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 Pan^{1}

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 R*_{02}-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 ha**i**, xi = b**i**, 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) :=*¡

*x*^{2} *+ y*^{2}*+ (τ − 2)(x ◦ y)*¢_{1/2}

*− (x + y),* (4)

*where τ ∈ (0, 4) is an arbitrary but fixed parameter, x*^{2} *= 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*

*reduces to the vector-valued Fischer-Burmeister function given by*

_{τ}*φ*_{FB}*(x, y) := (x*^{2}*+ y*^{2})^{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

2*kφ*_{τ}*(x, y)k*^{2}*,* (6)

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

min*ζ∈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 ψ*

*τ*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*

*. These results generalize some recent important works in [4, 25, 28] under a unified framework, as well as improve the work*

^{−1}*[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

*f*b*τ**(ζ) := ψ*0*(G(ζ) ◦ F (ζ)) + ψ**τ**(G(ζ), F (ζ)),* (8)
*where ψ*_{0} *: V → R*_{+} is continuously differentiable and satisfies

*ψ*_{0}*(u) = 0 ∀u ∈ −K and ψ*_{0}*(u) ≥ βk(u)*_{+}*k ∀u ∈ V* (9)
*for some constant β > 0. Using the properties of ψ*0 in (9), it is not hard to verify that
*f*b* _{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 b*f** _{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 R*

_{02}-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*^{T}_{1}*, . . . , x*^{T}* _{m}*)

*as*

^{T}*(x*1

*, . . . , x*

*m*

*), where x*

*i*is a column vector from the subspace V

*i*

*. 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 ◦ (x*^{2}*◦ y) = x*^{2}*◦ (x ◦ y) for all x, y ∈ V, where x*^{2} *:= 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, x*^{2}*, . . . , x*^{k}*} are linearly dependent*ª
*.*

*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 := {x*^{2} *: 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 (S^{n}*, ◦, h·, ·i*_{S}* ^{n}*), where S

^{n}*is the space of n × n real symmetric*

*matrices with the inner product hX, Y i*

_{S}

^{n}*:= 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 (R

^{n}*, ◦, h·, ·i*

_{R}

*), where R*

^{n}*is the Euclidean*

^{n}*space of dimension n with the standard inner product hx, yi*

_{R}

^{n}*= x*

^{T}*y, and the Jordan*

*product is defined by x ◦ y := (hx, yi*

_{R}

^{n}*, x*

_{1}

*y*

_{2}

*+ y*

_{1}

*x*

_{2}

*) for any x = (x*

_{1}

*, x*

_{2}

*), y = (y*

_{1}

*, y*

_{2}

*) ∈*

*R × R*

*. The associate cone, called the Lorentz cone or the second-order cone, is*

^{n−1}*K :=*©

*x = (x*_{1}*, x*_{2}*) ∈ R × R*^{n−1}*: kx*_{2}*k ≤ x*_{1}ª
*.*

*Recall that an element c ∈ V is said to be idempotent if c*^{2} *= c. Two idempotents c*
*and d are said to be orthogonal if c ◦ d = 0. One says that {c*1*, c*2*, . . . , c**k**} is a complete*
system of orthogonal idempotents if

*c*^{2}_{j}*= c*_{j}*,* *c*_{j}*◦ c*_{i}*= 0 if j 6= i, j, i = 1, 2, . . . , k,* and P_{k}

*j=1**c*_{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*
*{c*_{1}*, c*_{2}*, . . . , c*_{r}*} and real numbers λ*_{1}*(x), λ*_{2}*(x), . . . , λ*_{r}*(x), arranged in the decreasing order*
*λ*1*(x) ≥ λ*2*(x) ≥ · · · ≥ λ**r**(x), such that x =*P_{r}

*j=1**λ**j**(x)c**j**.*

*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=1**λ*_{j}*(x); whereas the determinant of x is defined by det(x) :=*Q_{r}

*j=1**λ*_{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=1**λ*^{2}_{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*(kxk*^{2}*+ kyk*^{2}*) and kx + yk*^{2} *≤ 2(kxk*^{2}*+ kyk*^{2}*).* (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(λ*_{1}*(x))c*_{1}*+ ϕ(λ*_{2}*(x))c*_{2}*+ · · · + ϕ(λ*_{r}*(x))c*_{r}*,* (12)
*where x ∈ V has the spectral decomposition x =* P_{r}

