DOI 10.1007/s10589-008-9182-9

**A one-parametric class of merit functions**

**for the second-order cone complementarity problem**

**Jein-Shan Chen· Shaohua Pan**

Received: 11 January 2007 / Revised: 21 April 2008 / Published online: 8 May 2008

© Springer Science+Business Media, LLC 2008

**Abstract We investigate a one-parametric class of merit functions for the second-**
order cone complementarity problem (SOCCP) which is closely related to the popular
Fischer–Burmeister (FB) merit function and natural residual merit function. In fact, it
*will reduce to the FB merit function if the involved parameter τ equals 2, whereas as*
*τ*tends to zero, its limit will become a multiple of the natural residual merit function.

In this paper, we show that this class of merit functions enjoys several favorable prop-
erties as the FB merit function holds, for example, the smoothness. These properties
play an important role in the reformulation method of an unconstrained minimization
or a nonsmooth system of equations for the SOCCP. Numerical results are reported
for some convex second-order cone programs (SOCPs) by solving the unconstrained
minimization reformulation of the KKT optimality conditions, which indicate that
the FB merit function is not the best. For the sparse linear SOCPs, the merit function
*corresponding to τ= 2.5 or 3 works better than the FB merit function, whereas for*
*the dense convex SOCPs, the merit function with τ= 0.1, 0.5 or 1.0 seems to have*
better numerical performance.

**Keywords Second-order cone**· Complementarity · Merit function · Jordan product

J.-S. Chen (

^{)}

Department of Mathematics, National Taiwan Normal University, 88 Section 4, Ting-Chou Road, 11677 Taipei, Taiwan

e-mail:jschen@math.ntnu.edu.tw J.-S. Chen

Mathematics Division, National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwain S. Pan

School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China e-mail:shhpan@scut.edu.cn

**1 Introduction**

*We consider the conic complementarity problem of finding a vector ζ*∈ R* ^{n}*such that

*F (ζ )∈ K,*

*G(ζ )∈ K,*

*F (ζ ), G(ζ ) = 0,*(1) where

*·, · is the Euclidean inner product, F : R*

*→ R*

^{n}

^{n}*and G*: R

*→ R*

^{n}*are the mappings assumed to be continuously differentiable throughout this paper, and*

^{n}*K is*the Cartesian product of second-order cones (SOCs). In other words,

*K = K*^{n}^{1}*× K*^{n}^{2}*× · · · × K*^{n}^{N}*,* (2)
*where N, n*1*, . . . , n*_{N}*≥ 1, n*1*+ · · · + n**N**= n, and*

*K*^{n}* ^{i}*:=

*(x*1*, x*2*)*∈ R × R^{n}^{i}^{−1}*| x*2* ≤ x*1

*,*

with* · denoting the Euclidean norm and K*^{1} denoting the set of nonnegative re-
alsR+. We will refer to (1)–(2) as the second-order cone complementarity problem
(SOCCP).

*An important special case of the SOCCP corresponds to G(ζ )= ζ for all ζ ∈ R** ^{n}*.
Then (1) reduces to

*F (ζ )∈ K, ζ ∈ K,* *F (ζ ), ζ = 0,* (3)
which is a natural extension of the nonlinear complementarity problem (NCP) [7,8]

with*K = R*^{n}_{+}, the nonnegative orthant cone ofR* ^{n}*. Another important special case
corresponds to the KKT optimality conditions of the convex second-order cone pro-
gram (CSOCP):

*minimize g(x)*

subject to *Ax= b, x ∈ K,* (4)

*where g*: R^{n}*→ R is a convex twice continuously differentiable function, A ∈ R*^{m}^{×n}*has full row rank and b*∈ R* ^{m}*. From [4], we know that the KKT conditions of (4),
which are sufficient but not necessary for optimality, can be reformulated as (1) with

*F (ζ ):= ¯x +*

*I− A*^{T}*(AA*^{T}*)*^{−1}*A*

*ζ,* *G(ζ ):= ∇g(F (ζ )) − A*^{T}*(AA*^{T}*)*^{−1}*Aζ,* (5)
where *¯x ∈ R*^{n}*is any point such that A¯x = b. When g is linear, the CSOCP reduces*
to the linear SOCP which arises in numerous applications in engineering design,
finance, robust optimization, and includes as special cases convex quadratically con-
strained quadratic programs and linear programs; see [1,13] and references therein.

There have been various methods proposed for solving SOCPs and SOCCPs. They
include the interior-point methods [2,3,15,16,19], the non-interior smoothing New-
ton methods [6,9], and the smoothing-regularization method [11]. Recently, there
was an alternative method [4] based on reformulating the SOCCP as an unconstrained
*minimization problem. In that approach, it aims to find a function ψ*: R* ^{n}*×R

*→ R*

^{n}_{+}satisfying

*ψ (x, y)= 0 ⇐⇒ x ∈ K, y ∈ K, x, y = 0,* (6)

so that the SOCCP can be reformulated as an unconstrained minimization problem min

*ζ*∈R^{n}*f (ζ ):= ψ(F (ζ ), G(ζ )).*

*We call such ψ a merit function associated with the coneK.*

*A popular choice of ψ is the Fischer–Burmeister (FB) merit function*
*ψ*_{FB}*(x, y)*:=1

2*φ*FB*(x, y)*^{2}*,* (7)

*where φ*_{FB}: R* ^{n}*× R

*→ R*

^{n}*is the vector-valued FB function defined by*

^{n}*φ*_{FB}*(x, y):= (x*^{2}*+ y*^{2}*)*^{1/2}*− (x + y),* (8)
*with x*^{2}*= x ◦ x denoting the Jordan product between x and itself, x** ^{1/2}*being a vector

