3 Differentiability of the function ψ

Journal of Mathematical Analysis and Applications, vol. 355, pp. 195-215, 2009

A one-parametric class of merit functions for the symmetric cone complementarity problem

Shaohua Pan¹

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China E-mail: shhpan@scut.edu.cn

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

Taipei, Taiwan 11677 E-mail: jschen@math.ntnu.edu.tw

June 23, 2008

Abstract. In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [18] to the SCCP. By exploiting the Cartesian P -properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R₀₂-property. All of these results generalize some recent important works in [4, 25, 28] under a unified framework.

1The author’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.

2Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.

Key words. Symmetric cone complementarity problem, merit function, Jordan algebra, smoothness, Lipschitz continuity, Cartesian P -properties.

1 Introduction

Given a Euclidean Jordan algebra A = (V, ◦, h·, ·i) where V is a finite-dimensional vector space over the real field R endowed with the inner product h·, ·i and “◦” denotes the Jordan product. Let K be a symmetric cone in V and G, F : V → V be nonlinear trans- formations assumed to be continuously differentiable throughout this paper. Consider the symmetric cone complementarity problem (SCCP) of finding ζ ∈ V such that

G(ζ) ∈ K, F (ζ) ∈ K, hG(ζ), F (ζ)i = 0. (1) The model provides a simple, natural and unified framework for various existing comple- mentarity problems such as the nonnegative orthant nonlinear complementarity problem (NCP), the second-order cone complementarity problem (SOCCP), and the semidefinite complementarity problem (SDCP). In addition, the model itself is closely related to the KKT optimality conditions for the convex symmetric cone program (CSCP):

minimize g(x)

subject to hai, xi = bi, i = 1, 2, . . . , m, x ∈ K,

(2)

where a_i ∈ V, b_i ∈ R for i = 1, 2, . . . , m, and g : V → R is a convex twice continu- ously differentiable function. Therefore, the SCCP has wide applications in engineering, economics, management science and other fields; see [1, 11, 20, 29] and references therein.

During the past several years, interior-point methods have been well used for solving the symmetric cone linear programming problem (SCLP), i.e., the CSCP with g being a linear function (see [7, 8, 23, 24]). However, in view of the wide applications of the SCCP, it is worthwhile to explore other solution methods for the more general CSCP and SCCP. Recently, motivated by the successful applications of the merit function approach in the solution of NCPs, SOCCPs and SDCPs (see, e.g., [4, 10, 22, 28]), some researchers started with the investigation of merit functions or complementarity functions associated with symmetric cones. For example, Liu, Zhang and Wang [21] extended a class of merit functions proposed in [18] to the following special SCCP:

ζ ∈ K, F (ζ) ∈ K, hζ, F (ζ)i = 0; (3)

Kong, Tuncel and Xiu [17] studied the extension of the implicit Lagrangian function proposed by Mangasarian and Solodov [22] to symmetric cones; and Kong, Sun and Xiu [16] proposed a regularized smoothing method for the SCCP (3) based on the natural

residual complementarity function associated with symmetric cones. Following this line, in this paper we will consider the extension of the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] and a class of regularized functions based on it.

We define the one-parametric class of vector-valued functions φ_τ : V × V → V by φ_τ(x, y) :=¡

x² + y²+ (τ − 2)(x ◦ y)¢_1/2

− (x + y), (4)

where τ ∈ (0, 4) is an arbitrary but fixed parameter, x² = x ◦ x, x^1/2 is a vector such that (x^1/2)² = x, and x + y means the usual componentwise addition of vectors. When τ = 2, φ_τ reduces to the vector-valued Fischer-Burmeister function given by

φ_FB(x, y) := (x²+ y²)^1/2− (x + y); (5) whereas as τ → 0 it will become a multiple of the vector-valued residual function

ψ_NR(x, y) := x − (x − y)+

where (·)₊ denotes the metric projection on K. In this sense, the one-parametric class of vector-valued functions covers the two popular complementarity functions associated with the symmetric cone K. In fact, from Lemma 3.1 later, it follows that the function φ_τ with any τ ∈ (0, 4) is a complementarity function associated with K, that is,

φ_τ(x, y) = 0 ⇐⇒ x ∈ K, y ∈ K, hx, yi = 0.

Consequently, its squared norm yields a merit function associated with K ψ_τ(x, y) := 1

2kφ_τ(x, y)k², (6)

where k · k is the norm induced by h·, ·i, and the SCCP can be reformulated as

minζ∈V f_τ(ζ) := ψ_τ(G(ζ), F (ζ)). (7)

To apply the effective unconstrained optimization methods, such as the quasi-Newton method, the trust-region method and the conjugate gradient method, for solving the un- constrained minimization reformulation (7) of the SCCP, the smoothness of the merit function ψ_τ and the Lipschitz continuity of its gradient will play an important role. In Section 3 and Section 4, we show that the function ψτ defined by (6) is continuously differentiable everywhere and has a globally Lipschitz continuous gradient with the Lip- schitz constant being a positive multiple of 1 + τ⁻¹. These results generalize some recent important works in [4, 25, 28] under a unified framework, as well as improve the work [21] greatly in which only the differentiability of the merit function ψ_FB was given.