*j=1**λ*_{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)*+ :=

X*r*
*j=1*

*max{0, λ**j**(x)}c**j* *and (x)**−* :=

X*r*
*j=1*

*min{0, λ**j**(x)}c**j**.* (13)

*Lemma 2.1 [26, Theorem 13] For any x =* P_{r}

*j=1**λ**j**(x)c**j**, 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*

*ϕ*^{0}_{V}*(x)h =*
X*r*

*j=1*

*[ϕ*^{[1]}*(λ(x))]**jj**hc**j**, hic**j*+ X

*1≤j<l≤r*

*4[ϕ*^{[1]}*(λ(x))]**jl**c**j**◦ (c**l**◦ 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)*
*ϕ*^{0}*(λ*_{i}*(x))* *if λ*_{i}*(x) = λ*_{j}*(x)*

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

*In fact, the Jacobian ϕ*^{0}_{V}*(·) is a linear and symmetric operator, which can be written as*
*ϕ*^{0}_{V}*(x) =*

X*r*
*j=1*

*ϕ*^{0}*(λ*_{j}*(x))Q(c** _{j}*) + 2
X

*r*

*i,j=1,i6=j*

*[ϕ*^{[1]}*(λ(x))]*_{ij}*L(c*_{j}*)L(c** _{i}*) (14)

*where Q(x) := 2L*^{2}*(x) −L(x*^{2}*) 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*
*2L*^{3}*(c) − 3L*^{2}*(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,*^{1}_{2}*) 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,*^{1}_{2}*) 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 {c*1*, c*2*, . . . , c**r**} be a Jordan frame of A. For i, j ∈ {1, . . . , r}, define the eigenspaces*
V*ii* *:= V(c**i**, 1) = Rc**i**,*

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 V**ij* *(1 ≤ i ≤ j ≤ r),*
*i.e., V = ⊕*_{i≤j}*V*_{ij}*. Furthermore,*

V_{ij}*◦ V*_{ij}*⊂ V** _{ii}*+ V

_{jj}*,*V

*ij*

*◦ V*

*jk*

*⊂ V*

*ik*

*, 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=1**λ*_{j}*(x)c*_{j}*. For i, j ∈ {1, 2, . . . , r},*
*let C**ij**(x) be the orthogonal projection operator onto V**ij*. Then,

*C*_{ij}*(x) = C*_{ij}^{∗}*(x), C*_{ij}^{2}*(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≤r**C*_{ij}*(x) = I,* (16)

*where C*_{ij}^{∗}*is the adjoint (operator) of C**ij*. 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*

*instead.*

_{ij}*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(x*^{2}); see [15, Chapters VI–V].

*Lemma 2.2 Let x ∈ V have the spectral decomposition x =* P_{r}

*j=1**λ*_{j}*(x)c*_{j}*. Then the*
*symmetric operator L(x) has the spectral decomposition*

*L(x) =*
X*r*
*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 {*^{1}_{2}*(λ**j**(x) + λ**l**(x)) : j, l =*
*1, 2, . . . , r}, and L(x*^{2}*) has the spectral decomposition*

*L(x*^{2}) =
X*r*

*j=1*

*λ*^{2}_{j}*(x)C*_{jj}*(x) +* X

*1≤j<l≤r*

1 2

¡*λ*^{2}_{j}*(x) + λ*^{2}_{l}*(x)*¢

*C*_{jl}*(x)* (17)

*with the spectrum σ(L(x*^{2}*)) consisting of all distinct numbers in {*^{1}_{2}¡

*λ*^{2}_{j}*(x) + λ*^{2}_{l}*(x)*¢
:
*j, l = 1, 2, . . . , r}.*

*Proposition 2.1 For any x ∈ V, the operator L(x*^{2}*) − L*^{2}*(x) is positive semidefinite.*

*Proof. By Lemma 2.2 and (15), we can verify that L*^{2}*(x) has the spectral decomposition*
*L*^{2}*(x) =*

X*r*
*j=1*

*λ*^{2}_{j}*(x)C**jj**(x) +* X

*1≤j<l≤r*

1

4*(λ**j**(x) + λ**l**(x))*^{2}*C**jl**(x).* (18)
*This means that the operator L(x*^{2}*) − L*^{2}*(x) has the spectral decomposition*