*such that (x*

^{1/2}*)*

^{2}

*= x, and x + y meaning the usual componentwise addition of vec-*

*tors. The function ψ*

_{FB}was studied in [4] and particularly shown to be continuously

*differentiable (smooth). Another popular choice of ψ is the natural residual merit*function

*ψ*_{NR}*(x, y)*:=1

2*φ*NR*(x, y)*^{2}*,*

*where φ*_{NR}: R* ^{n}*× R

*→ R*

^{n}*is the vector-valued natural residual function given by*

^{n}*φ*

_{NR}

*(x, y):= x − (x − y)*

_{+}

*with (·)*_{+}meaning the projection in the Euclidean norm onto *K. The function φ*NR

was studied in [9,11] which is involved in smoothing methods for the SOCCP. Com-
*pared with the FB merit function ψ*_{FB}*, the function ψ*_{NR} has a drawback, i.e., its
non-differentiability.

In this paper, we will investigate the following one-parametric class of functions
*ψ**τ**(x, y)* := 1

2*φ**τ**(x, y)*^{2}*,* (9)

*where τ is a fixed parameter from (0, 4) and φ**τ*: R* ^{n}*× R

*→ R*

^{n}*is defined by*

^{n}*φ*

_{τ}*(x, y)*:=

*(x− y)*^{2}*+ τ(x ◦ y)**1/2*

*− (x + y).* (10)

*Specifically, we prove that ψ**τ* is a merit function associated with*K which is continu-*
ously differentiable everywhere with computable gradient formulas (see Propositions
3.1–3.3), and hence the SOCCP can be reformulated as an unconstrained smooth
minimization

*ζ*min∈R^{n}*f**τ**(ζ ):= ψ**τ**(F (ζ ), G(ζ )).* (11)
*Also, we show that every stationary point of f**τ* solves the SOCCP under the condi-
tion that*∇F and −∇G are column monotone (see Proposition*4.1). Observe that φ*τ*

*reduces to φ*_{FB} *when τ= 2, whereas its limit as τ → 0 becomes a multiple of φ*NR.

Thus, this class of merit functions has a close relation to two of the most important
merit functions so that a closer look and study for it is worthwhile. In addition, this
study is motivated by the work [12] where φ*τ* was used to develop a nonsmooth
Newton method for the NCP. This paper is mainly concerned with the merit function
approach based on the unconstrained minimization problem (11). Numerical results
*are also reported for some convex SOCPs, which indicate that ψ**τ* can be an alterna-
*tive for ψ*_{FB}*if a suitable τ is selected.*

Throughout this paper,R^{n}*denotes the space of n-dimensional real column vec-*
tors, andR^{n}^{1}× · · · × R^{n}* ^{m}* is identified withR

^{n}^{1}

^{+···+n}

^{m}*. Thus, (x*1

*, . . . , x*

_{m}*)*∈ R

^{n}^{1}×

· · · × R^{n}* ^{m}* is viewed as a column vector inR

^{n}^{1}

^{+···+n}

^{m}*. The notation I denotes an*

*identity matrix of suitable dimension, and int(K*

^{n}*)*denotes the interior of

*K*

*. For any*

^{n}*differentiable mapping F*: R

*→ R*

^{n}*,*

^{m}*∇F (x) ∈ R*

^{n}*denotes the transposed Jaco-*

^{×m}*bian of F at x. For a symmetric matrix A, we write A O (respectively, A O)*

*to mean A is positive semidefinite (respectively, positive definite). For nonnegative*

*αand β, we write α= O(β) to mean α ≤ Cβ, with C > 0 independent of α and β.*

Without loss of generality, in the rest of this paper we assume that*K = K*^{n}*(n >1).*

All analysis can be carried over to the general case where*K has the structure as (2).*

*In addition, we always assume that τ satisfies 0 < τ < 4.*

**2 Preliminaries**

It is known that*K*^{n}*(n >1) is a closed convex self-dual cone with nonempty interior*
*int(K*^{n}*)*:=

*x= (x*1*, x*2*)*∈ R × R^{n}^{−1}*| x*1*>x*2
*.*

*For any x= (x*1*, x*_{2}*), y= (y*1*, y*_{2}*)*∈ R × R^{n}^{−1}*, the Jordan product of x and y is*
defined by

*x◦ y := (x, y, y*1*x*2*+ x*1*y*2*).* (12)
The Jordan product, unlike scalar or matrix multiplication, is not associative, which
is a main source on complication in the analysis of SOCCP. The identity element
*under this product is e:= (1, 0, . . . , 0)*^{T}∈ R^{n}*. For any x= (x*1*, x*2*)*∈ R × R^{n}^{−1}, the
*determinant of x is defined by det(x):= x*_{1}^{2}*−x*2^{2}*. If det(x)= 0, then x is said to be*
*invertible. If x is invertible, there exists a unique y= (y*1*, y*_{2}*)*∈ R × R^{n}^{−1}satisfying
*x◦ y = y ◦ x = e. We call this y the inverse of x and denote it by x*^{−1}. For each
*x= (x*1*, x*2*)*∈ R × R^{n}^{−1}, let

*L**x*:=

*x*1 *x*_{2}^{T}
*x*2 *x*1*I*

*.* (13)

*It is easily verified that L**x**y* *= x ◦ y and L**x**+y**= L**x**+ L**y* *for any x, y*∈ R* ^{n}*, but

