × IRl−1 admits a spectral factorization, associated with Kl, 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)ikx2k, u(i)x = 1 2

¡1, (−1)ix¯2¢

(15) with ¯x2 = x2/kx2k if x2 6= 0, and otherwise ¯x2 being any vector in IRl−1 such that k¯x2k = 1. If x2 6= 0, then the factorization is unique. The spectral decomposition of x, x2 and x1/2 has some basic properties as below, whose proofs can be found in [9].

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Property 2.1 For any x = (x1, x2) ∈ IR × IRl−1 with the spectral values λ1(x), λ2(x) and spectral vectors u(1)x , u(2)x given as above, the following results hold:

(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 =p

λ1(x) u(1)x +p

λ2(x) u(2)x ∈ Kl if x ∈ Kl.

(d) det(x) = λ1(x)λ2(x), tr(x) = λ1(x) + λ2(x) and kxk2 = [λ21(x) + λ22(x)]/2.

For the sake of notation, throughout the rest of this paper, we always write 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) ∈ IR × IRl−1, and let ¯w2 = w2/kw2k if w2 6= 0, and otherwise ¯w2 be any vector in IRl−1 satisfying k ¯w2k = 1. We have

w1 = kxk2+ kyk2+ (τ − 2)xTy, w2 = 2(x1x2+ y1y2) + (τ − 2)(x1y2+ y1x2).

Moreover, w ∈ Kl and z ∈ Kl hold by noting that w = x2+ y2+ (τ − 2)(x ◦ y) =

µ

x +τ − 2 2 y

2

+τ (4 − τ ) 4 y2

= µ

y +τ − 2 2 x

2

+τ (4 − τ )

4 x2. (17) In addition, using Property 2.1 (b) and (c), it is not hard to compute that

z =

Ãpλ2(w) +p λ1(w)

2 ,

pλ2(w) −p λ1(w)

2 w¯2

!

∈ Kl. (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(Kl), and furthermore,

xz(x, y) = Lx+τ −2

2 yL−1z , ∇yz(x, y) = Ly+τ −2

2 xL−1z , where

L−1z =





µ b c ¯w2T

c ¯w2 aI + (b − a) ¯w2w¯2T

if w2 6= 0;

³ 1/√

w1

´

I if w2 = 0,

(19)

with a = 2

λ2(w)+

λ1(w), b = 12 µ

1

λ2(w) + 1

λ1(w)

and c = 12 µ

1

λ2(w) 1

λ1(w)

.

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Lemma 2.2 [6, Lemma 3.1] For any x = (x1, x2), y = (y1, y2) ∈ IR × IRl−1, let w = (w1, w2) be given as in (16). If (x − y)2+ τ (x ◦ y) /∈ int(Kl), then

x21 = kx2k2, y12 = ky2k2, x1y1 = xT2y2, x1y2 = y1x2; (20) x21+ y21+ (τ − 2)x1y1 = kx1x2+ y1y2+ (τ − 2)x1y2k

= kx2k2+ ky2k2+ (τ − 2)xT2y2. (21) If, in addition, (x, y) 6= (0, 0), then w2 6= 0, and moreover,

xT2w¯2 = x1, x1w¯2 = x2, y2Tw¯2 = y1, y1w¯2 = y2. (22)

Lemma 2.3 [6, Lemma 3.2] For any x = (x1, x2), y = (y1, y2) ∈ IR × IRl−1, let w = (w1, w2) be defined as in (16). If w2 6= 0, then for i = 1, 2,

x1+ τ − 2 2 y1

+ (−1)i µ

x2+ τ − 2 2 y2

T

¯ w2

#2

°°

°° µ

x2+ τ − 2 2 y2

+ (−1)i µ

x1+ τ − 2 2 y1

¯ w2

°°

°°

2

≤ λi(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 : IRn→ IRm, if H is locally Lipschitz continuous, the following set

BH(z) :=©

V ∈ IRm×n| ∃{zk} ⊆ DH : zk→ z, H0(zk) → Vª

is nonempty and is called the B-subdifferential of H at z, where DH ⊆ IRn denotes 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 [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 ∈ IRn×n, which are in fact special cases of those introduced by Chen and Qi [3] for a linear transformation.