In addition, we also consider a class of regularized functions for f_τ defined as

fbτ(ζ) := ψ0(G(ζ) ◦ F (ζ)) + ψτ(G(ζ), F (ζ)), (8) where ψ₀ : V → R₊ is continuously differentiable and satisfies

ψ₀(u) = 0 ∀u ∈ −K and ψ₀(u) ≥ βk(u)₊k ∀u ∈ V (9) for some constant β > 0. Using the properties of ψ0 in (9), it is not hard to verify that fb_τ is a merit function for the SCCP. The class of functions will reduce to the one studied in [21] if τ = 2 and G degenerates into an identity transformation. In Section 5, we show that the class of merit functions can provide a global error bound for the solution of the SCCP under the condition that G and F have the joint uniform Cartesian P -property.

In Section 6, we establish the boundedness of the level sets of bf_τ under a weaker con- dition than the one used by [21], which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with G and F having the joint Cartesian R₀₂-property.

Throughout this paper, I denotes an identity operator, k · k represents the norm induced by the inner product h·, ·i, and int(K) denotes the interior of the symmetric cone K. All vectors are column ones and write the column vector (x^T₁, . . . , x^T_m)^T as (x1, . . . , xm), where xi is a column vector from the subspace Vi. For any x ∈ V, we denote (x)₊ and (x)₋ by the metric projection of x onto K and −K, respectively, i.e., (x)₊ := argmin_y∈K{kx − yk}. For any symmetric matrix A, the notation A º O means that A is positive semidefinite. For a differentiable mapping F : V → V, the notation

∇F (x) denotes the transposed Jacobian operator of F at a point x. We write x = o(α) (respectively, x = O(α)) if kxk/|α| → 0 (respectively, uniformly bounded) as α → 0.

2 Preliminaries

In this section, we recall some concepts and materials of Euclidean Jordan algebras that will be used in the subsequent sections. More detailed expositions of Euclidean Jordan algebras can be found in the monograph by Faraut and Kor´anyi [9]. Besides, one can find excellent summaries in the articles [2, 12, 24, 26].

A Euclidean Jordan algebra is a triple (V, ◦, h·, ·i_V), where V is a finite-dimensional inner product space over the real field R and (x, y) 7→ x ◦ y : V × V → V is a bilinear mapping satisfying the following conditions:

(i) x ◦ y = y ◦ x for all x, y ∈ V,

(ii) x ◦ (x²◦ y) = x²◦ (x ◦ y) for all x, y ∈ V, where x² := x ◦ x, and (iii) hx ◦ y, zi_V = hx, y ◦ zi_V for all x, y, z ∈ V.

We call x ◦ y the Jordan product of x and y. We also assume that there is an element e ∈ V, called the unit element, such that x ◦ e = x for all x ∈ V. For x ∈ V, let ζ(x) be the degree of the minimal polynomial of x, which can be equivalently defined as

ζ(x) := min©

k : {e, x, x², . . . , x^k} are linearly dependentª .

Since ζ(x) ≤ dim(V) where dim(V) denotes the dimension of V, the rank of V is well defined by r := max{ζ(x) : x ∈ V}. In a Euclidean Jordan algebra A = (V, ◦, h·, ·i_V), we define the set of squares as K := {x² : x ∈ V}. Then, by Theorem III. 2.1 of [9], K is a symmetric cone. This means that K is a self-dual closed convex cone with nonempty interior int(K) and for any two elements x, y ∈ int(K), there exists an invertible linear transformation T : V → V such that T (K) = K and T (x) = y.

A Euclidean Jordan algebra is said to be simple if it is not the direct sum of two Euclidean Jordan algebras. By Proposition III. 4.4 of [9], each Euclidean Jordan algebra is, in a unique way, a direct sum of simple Euclidean Jordan algebras. A common simple Euclidean Jordan algebra is (Sⁿ, ◦, h·, ·i_Sⁿ), where Sⁿ is the space of n × n real symmetric matrices with the inner product hX, Y i_Sⁿ := Tr(XY ), and the Jordan product is defined by X ◦ Y := (XY + Y X)/2. Here, XY is the usual matrix multiplication of X and Y and Tr(X) is the trace of X. The associate cone K is the set of all positive semidefinite matrices. Another one is the Lorentz algebra (Rⁿ, ◦, h·, ·i_Rⁿ), where Rⁿ is the Euclidean space of dimension n with the standard inner product hx, yi_Rⁿ = x^Ty, and the Jordan product is defined by x ◦ y := (hx, yi_Rⁿ, x₁y₂+ y₁x₂) for any x = (x₁, x₂), y = (y₁, y₂) ∈ R × Rⁿ⁻¹. The associate cone, called the Lorentz cone or the second-order cone, is

K :=©

x = (x₁, x₂) ∈ R × Rⁿ⁻¹: kx₂k ≤ x₁ª .

Recall that an element c ∈ V is said to be idempotent if c² = c. Two idempotents c and d are said to be orthogonal if c ◦ d = 0. One says that {c1, c2, . . . , ck} is a complete system of orthogonal idempotents if

c²_j = c_j, c_j◦ c_i = 0 if j 6= i, j, i = 1, 2, . . . , k, and P_k

j=1c_j = e.

A nonzero idempotent is said to be primitive if it cannot be written as the sum of two other nonzero idempotents. We call a complete system of orthogonal primitive idempotents a Jordan frame. Then, we have the following spectral decomposition theorem.

Theorem 2.1 [9, Theorem III. 1.2] Suppose that A = (V, ◦, h·, ·i_V) is a Euclidean Jordan algebra and the rank of A is r. Then for any x ∈ V, there exist a Jordan frame {c₁, c₂, . . . , c_r} and real numbers λ₁(x), λ₂(x), . . . , λ_r(x), arranged in the decreasing order λ1(x) ≥ λ2(x) ≥ · · · ≥ λr(x), such that x =P_r

j=1λj(x)cj.