*generally L*

^{2}

_{x}*= L*

*x*

*L*

*x*

*= L*

_{x}^{2}

*and L*

^{−1}

_{x}*= L*

*−1*

_{x}*. If L*

*x*is invertible, then the inverse of

*L*

*is given by*

_{x}*L*^{−1}* _{x}* = 1

*det(x)*

⎡

⎣ *x*1 *−x*_{2}^{T}

*−x*2

*det(x)*
*x*_{1} *I*+ 1

*x*_{1}*x*2*x*_{2}^{T}

⎤

*⎦ .* (14)

We next recall from [9] that each x*= (x*1*, x*_{2}*)*∈ R × R^{n}^{−1}admits a spectral fac-
torization, associated with*K** ^{n}*, of the form

*x= λ*1*(x)· u*^{(1)}_{x}*+ λ*2*(x)· u*^{(2)}_{x}*,*

*where λ*1*(x), λ*2*(x)and u*^{(1)}*x* *, u*^{(2)}*x* are the spectral values and the associated spectral
*vectors of x given by*

*λ*_{i}*(x)= x*1*+ (−1)*^{i}*x*2*,* *u*^{(i)}* _{x}* =1

2*(1, (−1)*^{i}*¯x*2*)* *for i= 1, 2,*

with *¯x*2= _{x}^{x}^{2}_{2}_{} *if x*2*= 0, and otherwise ¯x*2 being any vector in R^{n}^{−1} such that

* ¯x*2* = 1. If x*2*= 0, the factorization is unique. The spectral factorization of x has*
various interesting properties; see [9]. We list three properties that will be used later.

**Property 2.1**

*(a) x*^{2}*= λ*^{2}_{1}*(x)· u*^{(1)}*x* *+ λ*^{2}_{2}*(x)· u*^{(2)}*x* *∈ K*^{n}*for any x*∈ R* ^{n}*.

*(b) If x∈ K*

^{n}*, then x*

*=√*

^{1/2}*λ*1*(x)· u*^{(1)}*x* +√

*λ*2*(x)· u*^{(2)}*x* *∈ K** ^{n}*.

*(c) x∈ K*^{n}*⇐⇒ λ*1*(x)≥ 0 ⇐⇒ L**x**
O, x ∈ int(K*^{n}*)⇐⇒ λ*1*(x) >*0*⇐⇒ L**x** O.*

**3 Smoothness of the function ψ****τ**

*In this section we will show that ψ**τ* defined by (9) is a smooth merit function. First,
by Properties2.1(a) and (b), φ*τ* *and ψ**τ* *are well-defined since for any x, y*∈ R* ^{n}*, we
can verify that

*(x− y)*^{2}*+ τ(x ◦ y) =*

*x*+*τ*− 2
2 *y*

2

+*τ (4− τ)*
4 *y*^{2}

=

*y*+*τ*− 2
2 *x*

2

+*τ (4− τ)*

4 *x*^{2}*∈ K*^{n}*.* (15)
*The following proposition shows that ψ**τ* is indeed a merit function associated
with*K** ^{n}*.

**Proposition 3.1 Let ψ***τ* *and φ**τ**be given as in (9) and (10), respectively. Then,*

*ψ**τ**(x, y)= 0 ⇐⇒ φ**τ**(x, y)= 0 ⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *x, y = 0.*

*Proof The first equivalence is clear by the definition of ψ**τ*. We consider the second
one.

“*⇐”. Since x ∈ K, y ∈ K and x, y = 0, we have x ◦ y = 0. Substituting it*
*into the expression of φ**τ**(x, y)then yields that φ**τ**(x, y)= (x*^{2}*+ y*^{2}*)*^{1/2}*− (x + y) =*
*φ*_{FB}*(x, y). From Proposition 2.1 of [9], we immediately obtain φ**τ**(x, y)*= 0.

“⇒”. Suppose that φ*τ**(x, y)= 0. Then, x + y =*

*(x− y)*^{2}*+ τ(x ◦ y)**1/2*

. Squar-
*ing both sides yields x◦ y = 0. This implies that x + y = (x*^{2}*+ y*^{2}*)** ^{1/2}*, i.e.

*φ*_{FB}*(x, y)*= 0. From Proposition 2.1 of [9], it then follows that x*∈ K*^{n}*, y∈ K** ^{n}*and

*x, y = 0.*

*In what follows, we focus on the proof of the smoothness of ψ**τ*. We first introduce
*some notation that will be used in the sequel. For any x= (x*1*, x*2*), y= (y*1*, y*2*)*∈
R × R^{n}^{−1}, let

*w= (w*1*, w*_{2}*)= w(x, y) := (x − y)*^{2}*+ τ(x ◦ y),*

(16)
*z= (z*1*, z*2*)= z(x, y) :=*

*(x− y)*^{2}*+ τ(x ◦ y)**1/2*

*.*

*Then, w∈ K*^{n}*and z∈ K** ^{n}*. Moreover, by the definition of Jordan product,

*w*1

*= w*1

*(x, y)= x*

^{2}

*+ y*

^{2}

*+ (τ − 2)x*

^{T}

*y,*

(17)
*w*2*= w*2*(x, y)= 2(x*1*x*2*+ y*1*y*2*)+ (τ − 2)(x*1*y*2*+ y*1*x*2*).*

*Let λ*1*(w)and λ*2*(w)be the spectral values of w. By Property*2.1(b), we have that
*z*1*= z*1*(x, y)*=

√*λ*_{2}*(w)*+√
*λ*_{1}*(w)*

2 *,*

(18)
*z*_{2}*= z*2*(x, y)*=

√*λ*2*(w)*−√
*λ*1*(w)*

2 *¯w*2*,*

where *¯w*2:= _{w}^{w}^{2}_{2}_{} *if w*2*= 0 and otherwise ¯w*2 is any vector in R^{n}^{−1} satisfying

* ¯w*2 = 1.