Definition 2.1 A matrix M ∈ IRn×n is said to have

(a) the Cartesian P -property if for any 0 6= x = (x1, . . . , xm) ∈ IRn with xi ∈ IRni, there exists an index ν ∈ {1, 2, . . . , m} such that hxν, (Mx)νi > 0;

(b) the Cartesian P0-property if for any 0 6= x = (x1, . . . , xm) ∈ IRn with xi ∈ IRni, there exists an index ν ∈ {1, 2, . . . , m} such that xν 6= 0 and hxν, (Mx)νi ≥ 0.

Some nonlinear generalizations of these concepts in the setting of K are defined as follows.

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Definition 2.2 Given a mapping F = (F1, . . . , Fm) with Fi : IRn → IRni, F is said to (a) have the uniform Cartesian P -property if for any x = (x1, . . . , xm), y = (y1, . . . , ym) ∈

IRn, there is an index ν ∈ {1, 2, . . . , m} and a constant ρ > 0 such that hxν − yν, Fν(x) − Fν(y)i ≥ ρkx − yk2;

(b) have the Cartesian P0-property if for any x = (x1, . . . , xm), y = (y1, . . . , ym) ∈ IRn and x 6= y, there exists an index ν ∈ {1, 2, . . . , m} such that

xν 6= yν and hxν − yν, Fν(x) − Fν(y)i ≥ 0.

3 Properties of the functions φ

τ

and Φ

τ

First, we study the favorable properties of φτ, including the globally Lipschitz continuity, the strong semismoothness and the characterization of the B-subdifferential at any point.

Proposition 3.1 The function φτ defined as in (9) has the following properties.

(a) φτ is (continuously) differentiable at (x, y) if and only if w(x, y) ∈ int(Kl). Also,

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) ∈ IRl× IRl.

(d) The squared norm of φτ, i.e. ψτ, is continuously differentiable everywhere.

Proof. (a) The proof directly follows from Lemma 2.1 and the following fact that φτ(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 = (x1, x2), y = (y1, y2) ∈ IR × IRl−1. Then, applying Lemma A.1 in the appendix and the Mean-Value Theorem, we have

°°

°z(x, y) − z(a, b)

°°

° =

°°

°° lim²→0+z(x, y, ²) − limˆ

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

xz(a + t(x − a), y, ²)(x − a)dtˆ

°°

°°

+ lim

²→0+

°°

°° Z 1

0

yz(a, b + t(y − b), ²)(y − b)dtˆ

°°

°°

2Ck(x, y) − (a, b)k

(9)

for any (x, y), (a, b) ∈ IRl× IRl, 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 Ã

x +τ − 2 2 y,

pτ (4 − τ )

2 y

! +1

2

³

τ − 4 +p

τ (4 − τ )

´ y.

Notice that φFBis strongly semismooth by [23, Corollary 3.3], and the functions x+τ −22 y,

1 2

pτ (4 − τ )y and 12(τ − 4 +p

τ (4 − τ ))y are also strongly semismooth. Therefore, φτ 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 ∂Bφτ(x, y) at a general point (x, y). The discussion of B-subdifferential for φFB was given in [19], and we here generalize it to φτ 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 × IRl−1, each element in ∂Bφτ(x, y) is of the form V = [Vx− I Vy− I] with Vx and Vy having the following representation:

(a) If (x − y)2 + τ (x ◦ y) ∈ int(Kl), then Vx = L−1z Lx+τ −2

2 y and Vy = L−1z Ly+τ −2

2 x. (b) If (x − y)2+ τ (x ◦ y) ∈ bd(Kl) and (x, y) 6= (0, 0), then

Vx

½ 1

2 2w1

µ 1 w¯T2

¯

w2 4I − 3 ¯w2w¯T2

¶ µ

Lx+τ − 2 2 Ly

¶ +1

2

µ 1

− ¯w2

uT

¾

Vy

½ 1

2 2w1

µ 1 w¯2T

¯

w2 4I − 3 ¯w2w¯T2

¶ µ

Ly+ τ − 2 2 Lx

¶ +1

2

µ 1

− ¯w2

vT

¾ (25) for some u = (u1, u2), v = (v1, v2) ∈ IR × IRl−1 satisfying |u1| ≤ ku2k ≤ 1 and

|v1| ≤ kv2k ≤ 1, where ¯w2 = kww22k.

