Computational Optimization and Applications, vol. 45, pp. 581-606, 2010

**A one-parametric class of merit functions for the second-order** **cone complementarity problem**

Jein-Shan Chen ^{1}
Department of Mathematics
National Taiwan Normal University

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

Shaohua Pan

School of Mathematical Sciences South China University of Technology

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

January 10, 2007

(ﬁrst revised September 3, 2007) (ﬁnal revised April 21, 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 properties 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.*

**Key words. Second-order cone, complementarity, merit function, Jordan product.**

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Oﬃce. The author’s work is partially supported by National Science Council of Taiwan.

**AMS subject classifications. 26B05, 26B35, 90C33, 65K05**

**1** **Introduction**

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

*F (ζ)∈ K, G(ζ) ∈ K, ⟨F (ζ), G(ζ)⟩ = 0,*(1) where

*⟨·, ·⟩ is the Euclidean inner product, F : IR*

^{n}*→ IR*

^{n}*and G : IR*

^{n}*→ IR*

*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})

*∈ IR × IR*

^{n}

^{i}

^{−1}*| ∥x*2

*∥ ≤ x*1

}

*,*

with*∥ · ∥ denoting the Euclidean norm and K*^{1} denoting the set of nonnegative reals IR_{+}.
*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* *∈ IR** ^{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) [9, 10]

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

minimize *g(x)*

*subject to Ax = b,* *x∈ K,* (4)

*where g : IR*^{n}*→ IR is a convex twice continuously diﬀerentiable function, A ∈ IR*^{m}* ^{×n}* has

*full row rank and b∈ IR*

*. From [6], we know that the KKT conditions of (4), which are suﬃcient but not necessary for optimality, can be reformulated as (1) with*

^{m}*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∈ IR*

^{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 constrained quadratic programs and linear programs; see [1, 15] and references therein.

There have been various methods proposed for solving SOCPs and SOCCPs. They
include the interior-point methods [2, 3, 16, 17, 21], the non-interior smoothing Newton
methods [5, 8], and the smoothing-regularization method [12]. Recently, there was an
alternative method [6] based on reformulating the SOCCP as an unconstrained mini-
*mization problem. In that approach, it aims to ﬁnd a function ψ : IR*^{n}*× IR*^{n}*→ IR*+

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

*ζ*min*∈IR*^{n}*f (ζ) := ψ(F (ζ), G(ζ)).*

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

*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} : IR^{n}*× IR*^{n}*→ IR** ^{n}* is the vector-valued FB function deﬁned by

*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 vectors. The*

*function ψ*

_{FB}was studied in [6] 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} : IR^{n}*× IR*^{n}*→ IR** ^{n}* is the vector-valued natural residual function given by

*ϕ*

_{NR}

*(x, y) := x− (x − y)*+

with (*·)*+ meaning the projection in the Euclidean norm onto *K. The function ϕ*NR was
studied in [8, 12] which is involved in smoothing methods for the SOCCP. Compared 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}* : IR

^{n}*× IR*

^{n}*→ IR*

*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 continuously*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*∈IR*^{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 condition
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 [13] 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 alternative for ψ*

_{FB}

*if a suitable τ is*selected.

Throughout this paper, IR^{n}*denotes the space of n-dimensional real column vectors,*
and IR^{n}^{1}*× · · · × IR*^{n}* ^{m}* is identiﬁed with IR

^{n}^{1}

^{+}

^{···+n}

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

*, . . . , x*

*m*)

*∈ IR*

^{n}^{1}

*× · · · × IR*

^{n}*is viewed as a column vector in IR*

^{m}

^{n}^{1}

^{+}

^{···+n}

^{m}*. The notation I denotes an identity matrix*of suitable dimension, and int(

*K*

*) denotes the interior of*

^{n}*K*

*. For any diﬀerentiable*

^{n}*mapping F : IR*

^{n}*→ IR*

*,*

^{m}*∇F (x) ∈ IR*

^{n}

^{×m}*denotes the transposed Jacobian 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 α and 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})

*∈ IR × IR*

^{n}

^{−1}*| x*1

*>∥x*2

*∥*

^{}}

*.*

*For any x = (x*_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR × IR*^{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}*∈ IR*^{n}*. For any x = (x*_{1}*, x*_{2})*∈ IR × IR*^{n}^{−1}*, the determinant*

*of x is deﬁned by det(x) := x*^{2}_{1}*− ∥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}) *∈ IR × IR*^{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})*∈ IR × IR** ^{n−1}*,
let