*The following technical lemma describes the behavior of x, y when w= (x −*
*y)*^{2}*+ τ(x ◦ y) is on the boundary of K** ^{n}*. In fact, it may be viewed as an extension of
[4, Lemma 3.2].

* Lemma 3.1 For any x= (x*1

*, x*2

*), y= (y*1

*, y*2

*)*∈ R × R

^{n−1}*, if w /∈ int(K*

^{n}*), then*

*x*

_{1}

^{2}

*= x*2

^{2}

*,*

*y*

_{1}

^{2}

*= y*2

^{2}

*,*

*x*

_{1}

*y*

_{1}

*= x*

_{2}

^{T}

*y*

_{2}

*,*

*x*

_{1}

*y*

_{2}

*= y*1

*x*

_{2}; (19)

*x*

_{1}

^{2}

*+ y*1

^{2}

*+ (τ − 2)x*1

*y*1

*= x*1

*x*2

*+ y*1

*y*2

*+ (τ − 2)x*1

*y*2

*= x*2^{2}*+ y*2^{2}*+ (τ − 2)x*_{2}^{T}*y*_{2}*.* (20)
*If, in addition, (x, y)= (0, 0), then w*2*= 0, and furthermore,*

*x*^{T}_{2} *w*2

*w*2*= x*1*,* *x*_{1} *w*2

*w*2 *= x*2*,* *y*_{2}^{T} *w*2

*w*2*= y*1*,* *y*_{1} *w*2

*w*2 *= y*2*.* (21)
*Proof Since w= (x −y)*^{2}*+τ(x ◦y) /∈ int(K*^{n}*), using (15) and [4, Lemma 3.2] yields*

*x*_{1}+*τ*− 2
2 *y*_{1}

2

=

*x*^{2}+*τ*− 2
2 *y*_{2}

^{2}*,* *y*_{1}^{2}*= y*2^{2}*,*

*x*_{1}+*τ*− 2
2 *y*_{1}

*y*_{2}=

*x*_{2}+*τ*− 2
2 *y*_{2}

*y*_{1}*,*

*x*1+*τ*− 2
2 *y*1

*y*1=

*x*2+*τ*− 2
2 *y*2

T

*y*2;

*y*1+*τ*− 2
2 *x*1

2

=

*y*^{2}+*τ*− 2
2 *x*2

^{2}*,* *x*_{1}^{2}*= x*2^{2}*,*

*y*1+*τ*− 2
2 *x*1

*x*2=

*y*2+*τ*− 2
2 *x*2

*x*1*,*

*y*_{1}+*τ*− 2
2 *x*_{1}

*x*_{1}=

*y*_{2}+*τ*− 2
2 *x*_{2}

T

*x*_{2}*.*

From these equalities, we readily get the results in (19). Since w *∈ K*^{n}*but w /*∈
*int(K*^{n}*), we havex*^{2}*+ y*^{2}*+ (τ − 2)x*^{T}*y= 2x*1*x*2*+ 2y*1*y*2*+ (τ − 2)(x*1*y*2+
*y*1*x*2*) by λ*1*(w)*= 0. Applying the relations in (19) then gives the equalities in (20).

*If, in addition, (x, y)= (0, 0), then it is clear that x*1*x*_{2}*+ y*1*y*_{2}*+ (τ − 2)x*1*y*_{2} =
*x*_{1}^{2}*+ y*_{1}^{2}*+ (τ − 2)x*1*y*1= 0. To prove the equalities in (21), it suffices to verify that
*x*_{2}^{T}_{w}^{w}^{2}

2*= x*1*and x*1 *w*_{2}

*w*2*= x*2 *by the symmetry of x and y in w. The verifications*

are straightforward by (20) and x1*y*2*= y*1*x*2.

By Lemma3.1, when w /*∈ int(K*^{n}*), the spectral values of w are calculated as fol-*
lows:

*λ*_{1}*(w)= 0,* *λ*_{2}*(w)*= 4

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

*.* (22)

*If (x, y)= (0, 0) also holds, then using (18), (20) and (22) yields that*

*z*_{1}*(x, y)*=

*x*_{1}^{2}*+ y*_{1}^{2}*+ (τ − 2)x*1*y*_{1}*,* *z*_{2}*(x, y)*=*x*1*x*2*+ y*1*y*2*+ (τ − 2)x*1*y*2

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

*.*

*Thus, if (x, y)= (0, 0) and (x − y)*^{2}*+ τ(x ◦ y) /∈ int(K*^{n}*), φ**τ**(x, y)*can be rewritten
as

*φ*_{τ}*(x, y)= z(x, y) − (x + y) =*

⎛

⎜⎝

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

*x*1*x*2*+y*1*y*2*+(τ−2)x*1*y*2

*x*_{1}^{2}*+y*1^{2}*+(τ−2)x*1*y*1

*− (x*2*+ y*2*)*

⎞

⎟*⎠ .* (23)

This specific expression will be employed in the proof of the following main result.