The numbers λ_j(x) (counting multiplicities), which are uniquely determined by x, are called the eigenvalue, and we write the maximum eigenvalue and the minimum eigenvalue of x as λ_max(x) and λ_min(x), respectively. The trace of x, denoted as tr(x), is defined by tr(x) :=P_r

j=1λ_j(x); whereas the determinant of x is defined by det(x) :=Q_r

j=1λ_j(x).

By Proposition III. 1.5 of [9], a Jordan algebra over R with a unit element e ∈ V is Euclidean if and only if the symmetric bilinear form tr(x ◦ y) is positive definite. Hence, we may define an inner product h·, ·i on V by

hx, yi := tr(x ◦ y), ∀ x, y ∈ V. (10)

Unless otherwise states, the inner product h·, ·i appearing in this paper always means the one defined by (10). By the associativity of tr(·) (see [9, Proposition II. 4.3]), the inner product h·, ·i is associative, i.e., hx, y ◦ zi = hy, x ◦ zi for all x, y, z ∈ V. Let

L(x)y := x ◦ y for every y ∈ V.

Then, the linear operator L(x) for each x ∈ V is symmetric with respect to the inner product h·, ·i in the sense that hL(x)y, zi = hy, L(x)zi for any y, z ∈ V. Let k · k be the norm on V induced by the inner product h·, ·i, namely,

kxk := p

hx, xi = ³P_r

j=1λ²_j(x)

´_1/2

, ∀ x ∈ V.

It is not difficult to verify that for any x, y ∈ V, there always holds that hx, yi ≤ 1

2(kxk²+ kyk²) and kx + yk² ≤ 2(kxk²+ kyk²). (11) Let ϕ : R → R be a scalar valued function. Then, it is natural to define a vector- valued function associated with the Euclidean Jordan algebra A = (V, ◦, h·, ·i) by

ϕ_V(x) := ϕ(λ₁(x))c₁+ ϕ(λ₂(x))c₂+ · · · + ϕ(λ_r(x))c_r, (12) where x ∈ V has the spectral decomposition x = P_r

j=1λ_j(x)c_j. The function ϕ_V is also called the L¨owner operator [26]. When ϕ(t) is chosen as max{0, t} and min{0, t} for t ∈ R, respectively, ϕ_V becomes the metric projection operator onto K and −K:

(x)+ :=

Xr j=1

max{0, λj(x)}cj and (x)− :=

Xr j=1

min{0, λj(x)}cj. (13)

Lemma 2.1 [26, Theorem 13] For any x = P_r

j=1λj(x)cj, let ϕ_V be given as in (12).

Then ϕ_V is (continuously) differentiable at x if and only if ϕ is (continuously) differen- tiable at each λ_j(x), j = 1, 2, . . . , r. The derivative of ϕ_V at x, for any h ∈ V, is

ϕ⁰_V(x)h = Xr

j=1

[ϕ^[1](λ(x))]jjhcj, hicj+ X

1≤j<l≤r

4[ϕ^[1](λ(x))]jlcj◦ (cl◦ h)

where

[ϕ^[1](λ(x))]_ij :=





ϕ(λ_i(x)) − ϕ(λ_j(x))

λ_i(x) − λ_j(x) if λ_i(x) 6= λ_j(x) ϕ⁰(λ_i(x)) if λ_i(x) = λ_j(x)

, i, j = 1, 2, . . . , r.

In fact, the Jacobian ϕ⁰_V(·) is a linear and symmetric operator, which can be written as ϕ⁰_V(x) =

Xr j=1

ϕ⁰(λ_j(x))Q(c_j) + 2 Xr i,j=1,i6=j

[ϕ^[1](λ(x))]_ijL(c_j)L(c_i) (14)

where Q(x) := 2L²(x) −L(x²) for any x ∈ V is called the quadratic representation of V.

In the sequel, unless otherwise stated, we assume that A = (V, ◦, h·, ·i) is a simple Euclidean Jordan algebra of rank r and dim(V) = n.

An important part in the theory of Euclidean Jordan algebras is the Peirce decompo- sition. Let c be a nonzero idempotent in A. Then, by [9, Proposition III. 1.3], c satisfies 2L³(c) − 3L²(c) + L(c) = 0 and the distinct eigenvalues of the symmetric operator L(c) are 0,1

2 and 1. Let V(c, 1), V(c,¹₂) and V(c, 0) be the three corresponding eigenspaces, i.e.,

V(c, α) :=

n

x ∈ V : L(c)x = αx o

, α = 1, 1 2, 0.

Then V is the orthogonal direct sum of V(c, 1), V(c,¹₂) and V(c, 0). The decomposition V = V(c, 1) ⊕ V(c,1

2) ⊕ V(c, 0)

is called the Peirce decomposition of V with respect to the nonzero idempotent c.

Let {c1, c2, . . . , cr} be a Jordan frame of A. For i, j ∈ {1, . . . , r}, define the eigenspaces Vii := V(ci, 1) = Rci,

V_ij := V(c_i,1

2) ∩ V(c_j,1

2), i 6= j.

Then, from [9, Theorem IV. 2.1], it follows that the following conclusion holds.