*L(x*^{2}*) − L*^{2}*(x) =* X

*1≤j<l≤r*

·1 2

¡*λ*^{2}_{j}*(x) + λ*^{2}_{l}*(x)*¢

*−* 1

4*(λ**j**(x) + λ**l**(x))*^{2}

¸

*C**jl**(x).*

Noting that the orthogonal projection operator is positive semidefinite on V and
*λ*^{2}_{j}*(x) + λ*^{2}_{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(x*^{2}*) − L*^{2}*(x).* *2*

## 3 *Differentiability of the function ψ*

_{τ}*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,*

*x*^{2}*+ y*^{2} *+ (τ − 2)(x ◦ y) =*
µ

*x +τ − 2*
2 *y*

¶_{2}

+*τ (4 − τ )*
4 *y*^{2}

= µ

*y +τ − 2*
2 *x*

¶_{2}

+*τ (4 − τ )*

4 *x*^{2} *∈ 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*

*is respectively a complementarity function and a merit*

_{τ}*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 φ*

*τ*

*(x, y) = 0. Then,*

£*x*^{2}*+ y*^{2}*+ (τ − 2)(x ◦ y)*¤_{1/2}

*= (x + y).* (20)

Squaring the two sides of (20) yields that

*x*^{2} *+ y*^{2}*+ (τ − 2)(x ◦ y) = x*^{2} *+ y*^{2}*+ 2(x ◦ y),*

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

*x*^{2}*+ y*^{2}¢_{1/2}

*− y and y =* ¡

*x*^{2}*+ y*^{2}¢_{1/2}

*− x.*

*Since x*^{2}*+ y*^{2} *∈ K, x*^{2} *∈ K and y*^{2} *∈ 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} *= x*^{2}*+ y*^{2}*. This, together with x ◦ y = 0, implies*

that £

*x*^{2}*+ y*^{2} *+ (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) := (x*^{2}*+ y*^{2})^{1/2}*. Then, the function u(x, y) is*
*continuously differentiable at any point (x, y) such that x*^{2}*+ y*^{2} *∈ int(K). Furthermore,*

*∇*_{x}*u(x, y) = L(x)L*^{−1}*(u(x, y)) and ∇*_{y}*u(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*

*u*^{2}*(x, y) = x*^{2}*+ y*^{2}*,* *∀ x, y ∈ V.* (22)
*By the formula (14), it is easy to verify that ∇*_{x}*(x*^{2}*) = 2L(x). Differentiating on both*
*sides of (22) with respect to x then yields that*

*2∇*_{x}*u(x, y)L(u(x, y)) = 2L(x).*

*This implies that ∇*_{x}*u(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 ∇**y**u(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=1**λ*_{j}*(z)c** _{j}*,

*where {c*1

*, c*2

*, . . . , c*

*r*

*} is a Jordan frame and λ*1

*(z), . . . , λ*

*r*

*(z) are the eigenvalues arranged*

*in the decreasing order λ*

_{1}

*(z) ≥ λ*

_{2}

*(z) ≥ · · · ≥ λ*

_{r}*(z) = 0. Define the index*

*j** ^{∗}* := min
n

*j | λ**j**(z) = 0, j = 1, 2, . . . , r*
o

(23) and let

*c**J* := P_{j}^{∗}_{−1}

*l=1* *c**l**.* (24)

*Clearly, j*^{∗}*and c**J* *are well-defined since 0 6= z ∈ K and z /∈ int(K). Since c**J* 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}*,*^{1}_{2}*) and V(c*_{J}*, 0). In the sequel, we write P*_{1}*(c*_{J}*), P*^{1}

2*(c** _{J}*)

*and P*0

*(c*

*J*

*) as the orthogonal projection onto V(c*

*J*

*, 1), V(c*

*J*

*,*

^{1}

_{2}

*) and V(c*

*J*

*, 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) :=* £

*x*^{2}*+ y*^{2}*+ (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 z*^{2}*(x, y)+h ∈ K, let w = w(x, y) := [z*^{2}*(x, y)+h]*^{1/2}*−z(x, y).*

*Then, P*_{1}*(c*_{J}*)w =* ^{1}_{2}*L*^{−1}*(z(x, y))[P*_{1}*(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, ∇*_{x}*ψ*_{τ}*(0, 0) = ∇*_{y}*ψ*_{τ}*(0, 0) = 0, and if (x, y) 6= (0, 0), then*