**Proposition 3.2 The function ψ***τ**given by (9) is differentiable at every (x, y)*∈ R* ^{n}*×
R

^{n}*. Moreover,*∇

*x*

*ψ*

*τ*

*(0, 0)*= ∇

*y*

*ψ*

*τ*

*(0, 0)= 0; if (x − y)*

^{2}

*+ τ(x ◦ y) ∈ int(K*

^{n}*), then*

∇*x**ψ*_{τ}*(x, y)*=
*L*_{x}_{+}*τ−2*

2 *y**L*^{−1}_{z}*− I*

*φ*_{τ}*(x, y),*

∇*y**ψ*_{τ}*(x, y)*= (24)
*L*_{y}_{+}*τ*−2

2 *x**L*^{−1}_{z}*− I*

*φ*_{τ}*(x, y)*;

*if (x, y)= (0, 0) and (x − y)*^{2}*+ τ(x ◦ y) ∈ int(K*^{n}*), then x*_{1}^{2}*+ y*_{1}^{2}*+ (τ − 2)x*1*y*_{1}= 0
*and*

∇*x**ψ**τ**(x, y)*=

*x*1+^{τ}^{−2}_{2} *y*1

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

− 1

*φ**τ**(x, y),*

(25)

∇*y**ψ*_{τ}*(x, y)*=

*y*1+^{τ}^{−2}_{2} *x*1

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

− 1

*φ*_{τ}*(x, y).*

*Proof Case (1): (x, y)= (0, 0). For any u = (u*1*, u*_{2}*), v= (v*1*, v*_{2}*)*∈ R × R^{n}^{−1}, let
*μ*_{1}*, μ*_{2} *be the spectral values of (u− v)*^{2}*+ τ(u ◦ v) and ξ*^{(1)}*, ξ** ^{(2)}* be the spectral
vectors. Then,

2

*ψ*_{τ}*(u, v)− ψ**τ**(0, 0)*

=u^{2}*+ v*^{2}*+ (τ − 2)(u ◦ v)**1/2*

*− u − v*^{2}

=√*μ*_{1}*ξ** ^{(1)}*+√

*μ*_{2}*ξ*^{(2)}*− u − v*^{2}

≤

*2μ*2*+ u + v*2

*.*

In addition, from the definition of spectral value, it follows that

*μ*2*= u*^{2}*+ v*^{2}*+ (τ − 2)u*^{T}*v+ 2(u*1*u*2*+ v*1*v*2*)+ (τ − 2)(u*1*v*2*+ v*1*u*2*)*

*≤ 2u*^{2}*+ 2v*^{2}*+ 3|τ − 2|uv ≤ 5(u*^{2}*+ v*^{2}*).*

*Now combining the last two equations, we have ψ**τ**(u, v)− ψ**τ**(0, 0)= O(u*^{2}+

*v*^{2}*). This shows that ψ**τ* *is differentiable at (0, 0) with* ∇*x**ψ**τ**(0, 0)* =

∇*y**ψ*_{τ}*(0, 0)*= 0.

*Case (2): (x− y)*^{2}*+ τ(x ◦ y) ∈ int(K*^{n}*). By [5, Proposition 5], z(x, y) defined by*
(18) is continuously differentiable at such (x, y), and consequently φ*τ**(x, y)*is also
*continuously differentiable at such (x, y) since φ**τ**(x, y)= z(x, y) − (x + y). Notice*
that

*z*^{2}*(x, y)*=

*x*+*τ* − 2
2 *y*

2

+*τ (4− τ)*
4 *y*^{2}*,*
which leads to ∇*x**z(x, y)L*_{z}*= L*_{x+}*τ−2*

2 *y* by taking differentiation on both sides
*about x. Since L**z** O by Property*2.1(c), it follows that∇*x**z(x, y)= L*_{x}_{+}* ^{τ}*−2

2 *y**L*^{−1}* _{z}* .
Consequently,

∇*x**φ*_{τ}*(x, y)*= ∇*x**z(x, y)− I = L*_{x}_{+}*τ−2*

2 *y**L*^{−1}_{z}*− I.*

This together with∇*x**ψ**τ**(x, y)*= ∇*x**φ**τ**(x, y)φ**τ**(x, y)*proves the first formula of (24).

*For the symmetry of x and y in ψ**τ*, the second formula also holds.

*Case (3): (x, y)= (0, 0) and (x − y)*^{2}*+ τ(x ◦ y) /∈ int(K*^{n}*). For any x*^{} =
*(x*_{1}^{}*, x*_{2}^{}*), y*^{}*= (y*^{}_{1}*, y*_{2}^{}*)*∈ R × R^{n}^{−1}, it is easy to verify that

*2ψ**τ**(x*^{}*, y*^{}*)*=*x*^{2}*+ y*^{2}*+ (τ − 2)(x*^{}*◦ y*^{}*)**1/2*^{2}*+ x*^{}*+ y*^{}^{2}

− 2

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

*, x*^{}*+ y*^{}

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

− 2

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

*, x*^{}*+ y*^{}
*,*

where the second equality uses the fact that *z*^{2}*= z*^{2}*, e for any z ∈ R** ^{n}*. Since

*x*^{}^{2}*+ y*^{}^{2}*+ (τ − 2)x*^{}*, y*^{}* + x*^{} *+ y*^{}^{2} *is clearly differentiable in (x*^{}*, y*^{}*),*
it suffices to show that *[x*^{2}*+ y*^{2} *+ (τ − 2)(x*^{} *◦ y*^{}*)*]^{1/2}*, x*^{} *+ y*^{} is differen-
*tiable at (x*^{}*, y*^{}*)= (x, y). By Lemma*3.1, w2*= w*2*(x, y)= 0, which implies w*_{2}^{} =
*w*2*(x*^{}*, y*^{}*)= 2x*^{}_{1}*x*_{2}^{}*+2y*_{1}^{}*y*_{2}^{}*+(τ −2)(x*_{1}^{}*y*_{2}^{}*+y*_{1}^{}*x*_{2}^{}*)= 0 for all (x*^{}*, y*^{}*)*∈ R* ^{n}*×R