Theorem 2.2 The space V is the orthogonal direct sum of subspaces Vij (1 ≤ i ≤ j ≤ r), i.e., V = ⊕_i≤jV_ij. Furthermore,

V_ij ◦ V_ij ⊂ V_ii+ V_jj, Vij ◦ Vjk ⊂ Vik, if i 6= k,

V_ij ◦ V_kl = {0}, if {i, j} ∩ {k, l} = ∅.

Let x ∈ V have the spectral decomposition x =P_r

j=1λ_j(x)c_j. For i, j ∈ {1, 2, . . . , r}, let Cij(x) be the orthogonal projection operator onto Vij. Then,

C_ij(x) = C_ij^∗(x), C_ij²(x) = C_ij(x), C_ij(x)C_kl(x) = 0 if {i, j} 6= {k, l}, i, j, k, l = 1, . . . , r (15)

and P

1≤i≤j≤rC_ij(x) = I, (16)

where C_ij^∗ is the adjoint (operator) of Cij. In addition, by [9, Theorem IV. 2.1], C_jj(x) = Q(c_j) and C_ij(x) = 4L(c_i)L(c_j) = 4L(c_j)L(c_i) = C_ji(x), i, j = 1, 2, . . . , r.

Note that the original notation in [9] for orthogonal projection operator is P_ij. However, to avoid confusion with another orthogonal projector P_i(c_j) onto V(c, α) and orthogonal matrix P which will be used later (Sections 3–4), we adopt C_ij instead.

With the orthogonal projection operators {C_ij(x) : i, j = 1, 2, . . . , r}, we have the following spectral decomposition theorem for L(x) and L(x²); see [15, Chapters VI–V].

Lemma 2.2 Let x ∈ V have the spectral decomposition x = P_r

j=1λ_j(x)c_j. Then the symmetric operator L(x) has the spectral decomposition

L(x) = Xr j=1

λ_j(x)C_jj(x) + X

1≤j<l≤r

1

2(λ_j(x) + λ_l(x)) C_jl(x)

with the spectrum σ(L(x)) consisting of all distinct numbers in {¹₂(λj(x) + λl(x)) : j, l = 1, 2, . . . , r}, and L(x²) has the spectral decomposition

L(x²) = Xr

j=1

λ²_j(x)C_jj(x) + X

1≤j<l≤r

1 2

¡λ²_j(x) + λ²_l(x)¢

C_jl(x) (17)

with the spectrum σ(L(x²)) consisting of all distinct numbers in {¹₂¡

λ²_j(x) + λ²_l(x)¢ : j, l = 1, 2, . . . , r}.

Proposition 2.1 For any x ∈ V, the operator L(x²) − L²(x) is positive semidefinite.

Proof. By Lemma 2.2 and (15), we can verify that L²(x) has the spectral decomposition L²(x) =

Xr j=1

λ²_j(x)Cjj(x) + X

1≤j<l≤r

1

4(λj(x) + λl(x))²Cjl(x). (18) This means that the operator L(x²) − L²(x) has the spectral decomposition

L(x²) − L²(x) = X

1≤j<l≤r

·1 2

¡λ²_j(x) + λ²_l(x)¢

− 1

4(λj(x) + λl(x))²

¸

Cjl(x).

Noting that the orthogonal projection operator is positive semidefinite on V and λ²_j(x) + λ²_l(x)

2 ≥ (λ_j(x) + λ_l(x))²

4 for all j, l = 1, 2, . . . , r,

we readily obtain the conclusion from the spectral decomposition of L(x²) − L²(x). 2

3 Differentiability of the function ψ

In this section, we show that ψ_τ is a merit function associated with K, and moreover, it is differentiable everywhere on V × V. By the definition of Jordan product,

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 ∈ K (19)

for any x, y ∈ V, and consequently the function φ_τ in (4) is well defined. The following lemma states that φ_τ and ψ_τ is respectively a complementarity function and a merit function associated with K.

Lemma 3.1 For any x, y ∈ V, let φ_τ and ψ_τ be given by (4) and (6), respectively. Then, ψ_τ(x, y) = 0 ⇐⇒ φ_τ(x, y) = 0 ⇐⇒ x ∈ K, y ∈ K, hx, yi = 0,

Proof. The first equivalence is clear by the definition of ψ_τ, and we only need to prove the second equivalence. Suppose that φτ(x, y) = 0. Then,

£x²+ y²+ (τ − 2)(x ◦ y)¤_1/2

= (x + y). (20)

Squaring the two sides of (20) yields that

x² + y²+ (τ − 2)(x ◦ y) = x² + y²+ 2(x ◦ y),

which implies x ◦ y = 0 since τ ∈ (0, 4). Substituting x ◦ y = 0 into (20), we have that x =¡

x²+ y²¢_1/2

− y and y = ¡

x²+ y²¢_1/2

− x.

Since x²+ y² ∈ K, x² ∈ K and y² ∈ K, from [12, Proposition 8] or [19, Corollary 9] it follows that x, y ∈ K. Consequently, the necessity holds. For the other direction, suppose x, y ∈ K and x ◦ y = 0. Then, (x + y)² = x²+ y². This, together with x ◦ y = 0, implies

that £

x²+ y² + (τ − 2)(x ◦ y)¤_1/2

− (x + y) = 0.

Consequently, the sufficiency follows. The proof is thus completed. 2

In what follows, we concentrate on the differentiability of the merit function ψ_τ. For this purpose, we need the following two crucial technical lemmas.