*∇**x**ψ**τ**(x, y) =*

·
*L*

³

*x +τ − 2*
2 *y*

´

*L*^{−1}*(z(x, y)) − I*

¸

*φ**τ**(x, y),*

*∇**y**ψ**τ**(x, y) =*

·
*L*

³

*y +τ − 2*
2 *x*

´

*L*^{−1}*(z(x, y)) − I*

¸

*φ**τ**(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 u*^{2}*+ v*^{2}*+ (τ − 2)(u ◦ v) has the*
*spectrum decomposition u*^{2}*+ v*^{2} *+ (τ − 2)(u ◦ v) =* P_{r}

*j=1**µ*_{j}*d*_{j}*, where {d*_{1}*, d*_{2}*, . . . , d*_{r}*} is*
*the corresponding Jordan frame. Then, for j = 1, 2, . . . , r, we have*

*µ** _{j}* = 1

*kd*

_{j}*k*

^{2}

DP_{r}

*j=1**µ*_{j}*d*_{j}*, d** _{j}*E

=

*u*^{2}*+ v*^{2}*+ (τ − 2)(u ◦ v), d** _{j}*®

=

*µ

*u +* *τ − 2*
2 *v*

¶_{2}

+ *τ (4 − τ )*
4 *v*^{2}*, d*_{j}

+

*≤*

*µ

*u +* *τ − 2*
2 *v*

¶_{2}

+ *τ (4 − τ )*
4 *v*^{2}*, e*

+

*= kuk*^{2}*+ (τ − 2)hu, vi + kvk*^{2}

*≤ (τ /2)(kuk*^{2}*+ kvk*^{2}*),* (27)

*where the second equality is by kd*_{j}*k = 1, the first inequality is due to e =*P_{r}

*j=1**d** _{j}* and

*d*

*j*

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

*ψ**τ**(u, v) − ψ**τ**(0, 0) =* 1
2

°°

*°[u*^{2}*+ v*^{2}*+ (τ − 2)(u ◦ v)]*^{1/2}*− (u + v)*

°°

°^{2}

= 1 2

°°

°P_{r}

*j=1*

*√µ**j* *d**j**− (u + v)*

°°

°^{2}

*≤*

°°

°P_{r}

*j=1*

*√µ**j* *d**j*

°°

°^{2}*+ ku + vk*^{2}

*≤*
X*r*

*j=1*

*µ**j**kd**j**k*^{2}*+ 2(kuk*^{2}*+ kvk*^{2})

*≤*
µ1

2*τ r + 2*

¶¡

*kuk*^{2}*+ kvk*^{2}¢
*,*

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 ∇*_{x}*ψ*_{τ}*(0, 0) = ∇*_{y}*ψ*_{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 φ**τ* is continuously differentiable under this case. Notice that

*ψ*_{τ}*(x, y) =* 1
2

*e, φ*^{2}_{τ}*(x, y)*®
*,*

*and hence the function ψ** _{τ}* is continuously differentiable. Applying the chain rule yields

*∇*_{x}*ψ*_{τ}*(x, y) = ∇*_{x}*φ*_{τ}*(x, y)L(φ*_{τ}*(x, y))e = ∇*_{x}*φ*_{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 *y*^{2}

¸_{1/2}

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

*∇*_{x}*φ*_{τ}*(x, y) = L*

³

*x +τ − 2*
2 *y*

´

*L*^{−1}*(z(x, y)) − I.*

This, together with (28), immediately yields that

*∇*_{x}*ψ*_{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*

*∇*_{y}*ψ*_{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 + u*^{2}*+ v*^{2}*+ (τ − 2)u ◦ v*
with ˆ*x = x +* ^{τ −2}_{2} *y and ˆy = y +* ^{τ −2}_{2} *x. It is not difficult to verify that*

*z*^{2}*(x, y) + ˆz =*
µ

*(x + u) +* *τ − 2*

2 *(y + v)*

¶_{2}

+ *τ (4 − τ )*

4 *(y + v)*^{2}

*= z*^{2}*(x + u, y + v) ∈ K.*

Let

*w(x, y) :=* ¡

*z*^{2}*(x, y) + ˆz*¢_{1/2}

*− z(x, y).*

*From the definitions of ψ**τ* *and z(x, y), it then follows that*
*ψ*_{τ}*(x + u, y + v) − ψ*_{τ}*(x, y)*