*suf-*

^{n}*ficiently near to (x, y). Let μ*1

*, μ*

_{2}

*be the spectral values of x*

^{2}

*+y*

^{2}

*+(τ −2)(x*

^{}

*◦y*

^{}

*).*

Then we can compute that 2

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

*, x*^{}*+ y*^{}

=√
*μ*_{2}

*x*_{1}^{}*+ y*_{1}^{} +*[2(x*_{1}^{}*x*_{2}^{} *+ y*_{1}^{}*y*_{2}^{}*)+ (τ − 2)(x*_{1}^{}*y*_{2}^{} *+ y*_{1}^{}*x*_{2}^{}*)*]^{T}*(x*_{2}^{} *+ y*_{2}^{}*)*

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

+√
*μ*1

*x*^{}_{1}*+ y*1^{}−*[2(x*_{1}^{}*x*_{2}^{}*+ y*_{1}^{}*y*_{2}^{}*)+ (τ − 2)(x*_{1}^{}*y*_{2}^{}*+ y*_{1}^{}*x*_{2}^{}*)*]^{T}*(x*_{2}^{}*+ y*_{2}^{}*)*

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

*.*

(26)
*Since λ*2*(w) >0 and w*2*(x, y)*= 0, the first term on the right-hand side of (26) is
*differentiable at (x*^{}*, y*^{}*)= (x, y). Now, we claim that the second term is o(x*^{}−
*x + y*^{}*− y), i.e., it is differentiable at (x, y) with zero gradient. To see this, no-*
*tice that w*2*(x, y)= 0, and hence μ*1*= x*^{}^{2}*+ y*^{}^{2}*+ (τ − 2)x*^{}*, y*^{}* − 2(x*_{1}^{}*x*_{2}^{}+
*y*_{1}^{}*y*_{2}^{}*)+ (τ − 2)(x*^{}_{1}*y*^{}_{2}*+ y*_{1}^{}*x*^{}_{2}*), viewed as a function of (x*^{}*, y*^{}*), is differentiable at*
*(x*^{}*, y*^{}*)= (x, y). Moreover, μ*1*= λ*1*(w)= 0 when (x*^{}*, y*^{}*)= (x, y). Thus, the first-*
*order Taylor’s expansion of μ*1*at (x, y) yields*

*μ*1 *= O(x*^{}*− x + y*^{}*− y).*

*Also, since w*2*(x, y)*= 0, by the product and quotient rules for differentiation, the
function

*x*_{1}^{}*+ y*1^{}−*[2(x*_{1}^{}*x*_{2}^{}*+ y*^{}_{1}*y*_{2}^{}*)+ (τ − 2)(x*_{1}^{}*y*_{2}^{}*+ y*^{}_{1}*x*_{2}^{}*)*]^{T}*(x*_{2}^{}*+ y*^{}_{2}*)*

*2(x*_{1}^{}*x*_{2}^{} *+ y*^{}_{1}*y*_{2}^{}*)+ (τ − 2)(x*_{1}^{}*y*_{2}^{} *+ y*^{}_{1}*x*_{2}^{}*)* (27)
*is differentiable at (x*^{}*, y*^{}*)= (x, y), and it has value 0 at (x*^{}*, y*^{}*)= (x, y) due to*

*x*1*+ y*1−*[x*1*x*_{2}*+ y*1*y*_{2}*+ (τ − 2)x*1*y*_{2}]^{T}*(x*_{2}*+ y*2*)*

*x*1*x*2*+ y*1*y*2*+ (τ − 2)x*1*y*2

*= x*1*− x*_{2}^{T} *w*2

*w*2*+ y*1*− y*_{2}^{T} *w*2

*w*2*= 0.*

Hence, the function in (27) is O(x^{}*− x + y*^{}*− y) in magnitude, which together*
*with μ*1*= O(x*^{}*− x + y*^{}*− y) shows that the second term on the right-hand side*

of (26) is
*O*

*(x*^{}*− x + y*^{}*− y)*^{3/2}

*= o*

*x*^{}*− x + y*^{}*− y*
*.*

*Thus, we have shown that ψ**τ* *is differentiable at (x, y). Moreover, we see that*
2∇ψ*τ**(x, y)*is the sum of the gradient of*x*^{}^{2}*+ y*^{}^{2}*+ (τ − 2)x*^{}*, y*^{}* + x*^{}*+ y*^{}^{2}
and the gradient of the first term on the right-hand side of (26), evaluated at (x^{}*, y*^{}*)*=
*(x, y).*

The gradient of*x*^{}^{2}*+ y*^{}^{2}*+ (τ − 2)x*^{}*, y*^{}* + x*^{}*+ y*^{}^{2}*with respect to x*^{},
*evaluated at (x*^{}*, y*^{}*)= (x, y), is 2x + (τ − 2)y + 2(x + y). The derivative of the first*
term on the right-hand side of (26) with respect to x^{}_{1}*, evaluated at (x*^{}*, y*^{}*)= (x, y),*
works out to be

√ 1
*λ*2*(w)*

*x*1+*τ*− 2
2 *y*1

+

*x*2+*τ*− 2
2 *y*2

T

*w*_{2}

*w*2

×

*x*_{1}*+ y*1*+ (x*2*+ y*2*)*^{T} *w*2

*w*2

+
*λ*_{2}*(w)*

1+ *(x*2+^{τ}^{−2}_{2} *y*2*)*^{T}*(x*2*+ y*2*)*

*x*1*x*2*+ y*1*y*2*+ (τ − 2)x*1*y*2

− *w*^{T}_{2}*(x*2*+ y*2*)· w*_{2}^{T}*(x*2+^{τ}^{−2}_{2} *y*2*)*

*x*1*x*_{2}*+ y*1*y*_{2}*+ (τ − 2)x*1*y*_{2}* · w*2^{2}

= *2(x*1+^{τ}^{−2}_{2} *y*_{1}*)(x*_{1}*+ y*1*)*

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

*x*_{1}^{2}*+ y*_{1}^{2}*+ (τ − 2)x*1*y*1*,*

where the equality follows from Lemma3.1. Similarly, the gradient of the first term
on the right of (26) with respect to x^{}_{2}*, evaluated at (x*^{}*, y*^{}*)= (x, y), works out to be*

