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

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 ∈ 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 × · · · × K^nm, (2) where m, n₁, . . . , n_m ≥ 1, n₁+ n₂ + · · · + n_m = 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 ∈ IR^m×n has full row rank, b ∈ IR^m 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 − A^T(AA^T)⁻¹A)ζ, G(ζ) := ∇g(F (ζ)) − A^T(AA^T)⁻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 AA^T, 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

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) := (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 × IR^l−1, we define their Jordan product associated with K^l 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 ∈ IR^l. It is known that x² ∈ K^l for all x ∈ IR^l. Moreover, if x ∈ K^l, then there exists a unique vector in K^l, denoted by x^1/2, such that (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^l. The function φ_FB was proved in [9] to satisfy the equivalence (6), and its squared norm

ψ_FB(x, y) := 1

2kφ_FB(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 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

2kφ_NR(x, y)k²,

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 φτ(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 φNR 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(F₁(ζ), G₁(ζ)) ...

φ_τ(F_i(ζ), G_i(ζ)) ...

φτ(Fm(ζ), Gm(ζ))











= 0, (10)

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

Ψ_τ(ζ) = 1

2kΦ_τ(ζ)k² = Xm

i=1

ψ_τ(F_i(ζ), G_i(ζ), (11)

with

ψτ(x, y) = 1

2kφτ(x, y)k². (12)

In [6], we studied the continuous differentiability of ψ_τ 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 φ_τ, 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 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×· · ·×IR^nmis identified with IR^n1+···+nm. For a differentiable mapping F : IRⁿ → IR^m,

∇F (x) denotes the transpose of the Jacobian F⁰(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 Q₁, . . . , Q_m, we denote the block diagonal matrix with these matrices as block diagonals by diag(Q₁, . . . , Q_m) or by diag(Q_i, 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 ∈ IR^nj×nk of P with j ∈ J , k ∈ B, and by x_B a vector consisting of subvectors x_i ∈ IRⁿⁱ 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 = (x₁, x₂) ∈ IR × IR^l−1 | x₁ > kx₂kª and the boundary given by

bd(K^l) :=©

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

For each x = (x₁, x₂) ∈ IR × IR^l−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 ∈ IR^l 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 × IR^l−1, we often use the following symmetry matrix

L_x:=

· x₁ x^T₂ x₂ x₁I

¸

, (13)

which can be viewed as a linear mapping from IR^l to IR^l. It is easy to verify L_xy = 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  O. If x ∈ int(K^l), then L_x is invertible with

L⁻¹_x = 1 det(x)



 x1 −x^T₂

−x₂ det(x) x₁ I + 1

x₁x₂x^T₂



 . (14)

We recall from [9] that each x = (x₁, x₂) ∈ IR × IR^l−1 admits a spectral factorization, associated with K^l, of the form

x = λ₁(x) · u⁽¹⁾_x + λ₂(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) = 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 IR^l−1 such that k¯x2k = 1. If x2 6= 0, then the factorization is unique. The spectral decomposition of x, x² and x^1/2 has some basic properties as below, whose proofs can be found in [9].

Property 2.1 For any x = (x1, x2) ∈ IR × IR^l−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 ∈ K^l if and only if λ₁(x) ≥ 0, and x ∈ int(K^l) if and only if λ₁(x) > 0.

(b) x² = λ²₁(x)u⁽¹⁾x + λ²₂(x)u⁽²⁾x ∈ K^l; (c) x^1/2 =p

λ₁(x) u⁽¹⁾x +p

λ₂(x) u⁽²⁾x ∈ K^l if x ∈ K^l.

(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 × IR^l−1, and let ¯w₂ = w₂/kw₂k if w₂ 6= 0, and otherwise ¯w₂ be any vector in IR^l−1 satisfying k ¯w₂k = 1. We have

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

Moreover, w ∈ K^l and z ∈ K^l 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

!

∈ 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,

∇_xz(x, y) = L_x+^{τ −2}

2 yL⁻¹_z , ∇_yz(x, y) = L_y+^{τ −2}

2 xL⁻¹_z , where

L⁻¹_z =







µ b c ¯w₂^T

c ¯w₂ aI + (b − a) ¯w₂w¯₂^T

¶

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)

¶ .