= 1 2

h°*°[z*^{2}*(x, y) + ˆz]*^{1/2}*− (x + u + y + v)*°°^{2}*− kz(x, y) − (x + y)k*^{2}
i

= 1 2

£*hˆz, ei + ku + vk*^{2}¤

*− 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 = hP*_{1}*(c*_{J}*)w(x, y), x + yi*

=

¿1

2*L*^{−1}*(z(x, y))[P*1*(c**J*)ˆ*z] + o(kˆzk), x + y*
À

= 1

2

*P*_{1}*(c** _{J}*)ˆ

*z, L*

^{−1}*(z(x, y))[x + y]*®

*+ o(kˆzk)*

= D

*P*_{1}*(c** _{J}*) [ˆ

*x ◦ u + ˆy ◦ v] , L*

^{−1}*(z(x, y))[x + y]*

E

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

= D

ˆ

*x ◦ u + ˆy ◦ v, P*1*(c**J**)[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}*,*^{1}_{2}*) ⊕ V(c*_{J}*, 0), the fifth one is due*
*to P*_{1}*(c*_{J}*) = P*_{1}^{∗}*(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,*

*∇*_{x}*ψ*_{τ}*(x, y) = ˆx + x + y − z(x, y) −*£

*L*^{−1}*(z(x, y))(x + y)*¤

*◦ ˆx,*

*∇*_{y}*ψ*_{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*

= ˆ*x − 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,

*∇*_{x}*ψ*_{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

*∇*_{y}*ψ*_{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 (R^{n}*, ◦, h·, ·i*R* ^{n}*), the formula reduces to the one of [3, Proposition 3.2];

whereas when A is specified as (S^{n}*, ◦, h·, ·i*_{S}^{n}*) and τ = 2, the formula is same as the one*
*in [28, Lemma 6.3 (b)] by noting that z(x, y) = (x*^{2}*+ y*^{2})* ^{1/2}* and

*∇*_{x}*ψ*_{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, ∇**x**ψ**τ**(x, y)i + hy, ∇**y**ψ**τ**(x, y)i = kφ**τ**(x, y)k*^{2}*.*

*(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, ∇**x**ψ**τ**(x, y)i + hy, ∇**y**ψ**τ**(x, y)i*

=

¿
*x,*

µ

*x +τ − 2*
2 *y*

¶

*◦ [L*^{−1}*(z(x, y))φ**τ**(x, y)]*

À

*− hx, φ**τ**(x, y)i*
+

¿
*y,*

µ

*y +τ − 2*
2 *x*

¶

*◦ [L*^{−1}*(z(x, y))φ**τ**(x, y)]*

À

*− hy, φ**τ**(x, y)i*

=

¿
*x ◦*

µ

*x +τ − 2*
2 *y*

¶

*, L*^{−1}*(z(x, y))φ**τ**(x, y)*
À

*− hx, φ**τ**(x, y)i*
+

¿
*y ◦*

µ

*y +τ − 2*
2 *x*

¶

*, L*^{−1}*(z(x, y))φ**τ**(x, y)*
À

*− hy, φ**τ**(x, y)i*

*= hz*^{2}*(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)k*^{2}*,*

*where the next to last equality is by z*^{2} *= 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*

_{τ}*In this section, we investigate the continuity of the gradients ∇*_{x}*ψ*_{τ}*(x, y) and ∇*_{y}*ψ*_{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) :=* ¡

*x*^{2}*+ y*^{2}*+ (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 ∇*_{x}*ψ*_{τ}*(x, y) and ∇*_{y}*ψ*_{τ}*(x, y) are Lipschitz*
*continuous by arguing the Lipschitz continuity of z**²**(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 z**²**(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*

*∇*_{x}*z*_{²}*(x, y) = L*

³

*x +τ − 2*
2 *y*

´

*L*^{−1}*(z*_{²}*(x, y)),*

*∇*_{y}*z*_{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∇*_{x}*z*_{²}*(x, y)k ≤ C and k∇*_{y}*z*_{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 *y*^{2}*+ ²e*

¸_{1/2}

=

·³

*y +τ − 2*
2 *x*´_{2}

+*τ (4 − τ )*

4 *x*^{2}*+ ²e*

¸_{1/2}

*.* (34)