(c) If (x, y) = (0, 0), then Vx ∈ {Luˆ}, Vy ∈ {Lvˆ} for some ˆu = (ˆu1, ˆu2), ˆv = (ˆv1, ˆv2) ∈ IR × IRl−1 satisfying kˆuk, kˆvk ≤ 1 and ˆu1ˆv2+ ˆv1uˆ2 = 0, or

Vx

½1 2

µ 1

¯ w2

ξT +1 2

µ 1

− ¯w2

uT + 2

µ 0 0

(I − ¯w2w¯T2)s2 (I − ¯w2w¯2T)s1

¶¾

Vy

½1 2

µ 1

¯ w2

ηT + 1 2

µ 1

− ¯w2

vT + 2

µ 0 0

(I − ¯w2w¯2T2 (I − ¯w2w¯T21

¶¾ (26)

(10)

for some u = (u1, u2), v = (v1, v2), ξ = (ξ1, ξ2), η = (η1, η2) ∈ IR × IRl−1 satisfying

|u1| ≤ ku2k ≤ 1, |v1| ≤ kv2k ≤ 1, |ξ1| ≤ kξ2k ≤ 1 and |η1| ≤ kη2k ≤ 1,

¯

w2 ∈ IRl−1 satisfying k ¯w2k = 1, and s = (s1, s2), ω = (ω1, ω2) ∈ IR × IRl−1 such that ksk2+ kωk2 ≤ 1.

In what follows, we investigate the properties of the operator Φτ : IRn → IRn given by (10). We start with the semismoothness of Φτ. Since Φτ 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).

Proposition 3.3 The operator Φτ : IRn→ IRn given by (10) is semismooth. Moreover, it is strongly semismooth if F0 and G0 are locally Lipschitz continuous.

To characterize the B-subdifferential of Φτ, we write Fi(ζ) = (Fi1(ζ), Fi2(ζ)) and Gi(ζ) = (Gi1(ζ), Gi2(ζ)), and denote wi and zi for i = 1, 2, . . . , m by

wi = (wi1(ζ), wi2(ζ)) = w(Fi(ζ), Gi(ζ)), zi = (zi1(ζ), zi2(ζ)) = z(Fi(ζ), Gi(ζ)). (27) For convenience, we sometimes suppress in Fi(ζ) and Gi(ζ) the dependence on ζ.

Proposition 3.4 Let Φτ : IRn → IRn be defined as in (10). Then, for any ζ ∈ IRn,

BΦτ(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−1zi and Bi(ζ) = LGi+τ −2

2 FiL−1zi .

(b) If (Fi(ζ), Gi(ζ)) 6= (0, 0) and (Fi(ζ) − Gi(ζ))2+ τ (Fi(ζ) ◦ Gi(ζ)) ∈ bd(Kni), then Ai(ζ) ∈

½ 1

2 2wi1

µ

LFi +τ − 2 2 LGi

¶ µ 1 w¯i2T

¯

wi2 4I − 3 ¯wi2w¯Ti2

¶ +1

2ui(1, − ¯wTi2)

¾

Bi(ζ) ∈

½ 1

2 2wi1

µ

LGi+ τ − 2 2 LFi

¶ µ 1 w¯Ti2

¯

wi2 4I − 3 ¯wi2w¯i2T

¶ + 1

2vi(1, − ¯wTi2)

¾

for some ui = (ui1, ui2), vi = (vi1, vi2) ∈ IR × IRni−1 satisfying |ui1| ≤ kui2k ≤ 1 and |vi1| ≤ kvi2k ≤ 1, where ¯wi2= kwwi2i2k.