Lemma 3.2 For any x, y ∈ V, let u(x, y) := (x²+ y²)^1/2. Then, the function u(x, y) is continuously differentiable at any point (x, y) such that x²+ y² ∈ int(K). Furthermore,

∇_xu(x, y) = L(x)L⁻¹(u(x, y)) and ∇_yu(x, y) = L(y)L⁻¹(u(x, y)). (21) Proof. The first part is due to Lemma 2.1. It remains to derive the formulas in (21).

From the definition of u(x, y), it follows that

u²(x, y) = x²+ y², ∀ x, y ∈ V. (22) By the formula (14), it is easy to verify that ∇_x(x²) = 2L(x). Differentiating on both sides of (22) with respect to x then yields that

2∇_xu(x, y)L(u(x, y)) = 2L(x).

This implies that ∇_xu(x, y) = L(x)L⁻¹(u(x, y)) since, by u(x, y) ∈ int(K), L(u(x, y)) is positive definite on V. Similarly, we have that ∇yu(x, y) = L(y)L⁻¹(u(x, y)). 2

To present another lemma, we first introduce some related notations. For any 0 6=

z ∈ K and z /∈ int(K), suppose that z has the spectral decomposition z =P_r

j=1λ_j(z)c_j, where {c1, c2, . . . , cr} is a Jordan frame and λ1(z), . . . , λr(z) are the eigenvalues arranged in the decreasing order λ₁(z) ≥ λ₂(z) ≥ · · · ≥ λ_r(z) = 0. Define the index

j^∗ := min n

j | λj(z) = 0, j = 1, 2, . . . , r o

(23) and let

cJ := P_j^∗₋₁

l=1 cl. (24)

Clearly, j^∗ and cJ are well-defined since 0 6= z ∈ K and z /∈ int(K). Since cJ is an idem- potent and c_J 6= 0 (otherwise z = 0), V can be decomposed as the orthogonal direct sum of the subspaces V(c_J, 1), V(c_J,¹₂) and V(c_J, 0). In the sequel, we write P₁(c_J), P¹

2(c_J) and P0(cJ) as the orthogonal projection onto V(cJ, 1), V(cJ,¹₂) and V(cJ, 0), respectively.

From [21], we know that L(z) is positive definite on V(c_J, 1) and is a one-to-one mapping from V(c_J, 1) to V(c_J, 1). This means that L(z) an inverse L⁻¹(z) on V(c_J, 1), i.e., for any u ∈ V(c_J, 1), L⁻¹(z)u is the unique v ∈ V(c_J, 1) such that z ◦ v = u.

Lemma 3.3 For any x, y ∈ V, let z : V × V → V be the mapping defined as z = z(x, y) := £

x²+ y²+ (τ − 2)(x ◦ y)¤_1/2

. (25)

If (x, y) 6= (0, 0) such that z(x, y) /∈ int(K), then the following results hold:

(a) The vectors x, y, x + y, x + ^{τ −2}₂ y and y + ^{τ −2}₂ x belong to the subspace V(c_J, 1).

(b) For any h ∈ V such that z²(x, y)+h ∈ K, let w = w(x, y) := [z²(x, y)+h]^1/2−z(x, y).

Then, P₁(c_J)w = ¹₂L⁻¹(z(x, y))[P₁(c_J)h] + o(khk).

Proof. From (19) and the definition of z, it is clear that z(x, y) ∈ K for all x, y ∈ V.

Hence, using the similar arguments as Lemma 11 of [21] yields the desired result. 2

Now by Lemmas 3.2–3.3, we prove the differentiability of the merit function ψ_τ. Proposition 3.1 The function ψ_τ defined by (6) is differentiable everywhere on V × V.

Furthermore, ∇_xψ_τ(0, 0) = ∇_yψ_τ(0, 0) = 0, and if (x, y) 6= (0, 0), then

∇xψτ(x, y) =

· L

³

x +τ − 2 2 y

´

L⁻¹(z(x, y)) − I

¸

φτ(x, y),

∇yψτ(x, y) =

· L

³

y +τ − 2 2 x

´

L⁻¹(z(x, y)) − I

¸

φτ(x, y) (26)

where z(x, y) is given by (25).

Proof. We prove the conclusion by the following three cases.

Case (1): (x, y) = (0, 0). For any u, v ∈ V, suppose that u²+ v²+ (τ − 2)(u ◦ v) has the spectrum decomposition u²+ v² + (τ − 2)(u ◦ v) = P_r

j=1µ_jd_j, where {d₁, d₂, . . . , d_r} is the corresponding Jordan frame. Then, for j = 1, 2, . . . , r, we have

µ_j = 1 kd_jk²

DP_r

j=1µ_jd_j, d_jE

= ­

u²+ v²+ (τ − 2)(u ◦ v), d_j®

=

*µ

u + τ − 2 2 v

¶₂

+ τ (4 − τ ) 4 v², d_j

+

≤

*µ

u + τ − 2 2 v

¶₂

+ τ (4 − τ ) 4 v², e

+

= kuk²+ (τ − 2)hu, vi + kvk²

≤ (τ /2)(kuk²+ kvk²), (27)

where the second equality is by kd_jk = 1, the first inequality is due to e =P_r

j=1d_j and dj ∈ K for j = 1, 2, . . . , r, and the last inequality is due to (11). Therefore,

ψτ(u, v) − ψτ(0, 0) = 1 2

°°

°[u²+ v²+ (τ − 2)(u ◦ v)]^1/2− (u + v)

°°

°²

= 1 2

°°

°P_r

j=1

√µj dj− (u + v)

°°

°²

≤

°°

°P_r

j=1

õj dj

°°

°²+ ku + vk²

≤ Xr

j=1

µjkdjk²+ 2(kuk²+ kvk²)

≤ µ1

2τ r + 2

¶¡

kuk²+ kvk²¢ ,

where the first two inequalities are due to (11), and the last one is from (27). This shows that ψ_τ is differentiable at (0, 0) with ∇_xψ_τ(0, 0) = ∇_yψ_τ(0, 0) = 0.