√ 1
*λ*2*(w)*

*x*2+*τ*− 2
2 *y*2

+

*x*1+*τ*− 2
2 *y*1

*w*2

*w*2

×

*x*1*+ y*1*+ (x*2*+ y*2*)*^{T} *w*_{2}

*w*2

+
*λ*_{2}*(w)*

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

*x*1*x*2*+ y*1*y*2*+ (τ − 2)x*1*y*2

*.*− *w*^{T}_{2}*(x*2*+ y*2*)· (x*1+^{τ}^{−2}_{2} *y*1*)w*2

*x*1*x*2*+ y*1*y*2*+ (τ − 2)x*1*y*2* · w*2^{2}

= 2*(2x*1*+ (τ − 2)y*1*)x*2+^{τ}_{2}*(x*1*+ y*1*)y*2

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

*.*

Then, combining the last two gradient expressions yields that

2∇*x**ψ*_{τ}*(x, y)*

*= 2x + (τ − 2)y + 2(x + y) −*

2

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

0

− 2

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

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

*.*

*Using the fact that x*1*y*2*= y*1*x*2*and noting that φ**τ* can be simplified as the one in
(23) under this case, we readily rewrite the above expression for∇*x**ψ*_{τ}*(x, y)*in the
form of (25). By symmetry,∇*y**ψ*_{τ}*(x, y)*also holds as the form of (25).
Proposition3.2*shows that ψ**τ* is differentiable with a computable gradient. To
*establish the continuity of the gradient of ψ**τ* *or the smoothness of ψ**τ*, we need the
following two crucial technical lemmas whose proofs are provided in theAppendix.

* Lemma 3.2 For any x= (x*1

*, x*2

*), y= (y*1

*, y*2

*)*∈ R × R

^{n}^{−1}

*, if w*2

*= 0, then*

*x*_{1}+*τ* − 2
2 *y*_{1}

*+ (−1)*^{i}

*x*_{2}+*τ* − 2
2 *y*_{2}

T

*w*2

*w*2

2

≤

*x*_{2}+*τ*− 2
2 *y*_{2}

*+ (−1)*^{i}

*x*_{1}+*τ* − 2
2 *y*_{1}

*w*2

*w*2

^{2} *≤ λ**i**(w)*

*for i= 1, 2. Furthermore, these relations also hold when interchanging x and y.*

**Lemma 3.3 For all (x, y) satisfying (x**− y)^{2}*+ τ(x ◦ y) ∈ int(K*^{n}*), we have that*

L*x*+* ^{τ−2}*2

*y*

*L*

^{−1}

_{z}*F* *≤ C,* L

*y*+* ^{τ−2}*2

*x*

*L*

^{−1}

_{z}*F**≤ C,* (28)

*where C > 0 is a constant independent of x, y and τ , and*·*F* *denotes the Frobenius*
*norm.*

**Proposition 3.3 The function ψ***τ* *defined by (9) is smooth everywhere on*R* ^{n}*× R

*.*

^{n}*Proof By Proposition*3.2

*and the symmetry of x and y in∇ψ*

*τ*, it suffices to show that∇

*x*

*ψ*

_{τ}*is continuous at every (a, b)*∈ R

*× R*

^{n}

^{n}*. If (a− b)*

^{2}

*+ τ(a ◦ b) ∈ int(K*

^{n}*),*the conclusion has been shown in Proposition3.2. We next consider the other two cases.

*Case (1): (a, b)= (0, 0). By Proposition*3.2, we need to show that∇*x**ψ*_{τ}*(x, y)*

*→ 0 as (x, y) → (0, 0). If (x − y)*^{2}*+ τ(x ◦ y) ∈ int(K*^{n}*), then*∇*x**ψ*_{τ}*(x, y)*is given
by (24), whereas if (x, y)*= (0, 0) and (x −y)*^{2}*+τ(x ◦y) /∈ int(K*^{n}*), then*∇*x**ψ*_{τ}*(x, y)*
is given by (25). Notice that

*L*_{x}_{+}*τ*−2

2 *y**L*^{−1}* _{z}* and

*x*

_{1}+

^{τ}^{−2}

_{2}

*y*

_{1}

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

*are bounded with bound independent of x, y and τ , using the continuity of φ**τ**(x, y)*
immediately yields the desired result.

*Case (2): (a, b)= (0, 0) and (a − b)*^{2}*+ τ(a ◦ b) /∈ int(K*^{n}*). We will show that*

∇*x**ψ**τ**(x, y)*→ ∇*x**ψ**τ**(a, b)by the two subcases: (2a) (x, y)= (0, 0) and (x − y)*^{2}+
*τ (x◦y) /∈ int(K*^{n}*)and (2b) (x−y)*^{2}*+τ(x ◦y) ∈ int(K*^{n}*). In subcase (2a),*∇*x**ψ*_{τ}*(x, y)*
is given by (25). Noting that the right-hand side of (25) is continuous at (a, b), the
desired result follows.