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

x²₁ = kx₂k², y₁² = ky₂k², x₁y₁ = x^T₂y₂, x₁y₂ = y₁x₂; (20) x²₁+ y²₁+ (τ − 2)x₁y₁ = kx₁x₂+ y₁y₂+ (τ − 2)x₁y₂k

= kx₂k²+ ky₂k²+ (τ − 2)x^T₂y₂. (21) If, in addition, (x, y) 6= (0, 0), then w2 6= 0, and moreover,

x^T₂w¯₂ = x₁, x₁w¯₂ = x₂, y₂^Tw¯₂ = y₁, y₁w¯₂ = y₂. (22)

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

"µ

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

≤ 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 : IRⁿ→ IR^m, if H is locally Lipschitz continuous, the following set

∂BH(z) :=©

V ∈ IR^m×n| ∃{z^k} ⊆ DH : z^k→ z, H⁰(z^k) → Vª

is nonempty and is called the B-subdifferential of H at z, where D_H ⊆ 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 ∈ IR^n×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 ∈ IR^n×n is said to have

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

(b) the Cartesian P₀-property if for any 0 6= x = (x₁, . . . , x_m) ∈ IRⁿ with x_i ∈ IRⁿⁱ, there exists an index ν ∈ {1, 2, . . . , m} such that x_ν 6= 0 and hx_ν, (Mx)_νi ≥ 0.

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

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, . . . , x_m), y = (y₁, . . . , y_m) ∈

IRⁿ, 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 − yk²;

(b) have the Cartesian P₀-property if for any x = (x₁, . . . , x_m), y = (y₁, . . . , y_m) ∈ IRⁿ 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

and Φ

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,

∇_xφ_τ(x, y) = L_x+^{τ −2}

2 yL⁻¹_z − I, ∇_yφ_τ(x, y) = L_y+^{τ −2}

2 xL⁻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 = (x₁, x₂), y = (y₁, y₂) ∈ 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) − 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

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,

pτ (4 − τ )

2 y

! +1

2

³

τ − 4 +p

τ (4 − τ )

´ y.

Notice that φ_FBis strongly semismooth by [23, Corollary 3.3], and the functions x+^{τ −2}₂ y,

1 2

pτ (4 − τ )y and ¹₂(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 ∂_Bφ_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 ∂_Bφ_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 Vx = L⁻¹_z L_x+^{τ −2}

2 y and Vy = 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√ 2w₁

µ 1 w¯^T₂

¯

w2 4I − 3 ¯w2w¯^T₂

¶ µ

L_x+τ − 2 2 L_y

¶ +1

2

µ 1

− ¯w2

¶ u^T

¾

Vy ∈

½ 1

2√ 2w1

µ 1 w¯₂^T

¯

w₂ 4I − 3 ¯w₂w¯^T₂

¶ µ

Ly+ τ − 2 2 Lx

¶ +1

2

µ 1

− ¯w₂

¶ v^T

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

|v₁| ≤ kv₂k ≤ 1, where ¯w₂ = _kw^w2_2k.

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

Vx ∈

½1 2

µ 1

¯ w₂

¶

ξ^T +1 2

µ 1

− ¯w₂

¶

u^T + 2

µ 0 0

(I − ¯w₂w¯^T₂)s₂ (I − ¯w₂w¯₂^T)s₁

¶¾

V_y ∈

½1 2

µ 1

¯ w₂

¶

η^T + 1 2

µ 1

− ¯w₂

¶

v^T + 2

µ 0 0

(I − ¯w₂w¯₂^T)ω₂ (I − ¯w₂w¯^T₂)ω₁

¶¾ (26)

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

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

¯

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

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

Proposition 3.3 The operator Φ_τ : 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 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

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 F_i(ζ) and G_i(ζ) the dependence on ζ.