Case (2): z(x, y) ∈ int(K). Since φ_τ(x, y) = z(x, y) − (x + y), we have from Lemma 2.1 that φτ is continuously differentiable under this case. Notice that

ψ_τ(x, y) = 1 2

­e, φ²_τ(x, y)® ,

and hence the function ψ_τ is continuously differentiable. Applying the chain rule yields

∇_xψ_τ(x, y) = ∇_xφ_τ(x, y)L(φ_τ(x, y))e = ∇_xφ_τ(x, y)φ_τ(x, y). (28) On the other hand, from (19) it follows that

φ_τ(x, y) =

·³

x +τ − 2 2 y

´₂

+ τ (4 − τ ) 4 y²

¸_1/2

− (x + y), and therefore using the formulas in (21) gives that

∇_xφ_τ(x, y) = L

³

x +τ − 2 2 y

´

L⁻¹(z(x, y)) − I.

This, together with (28), immediately yields that

∇_xψ_τ(x, y) =

· L

µ

x + τ − 2 2 y

¶

L⁻¹(z(x, y)) − I

¸

φ_τ(x, y).

For symmetry of x and y in ψ_τ(x, y), we also have that

∇_yψ_τ(x, y) =

· L

µ

y + τ − 2 2 x

¶

L⁻¹(z(x, y)) − I

¸

φ_τ(x, y).

Case (3): (x, y) 6= (0, 0) and z(x, y) /∈ int(K). For any u, v ∈ V, define ˆ

z := 2ˆx ◦ u + 2ˆy ◦ v + u²+ v²+ (τ − 2)u ◦ v with ˆx = x + ^{τ −2}₂ y and ˆy = y + ^{τ −2}₂ x. It is not difficult to verify that

z²(x, y) + ˆz = µ

(x + u) + τ − 2

2 (y + v)

¶₂

+ τ (4 − τ )

4 (y + v)²

= z²(x + u, y + v) ∈ K.

Let

w(x, y) := ¡

z²(x, y) + ˆz¢_1/2

− z(x, y).

From the definitions of ψτ and z(x, y), it then follows that ψ_τ(x + u, y + v) − ψ_τ(x, y)

= 1 2

h°°[z²(x, y) + ˆz]^1/2− (x + u + y + v)°°²− kz(x, y) − (x + y)k² i

= 1 2

£hˆz, ei + ku + vk²¤

− hw(x, y), x + u + y + vi + hx + y − z(x, y), u + vi

= −hw(x, y), x + yi + hx + y − z(x, y), u + vi + hˆx, ui + hˆy, vi + o(k(u, v)k). (29) By Lemma 3.3 (a), x + y ∈ V(c_J, 1). Thus, using Lemma 3.3 (b), we have that

hw(x, y), x + yi = hP₁(c_J)w(x, y), x + yi

=

¿1

2L⁻¹(z(x, y))[P1(cJ)ˆz] + o(kˆzk), x + y À

= 1

2

­P₁(c_J)ˆz, L⁻¹(z(x, y))[x + y]®

+ o(kˆzk)

= D

P₁(c_J) [ˆx ◦ u + ˆy ◦ v] , L⁻¹(z(x, y))[x + y]

E

+ o(k(u, v)k)

= D

ˆ

x ◦ u + ˆy ◦ v, P1(cJ)[L⁻¹(z(x, y))(x + y)]

E

+ o(k(u, v)k)

= D

ˆ

x ◦ u + ˆy ◦ v, L⁻¹(z(x, y))(x + y) E

+ o(k(u, v)k)

= ­£

L⁻¹(z(x, y))(x + y)¤

◦ ˆx, u® +­£

L⁻¹(z(x, y))(x + y)¤

◦ ˆy, v®

+ o(k(u, v)k) (30)

where the first equality is since V = V(c_J, 1) ⊕ V(c_J,¹₂) ⊕ V(c_J, 0), the fifth one is due to P₁(c_J) = P₁^∗(c_J), and the sixth is from the fact that L⁻¹(z(x, y))(x + y) ∈ V(c_J, 1).

Combining (29) with (30), we obtain that ψ_τ(x + u, y + v) − ψ_τ(x, y)

= D

ˆ

x + x + y − z(x, y) −£

L⁻¹(z(x, y))(x + y)¤

◦ ˆx, u E

+ D

ˆ

y + x + y − z(x, y) −£

L⁻¹(z(x, y))(x + y)¤

◦ ˆy, v E

+ o(k(u, v)k).

This implies that the function ψ_τ is differentiable at (x, y), and furthermore,

∇_xψ_τ(x, y) = ˆx + x + y − z(x, y) −£

L⁻¹(z(x, y))(x + y)¤

◦ ˆx,

∇_yψ_τ(x, y) = ˆy + x + y − z(x, y) −£

L⁻¹(z(x, y))(x + y)¤

◦ ˆy.