Next, we prove that∇*x**ψ**τ**(x, y)*→ ∇*x**ψ**τ**(a, b)* in subcase (2b). From (24), we
have that

∇*x**ψ**τ**(x, y)*=

*x*+*τ*− 2
2 *y*

*− L*_{x}_{+}* ^{τ}*−2

2 *y**L*^{−1}_{z}*(x+ y) − φ**τ**(x, y).* (29)
*On the other hand, since (a, b)= (0, 0) and (a − b)*^{2}*+ τ(a ◦ b) /∈ int(K*^{n}*),*

*a*^{2}*+ b*^{2}*+ (τ − 2)a*^{T}*b= 2(a*1*a*2*+ b*1*b*2*)+ (τ − 2)(a*1*b*2*+ b*1*a*2*) = 0, (30)*
and moreover from (20) it follows that

*a*^{2}*+ b*^{2}*+ (τ − 2)a*^{T}*b*= 2

*a*^{2}_{1}*+ b*^{2}1*+ (τ − 2)a*1*b*1

= 2

*a*2^{2}*+ b*2^{2}*+ (τ − 2)a*_{2}^{T}*b*_{2}

= 2(a_{1}*a*_{2}*+ b*1*b*_{2}*)+ (τ − 2)a*1*b*_{2}. (31)

Using the equalities in (31), it is not hard to verify that
*a*_{1}+^{τ}^{−2}_{2} *b*_{1}

*a*_{1}^{2}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*_{1}

*(a− b)*^{2}*+ τ(a ◦ b)**1/2*

*= a +τ*− 2
2 *b.*

This together with the expression of∇*x**ψ*_{τ}*(a, b)*given by (25) yields

∇*x**ψ**τ**(a, b)*=

*a*+*τ*− 2
2 *b*

− *a*_{1}+^{τ}^{−2}_{2} *b*_{1}

*a*_{1}^{2}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*_{1}

*(a+ b) − φ**τ**(a, b).* (32)

Comparing (29) with (32), we see that if we wish to prove∇*x**ψ*_{τ}*(x, y)*→ ∇*x**ψ*_{τ}*(a, b)*
*as (x, y)→ (a, b), it suffices to show that*

*L*_{x}_{+}*τ*−2

2 *y**L*^{−1}_{z}*(x+ y) →* *a*_{1}+^{τ}^{−2}_{2} *b*_{1}

*a*^{2}_{1}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*_{1}

*(a+ b),* (33)

which is also equivalent to proving the following three relations

*L*_{x}_{+}*τ*−2
2 *y**L*^{−1}_{z}

*x*+*τ*− 2
2 *y*

→ *a*_{1}+^{τ−2}_{2} *b*_{1}

*a*_{1}^{2}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*_{1}

*a*+*τ*− 2
2 *b*

*,* (34)

*L*_{y}_{+}*τ*−2
2 *x**L*^{−1}_{z}

*y*+*τ*− 2
2 *x*

→ *b*1+^{τ}^{−2}_{2} *a*1

*a*_{1}^{2}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*1

*b*+*τ*− 2
2 *a*

*,* (35)

4*− τ*

2 *L*_{x}_{−y}*L*^{−1}_{z}

*y*+*τ*− 2
2 *x*

→

4*−τ*

2 *(a*1*− b*1*)*

*a*_{1}^{2}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*1

*b*+*τ*− 2
2 *a*

*.* (36)

*By the symmetry of x and y in (34) and (35), we only prove (34) and (36). Let*

*(ζ*_{1}*, ζ*_{2}*):= L*_{x}_{+}* ^{τ}*−2
2

*y*

*L*

^{−1}

_{z}

*x*+*τ*− 2
2 *y*

*,*

(37)
*(ξ*1*, ξ*2*):= L**x**−y**L*^{−1}_{z}

*y*+*τ* − 2
2 *x*

*.*

Then showing (34) and (36) reduces to proving the following relations hold as
*(x, y)→ (a, b):*

*ζ*_{1}→ *(a*1+^{τ}^{−2}_{2} *b*1*)*^{2}

*a*^{2}_{1}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*1

*,*

(38)
*ζ*_{2}→ *a*1+^{τ}^{−2}_{2} *b*1

*a*^{2}_{1}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*1

*a*_{2}+*τ*− 2
2 *b*_{2}

*,*

*ξ*1→ *(a*1*− b*1*)(b*1+^{τ}^{−2}_{2} *a*1*)*

*a*^{2}_{1}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*1

*,*

(39)
*ξ*_{2}→ *(a*1*− b*1*)*

*a*^{2}_{1}*+ b*^{2}_{1}*+ (τ − 2)a*1*b*1

*b*_{2}+*τ*− 2
2 *a*_{2}

*.*

To verify (38), we take (x, y) sufficiently near to (a, b). By (30), we may assume
*that w*2*= w*2*(x, y)= 0. Let s = (s*1*, s*_{2}*)= x +*^{τ}^{−2}_{2} *y. Using (14) and (37), we can*
calculate that

*ζ*1= 1
*det(z)*

*s*_{1}^{2}*z*1*− 2s*1*s*_{2}^{T}*z*2+*det(z)*

*z*_{1} *s*2^{2}+*(z*_{2}^{T}*s*2*)*^{2}
*z*_{1}

*,*

=*s*2^{2}

*z*1 +*(s*_{1}*z*_{1}*− s*_{2}^{T}*z*_{2}*)*^{2}

*z*1*det(z)* *,* (40)

*ζ*2= 1
*det(z)*

*s*1*z*1*s*2*− z*^{T}2*s*2*s*2*− s*^{2}1*z*2+*s*_{1}*det(z)*
*z*_{1} *s*2+*s*_{1}

*z*_{1}*z*^{T}_{2}*s*2*z*2

= *s*1

*z*1

*s*_{2}+*(s*_{1}*z*_{1}*− s*_{2}^{T}*z*_{2}*)*
*det(z)*

*s*_{2}−*s*1

*z*1

*z*_{2}

*.* (41)