*We next prove that the operator ∇*_{x}*z*_{²}*(x, y) is bounded for any x, y ∈ V and ² > 0. Let*
*{u*1*, u*2*, . . . , u**n**} be an orthonormal basis of V. For any x, y ∈ V, let L(z*^{2}*), 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(z*^{2}*), L(x +* ^{τ −2}_{2} *y), L(z*_{²}*) and L((x +* ^{τ −2}_{2} *y)*^{2}*) with respect to the basis {u*_{1}*, u*_{2}*, . . . , 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=1**λ**j**(z)c**j*, where
*λ*_{1}*(z) ≥ λ*_{2}*(z) ≥ · · · ≥ λ*_{r}*(z) ≥ 0 are the eigenvalue of z and {c*_{1}*, c*_{2}*, . . . , c*_{r}*} is the*
*corresponding Jordan frame. From Lemma 2.2, L(z) has the spectral decomposition*

*L(z) =*
X*r*

*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 {*^{1}_{2}*(λ*_{j}*(z) + λ*_{l}*(z)) : j, l =*
*1, 2, . . . , r}, and L(z*^{2}) has the spectral decomposition

*L(z*^{2}) =
X*r*

*j=1*

*λ*^{2}_{j}*(z)C**jj**(z) +* X

*1≤j<l≤r*

1 2

¡*λ*^{2}_{j}*(z) + λ*^{2}_{l}*(z)*¢

*C**jl**(z)* (36)

*with σ(L(z*^{2}*)) consisting of all distinct numbers in {*^{1}_{2}¡

*λ*^{2}_{j}*(z) + λ*^{2}_{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

*λ*^{2}_{j}*(z) + ² c** _{j}*, and

*consequently the symmetric operator L(z*

*) has the spectral decomposition*

_{²}*L(z** _{²}*) =
X

*r*

*j=1*

q

*λ*^{2}_{j}*(z) + ² C*_{jj}*(z) +* X

*1≤j<l≤r*

1 2

³q

*λ*^{2}_{j}*(z) + ² +*
q

*λ*^{2}_{l}*(z) + ²*

´

*C*_{jl}*(z)* (37)

*with the spectrum σ(L(z** _{²}*)) consisting of all distinct numbers in

½1 2

µq

*λ*^{2}_{j}*(z) + ² +*
q

*λ*^{2}_{l}*(z) + ²*

¶

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

¾
*.*

*We first prove that the matrix L(x +* ^{τ −2}_{2} *y) (L(z*^{2}*) + 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(z*^{2}*)P** ^{T}* = diag¡

*λ*_{1}*(L(z*^{2}*)), λ*_{2}*(L(z*^{2}*)), · · · , λ*_{n}*(L(z*^{2}))¢

(38)
*where λ*_{1}*(L(z*^{2}*)) ≥ λ*_{2}*(L(z*^{2}*)) · · · ≥ λ*_{n}*(L(z*^{2}*)) ≥ 0 are the eigenvalues of L(z*^{2}). Then, it
*is not hard to verify that for any ² > 0,*

*P* ¡

*L(z*^{2}*) + ²I*¢_{−1/2}

*P** ^{T}* = diag
Ã

p 1

*λ*_{1}*(L(z*^{2}*)) + ², · · · ,* 1

p*λ*_{n}*(L(z*^{2}*)) + ²*

!
*.*

Write e*U := P L*¡

*x +*^{τ −2}_{2} *y*¢

*P** ^{T}*. We can compute that

*L*

³

*x +τ − 2*
2 *y*

´³

*L(z*^{2}*) + ²I*

´_{−1/2}

*= P*^{T}*Udiag*e
Ã

p 1

*λ*_{1}*(L(z*^{2}*)) + ², · · · ,* 1

p*λ*_{n}*(L(z*^{2}*)) + ²*

!
*P*

*= P*^{T}

"

*U*e*ik*

p*λ**k**(L(z*^{2}*)) + ²*

#

*1≤i≤n*
*1≤k≤n*

*P.* (39)

*Since L(z*^{2}*) = L*¡

*(x +* ^{τ −2}_{2} *y)*^{2}¢
*+ L*³

*τ (4−τ )*
4 *y*^{2}´

*and L(y*^{2}) is positive semidefinite, we get

*L(z*^{2}*) − 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*
Ãµ

*x +* *τ − 2*
2 *y*

¶_{2}!

*− L*
µ

*x +* *τ − 2*
2 *y*

¶
*L*

µ

*x +τ − 2*
2 *y*

¶
*º O.*