Notice that ˆ

x + x + y − z(x, y) −£

L⁻¹(z(x, y))(x + y)¤

◦ ˆx

= ˆx − φ_τ(x, y) −£

L⁻¹(z(x, y))(x + y)¤

◦ µ

x +τ − 2 2 y

¶

= x + τ − 2

2 y − φ_τ(x, y) − L µ

x +τ − 2 2 y

¶ h

L⁻¹(z(x, y))(x + y) i

= L µ

x + τ − 2 2 y

¶

L⁻¹(z(x, y))[z(x, y) − x − y] − φ_τ(x, y)

=

· L

³

x + τ − 2 2 y

´

L⁻¹(z(x, y)) − I

¸

φ_τ(x, y),

where the third equality is due to L⁻¹(z(x, y))z(x, y) = e and the fact that x + τ − 2

2 y = L µ

x + τ − 2 2 y

¶ e = L

µ

x +τ − 2 2 y

¶

L⁻¹(z(x, y))z(x, y).

Therefore,

∇_xψ_τ(x, y) =

· L

³

x + τ − 2 2 y

´

L⁻¹(z(x, y)) − I

¸

φ_τ(x, y).

Similarly, we also have that

∇_yψ_τ(x, y) =

· L

µ

y + τ − 2 2 x

¶

L⁻¹(z(x, y)) − I

¸

φ_τ(x, y).

This shows that the conclusion holds under this case. The proof is thus completed. 2 It should be pointed out that the formula (26) is well-defined even if z(x, y) /∈ int(K) since in this case φ_τ(x, y) ∈ V(c_J, 1) by Lemma 3.3 (a). When A is specified as the Lorentz algebra (Rⁿ, ◦, h·, ·iRⁿ), the formula reduces to the one of [3, Proposition 3.2];

whereas when A is specified as (Sⁿ, ◦, h·, ·i_Sⁿ) and τ = 2, the formula is same as the one in [28, Lemma 6.3 (b)] by noting that z(x, y) = (x²+ y²)^1/2 and

∇_xψ_τ(x, y) = L(x)L⁻¹(z(x, y))φ_FB(x, y) − φ_FB(x, y)

= x ◦ [L⁻¹(z(x, y))φ_FB(x, y)] − L(z(x, y))L⁻¹(z(x, y))φ_FB(x, y)

= x ◦ [L⁻¹(z(x, y))φ_FB(x, y)] − z(x, y) ◦ [L⁻¹(z(x, y))φ_FB(x, y)]

= [L⁻¹(z(x, y))φ_FB(x, y)] ◦ (x − z(x, y)).

Thus, the formula (26) provides a unified framework for the SOCCP and the SDCP cases.

From Proposition 3.1, we readily obtain the following properties of ∇ψ_τ, which have been given in the setting of NCP [14] and the SOCCP [3], respectively.

Proposition 3.2 Let ψ_τ be given as in (6). Then, for any (x, y) ∈ V × V, we have (a) hx, ∇xψτ(x, y)i + hy, ∇yψτ(x, y)i = kφτ(x, y)k².

(b) ∇ψ_τ(x, y) = 0 if and only if x ∈ K, y ∈ K, hx, yi = 0.

Proof. (a) If (x, y) = (0, 0), the result is clear. Otherwise, from (26) it follows that hx, ∇xψτ(x, y)i + hy, ∇yψτ(x, y)i

=

¿ x,

µ

x +τ − 2 2 y

¶

◦ [L⁻¹(z(x, y))φτ(x, y)]

À

− hx, φτ(x, y)i +

¿ y,

µ

y +τ − 2 2 x

¶

◦ [L⁻¹(z(x, y))φτ(x, y)]

À

− hy, φτ(x, y)i

=

¿ x ◦

µ

x +τ − 2 2 y

¶

, L⁻¹(z(x, y))φτ(x, y) À

− hx, φτ(x, y)i +

¿ y ◦

µ

y +τ − 2 2 x

¶

, L⁻¹(z(x, y))φτ(x, y) À

− hy, φτ(x, y)i

= hz²(x, y), L⁻¹(z(x, y))φ_τ(x, y)i − hx + y, φ_τ(x, y)i

= hz(x, y), φ_τ(x, y)i − hx + y, φ_τ(x, y)i = kφ_τ(x, y)k²,

where the next to last equality is by z² = L(z)z and the symmetry of L(z).

(b) The proof is direct by part (a), Lemma 3.1 and Proposition 3.1. 2

4 Lipschitz continuity of ∇ψ

In this section, we investigate the continuity of the gradients ∇_xψ_τ(x, y) and ∇_yψ_τ(x, y).

To this end, for any ² > 0, we define the mapping z_²: V × V → V by z_² = z_²(x, y) := ¡

x²+ y²+ (τ − 2)(x ◦ y) + ²e¢_1/2

. (31)

From (19), clearly, z_²(x, y) ∈ int(K) for any x, y ∈ V, and hence the operator L(z_²(x, y)) is positive definite on V. Since the spectral function induced by ϕ(t) =√

t (t ≥ 0) is con- tinuous by Lemma 2.1, it follows that z_²(x, y) → z(x, y) as ² → 0⁺ for any (x, y) ∈ V × V, where z(x, y) is given by (25). This means that L(z_²(x, y)) → L(z(x, y)) as ² → 0⁺.

In what follows, we prove that the gradients ∇_xψ_τ(x, y) and ∇_yψ_τ(x, y) are Lipschitz continuous by arguing the Lipschitz continuity of z²(x, y) and the mapping

H_²(x, y) := L

³

x +τ − 2 2 y

´

L⁻¹(z_²(x, y))(x + y). (32) To establish the Lipschitz continuity of z²(x, y), we need the following crucial lemma.