(11)

(c) If (Fi(ζ), Gi(ζ)) = (0, 0), then Ai(ζ) ∈

n Luˆi

o

½1 2ξi¡

1, ¯wTi2¢ + 1

2ui¡

1, − ¯wTi2¢ +

µ 0 2sTi2(I − ¯wi2w¯i2T) 0 2si1(I − ¯wi2w¯i2T)

¶¾

Bi(ζ) ∈ n

Lˆvi o

½1 2ηi¡

1, ¯wi2T¢ +1

2vi¡

1, − ¯wTi2¢ +

µ 0 2ωi2T(I − ¯wi2w¯i2T) 0 2ωi1(I − ¯wi2w¯i2T)

¶¾

for some ˆui = (ˆui1, ˆui2), ˆvi = (ˆvi1, ˆvi2) ∈ IR × IRni−1 satisfying kˆuik, kˆvik ≤ 1 and ˆui1vˆi2 + ˆvi1uˆi2 = 0, some ui = (ui1, ui2), vi = (vi1, vi2), ξi = (ξi1, ξi2), ηi = i1, ηi2) ∈ IR × IRni−1 with |ui1| ≤ kui2k ≤ 1, |vi1| ≤ kvi2k ≤ 1, |ξi1| ≤ kξi2k ≤ 1 and |ηi1| ≤ kηi2k ≤ 1, ¯wi2 ∈ IRni−1 satisfying k ¯wi2k = 1, and si = (si1, si2), ωi = i1, ωi2) ∈ IR × IRni−1 such that ksik2+ kωik2 ≤ 1.

Proof. Let Φτ,i(ζ) denote the ith subvector of Φτ, i.e. Φτ,i(ζ) = φτ(Fi(ζ), Gi(ζ)) for all i = 1, 2, . . . , m. From Proposition 2.6.2 of [5], it follows that

BΦτ(ζ)T ⊆ ∂BΦτ,1(ζ)T × ∂BΦτ,2(ζ)T × · · · × ∂BΦτ,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 ∂BΦτ,i(ζ)T. Using the definition of B-subdifferential and the continuous differentiability of F and G, it is not difficult to verify that

BΦτ,i(ζ)T = [∇Fi(ζ) ∇Gi(ζ)]∂Bφτ(Fi(ζ), Gi(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 ∈ IRn, 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(Ai(ζ) − I)Φτ,i(ζ), (Bi(ζ) − I)Φτ,i(ζ)i ≥ 0,

and the equality holds if and only if Φτ,i(ζ) = 0. Particularly, for the index i such that (Fi(ζ) − Gi(ζ))2+ τ (Fi(ζ) ◦ Gi(ζ) ∈ int(Kni), we have

h(Ai(ζ) − I)υi, (Bi(ζ) − I)υii ≥ 0, for any υi ∈ IRni. Proof. From Theorem 2.6.6 of [5] and Proposition 3.1 (d), we have

∇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)).

(12)

In addition, from Propositions 3.2 and 3.4, 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)Φτ,i(ζ)

yψτ(Fi(ζ), Gi(ζ)) = (Bi(ζ) − I)Φτ,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 (Fi(ζ) − Gi(ζ))2+ τ (Fi(ζ) ◦ Gi(ζ) ∈ int(Kni) and any υi ∈ IRni,

h(Ai(ζ) − I)υi, (Bi(ζ) − I)υii

= D³

LFi+τ −2

2 Gi− Lzi

´ L−1zi υi,

³

LGi+τ −2

2 Fi− Lzi

´ L−1zi υi

E

= D³

LGi+τ −2

2 Fi− Lzi´³

LFi+τ −2

2 Gi − Lzi´

L−1zi υi, L−1zi υiE

. (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 ∂BΦτ(ζ) at a solution ζ of the SOCCP are nonsingular if ζ satisfies strict complementarity, i.e.,

Fi) + Gi) ∈ int(Kni) for all i = 1, 2, . . . , m. (33) First, we give a technical lemma which states that the multivalued matrix (Ai) − I) + (Bi) − 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 Fi) + Gi) ∈ int(Kni), we have that Φτ,i(ζ) is continuously differentiable at ζ and (Ai) − I) + (Bi) − I) is nonsingular.

Proof. Since ζ is a solution of the SOCCP, we have for all i = 1, 2, . . . , m Fi) ∈ Kni, Gi) ∈ Kni, hFi), Gi)i = 0.

It is not hard to verify that Fi) + Gi) ∈ int(Kni) if and only if one of the three cases shown as below holds.

參考文獻

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