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

∂_BΦ_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 Ai(ζ) and Bi(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(Kⁿⁱ), then A_i(ζ) = L_Fi+^{τ −2}

2 GiL⁻¹_zi and B_i(ζ) = L_Gi+^{τ −2}

2 FiL⁻¹_zi .

(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¯_i2^T

¯

w_i2 4I − 3 ¯w_i2w¯^T_i2

¶ +1

2ui(1, − ¯w^T_i2)

¾

B_i(ζ) ∈

½ 1

2√ 2w_i1

µ

L_Gi+ τ − 2 2 L_Fi

¶ µ 1 w¯^T_i2

¯

wi2 4I − 3 ¯wi2w¯_i2^T

¶ + 1

2v_i(1, − ¯w^T_i2)

¾

for some u_i = (u_i1, u_i2), v_i = (v_i1, v_i2) ∈ IR × IRⁿⁱ⁻¹ satisfying |u_i1| ≤ ku_i2k ≤ 1 and |v_i1| ≤ kv_i2k ≤ 1, where ¯w_i2= _kw^wi2_i2k.

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

n L_uˆi

o

∪

½1 2ξ_i¡

1, ¯w^T_i2¢ + 1

2u_i¡

1, − ¯w^T_i2¢ +

µ 0 2s^T_i2(I − ¯w_i2w¯_i2^T) 0 2s_i1(I − ¯w_i2w¯_i2^T)

¶¾

B_i(ζ) ∈ n

L_ˆvi o

∪

½1 2η_i¡

1, ¯w_i2^T¢ +1

2v_i¡

1, − ¯w^T_i2¢ +

µ 0 2ω_i2^T(I − ¯w_i2w¯_i2^T) 0 2ωi1(I − ¯wi2w¯_i2^T)

¶¾

for some ˆu_i = (ˆu_i1, ˆu_i2), ˆv_i = (ˆv_i1, ˆv_i2) ∈ IR × IRⁿⁱ⁻¹ satisfying kˆu_ik, kˆv_ik ≤ 1 and ˆu_i1vˆ_i2 + ˆv_i1uˆ_i2 = 0, some u_i = (u_i1, u_i2), v_i = (v_i1, v_i2), ξ_i = (ξ_i1, ξ_i2), η_i = (ηi1, ηi2) ∈ IR × IRⁿⁱ⁻¹ with |ui1| ≤ kui2k ≤ 1, |vi1| ≤ kvi2k ≤ 1, |ξi1| ≤ kξi2k ≤ 1 and |η_i1| ≤ kη_i2k ≤ 1, ¯w_i2 ∈ IRⁿⁱ⁻¹ satisfying k ¯w_i2k = 1, and s_i = (s_i1, s_i2), ω_i = (ω_i1, ω_i2) ∈ IR × IRⁿⁱ⁻¹ such that ks_ik²+ kω_ik² ≤ 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

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

∂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(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 (Fi(ζ) − Gi(ζ))²+ τ (Fi(ζ) ◦ Gi(ζ) ∈ int(Kⁿⁱ), we have

h(A_i(ζ) − I)υ_i, (B_i(ζ) − I)υ_ii ≥ 0, for any υ_i ∈ IRⁿⁱ. Proof. From Theorem 2.6.6 of [5] and Proposition 3.1 (d), we have

∇ψ_τ(x, y) = ∂_Bφ_τ(x, y)^Tφ_τ(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 Propositions 3.2 and 3.4, it is not hard to see that [A_i(ζ)^T − I B_i(ζ)^T − I] ∈ ∂_Bφ_τ(F_i(ζ), G_i(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)

∇_yψ_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(Kⁿⁱ) and any υ_i ∈ IRⁿⁱ,

h(A_i(ζ) − I)υ_i, (B_i(ζ) − I)υ_ii

= D³

L_Fi+^{τ −2}

2 Gi− L_zi

´ L⁻¹_zi υ_i,

³

L_Gi+^{τ −2}

2 Fi− L_zi

´ L⁻¹_zi υ_i

E

= D³

L_Gi+^{τ −2}

2 Fi− L_zi´³

L_Fi+^{τ −2}

2 Gi − L_zi´

L⁻¹_zi υ_i, L⁻¹_zi υ_iE

. (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 ∂_BΦ_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(Kⁿⁱ) 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 Fi(ζ^∗) + Gi(ζ^∗) ∈ int(Kⁿⁱ), we have that Φτ,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^∗) ∈ Kⁿⁱ, G_i(ζ^∗) ∈ Kⁿⁱ, hF_i(ζ^∗), G_i(ζ^∗)i = 0.

It is not hard to verify that F_i(ζ^∗) + G_i(ζ^∗) ∈ int(Kⁿⁱ) if and only if one of the three cases shown as below holds.