Lemma 4.1 For any (x, y) ∈ V × V and ² > 0, let z_²(x, y) be defined as in (31). Then the function z_²(x, y) is continuously differentiable everywhere with

∇_xz_²(x, y) = L

³

x +τ − 2 2 y

´

L⁻¹(z_²(x, y)),

∇_yz_²(x, y) = L

³

y + τ − 2 2 x

´

L⁻¹(z_²(x, y)). (33) Furthermore, there exists a constant C > 0, independent of x, y and ², τ , such that

k∇_xz_²(x, y)k ≤ C and k∇_yz_²(x, y)k ≤ C.

Proof. The first part follows from Lemma 3.2 and the following fact that z_²(x, y) =

·³

x +τ − 2 2 y

´₂

+τ (4 − τ )

4 y²+ ²e

¸_1/2

=

·³

y +τ − 2 2 x´₂

+τ (4 − τ )

4 x²+ ²e

¸_1/2

. (34)

We next prove that the operator ∇_xz_²(x, y) is bounded for any x, y ∈ V and ² > 0. Let {u1, u2, . . . , un} be an orthonormal basis of V. For any x, y ∈ V, let L(z²), L(x + ^{τ −2}₂ y), L(z_²) and L((x + ^{τ −2}₂ y)²) be the corresponding matrix representation of the operators L(z²), L(x + ^{τ −2}₂ y), L(z_²) and L((x + ^{τ −2}₂ y)²) with respect to the basis {u₁, u₂, . . . , u_n}.

Then, by the formula (33), it suffices to prove that the matrix L(x + ^{τ −2}₂ y)L⁻¹(z_²) is bounded for any x, y ∈ V and ² > 0. The verifications are given as below.

Suppose that z = z(x, y) has the spectral decomposition z = P_r

j=1λj(z)cj, where λ₁(z) ≥ λ₂(z) ≥ · · · ≥ λ_r(z) ≥ 0 are the eigenvalue of z and {c₁, c₂, . . . , c_r} is the corresponding Jordan frame. From Lemma 2.2, L(z) has the spectral decomposition

L(z) = Xr

j=1

λ_j(z)C_jj(z) + X

1≤j<l≤r

1

2(λ_j(z) + λ_l(z)) C_jl(z) (35) with the spectrum σ(L(z)) consisting of all distinct numbers in {¹₂(λ_j(z) + λ_l(z)) : j, l = 1, 2, . . . , r}, and L(z²) has the spectral decomposition

L(z²) = Xr

j=1

λ²_j(z)Cjj(z) + X

1≤j<l≤r

1 2

¡λ²_j(z) + λ²_l(z)¢

Cjl(z) (36)

with σ(L(z²)) consisting of all distinct numbers in {¹₂¡

λ²_j(z) + λ²_l(z)¢

: j, l = 1, 2, . . . , r}.

By the definition of z_²(x, y), it is easy to verify that z_² = P_r

j=1

q

λ²_j(z) + ² c_j, and consequently the symmetric operator L(z_²) has the spectral decomposition

L(z_²) = Xr

j=1

q

λ²_j(z) + ² C_jj(z) + X

1≤j<l≤r

1 2

³q

λ²_j(z) + ² + q

λ²_l(z) + ²

´

C_jl(z) (37)

with the spectrum σ(L(z_²)) consisting of all distinct numbers in

½1 2

µq

λ²_j(z) + ² + q

λ²_l(z) + ²

¶

: j, l = 1, 2, . . . , r

¾ .

We first prove that the matrix L(x + ^{τ −2}₂ y) (L(z²) + ²I)^−1/2 is bounded for any x, y ∈ V and ² > 0. For this purpose, let P be an n × n orthogonal matrix such that

P L(z²)P^T = diag¡

λ₁(L(z²)), λ₂(L(z²)), · · · , λ_n(L(z²))¢

(38) where λ₁(L(z²)) ≥ λ₂(L(z²)) · · · ≥ λ_n(L(z²)) ≥ 0 are the eigenvalues of L(z²). Then, it is not hard to verify that for any ² > 0,

P ¡

L(z²) + ²I¢_−1/2

P^T = diag Ã

p 1

λ₁(L(z²)) + ², · · · , 1

pλ_n(L(z²)) + ²

! .

Write eU := P L¡

x +^{τ −2}₂ y¢

P^T. We can compute that L

³

x +τ − 2 2 y

´³

L(z²) + ²I

´_−1/2

= P^TUdiage Ã

p 1

λ₁(L(z²)) + ², · · · , 1

pλ_n(L(z²)) + ²

! P

= P^T

"

Ueik

pλk(L(z²)) + ²

#

1≤i≤n 1≤k≤n

P. (39)

Since L(z²) = L¡

(x + ^{τ −2}₂ y)²¢ + L³

τ (4−τ ) 4 y²´

and L(y²) is positive semidefinite, we get

L(z²) − L µ

(x + τ − 2 2 y)²

¶ º O,

In addition, by Proposition 2.1 L[(x + ^{τ −2}₂ y)²] − L(x + ^{τ −2}₂ y)L(x + ^{τ −2}₂ y) is positive semidefinite, and hence we have that

L õ

x + τ − 2 2 y

¶₂!

− L µ

x + τ − 2 2 y

¶ L

µ

x +τ − 2 2 y

¶ º O.