*L** _{x}* :=

[ *x*_{1} *x*^{T}_{2}
*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* *∈ IR** ^{n}*, but generally

*L*

^{2}

_{x}*= L*

_{x}*L*

_{x}*̸= L*

*x*

^{2}

*and L*

^{−1}

_{x}*̸= L*

*x*

^{−1}*. If L*

_{x}*is invertible, then the inverse of L*

*is given by*

_{x}*L*^{−1}* _{x}* = 1

*det(x)*

*x*1 *−x** ^{T}*2

*−x*2

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

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

*.* (14)

*We next recall from [8] that each x = (x*1*, x*2)*∈ IR × IR*^{n}* ^{−1}* admits a spectral factor-
ization, associated with

*K*

*, of the form*

^{n}*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 IR^{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 [8]. 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∈ IR*^{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**

**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

*Property 2.1 (a) and (b), ϕ*

_{τ}*and ψ*

_{τ}*are well-deﬁned since for any x, y∈ IR*

*, there has*

^{n}*(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 [8], 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}*. Squaring 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 [8], it then follows that x∈ K*

^{n}*, y∈ K*

*and*

^{n}*⟨x, y⟩ = 0.*

*2*

*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})

*∈ IR × IR*

^{n}*, let*

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

*z = (z*_{1}*, z*_{2}*) = z(x, y) :=* ^{[}*(x− y)*^{2}*+ τ (x◦ y)*^{]}^{1/2}*.* (16)
*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,*

*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}*).* (17)
*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 *, z*_{2} *= z*_{2}*(x, y) =*

√

*λ*_{2}*(w)−*^{√}*λ*_{1}*(w)*

2 *w*¯_{2}*,* (18)
where ¯*w*_{2} := _{∥w}^{w}^{2}

2*∥* *if w*_{2} *̸= 0 and otherwise ¯w*_{2} is any vector in IR^{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 [6, Lemma
3.2].

**Lemma 3.1 For any x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR × IR*^{n}^{−1}*, if w /∈ int(K*^{n}*), then*
*x*^{2}_{1} = *∥x*2*∥*^{2}*, y*_{1}^{2} = *∥y*2*∥*^{2}*, x*_{1}*y*_{1} *= x*^{T}_{2}*y*_{2}*, x*_{1}*y*_{2} *= y*_{1}*x*_{2}; (19)
*x*^{2}_{1}*+ 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** ^{T}*2

*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 [6, Lemma 3.2] yields

(

*x*_{1} +*τ* *− 2*
2 *y*_{1}

)2

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

2

*, y*^{2}_{1} =*∥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*^{2}_{1} =*∥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 have

*∥x∥*

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

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

2*∥* *= x*_{2} by the
*symmetry of x and y in w. The veriﬁcations are straightforward by (20) and x*_{1}*y*_{2} *= y*_{1}*x*_{2}
*2*

*By Lemma 3.1, when w /∈ int(K*^{n}*), the spectral values of w are calculated as follows:*

*λ*1*(w) = 0,* *λ*2*(w) = 4*^{(}*x*^{2}_{1}*+ y*^{2}_{1} *+ (τ* *− 2)x*1*y*1

)

*.* (22)

*If (x, y)̸= (0, 0) also holds, then using equations (18), (20) and (22) yields that*
*z*_{1}*(x, y) =*

√

*x*^{2}_{1}*+ 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*^{2}_{1}*+ y*^{2}_{1}*+ (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*^{2}_{1}*+ y*^{2}_{1} *+ (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*^{2}_{1}*+ y*^{2}_{1} *+ (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)* *∈ IR*^{n}*×*
IR^{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) =* ^{[}*L*_{y+}^{τ}*−2*

2 *x**L*^{−1}_{z}*− I*^{]}*ϕ*_{τ}*(x, y);* (24)

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

*∇**x**ψ*_{τ}*(x, y) =*

*x*_{1}+^{τ}^{−2}_{2} *y*_{1}

√

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

*ϕ*_{τ}*(x, y),*

*∇**y**ψ*_{τ}*(x, y) =*

*y*_{1}+ ^{τ}^{−2}_{2} *x*_{1}

√

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

*ϕ*_{τ}*(x, y).* (25)

**Proof. Case (1): (x, y) = (0, 0). For any u = (u**_{1}*, u*_{2}*), v = (v*_{1}*, v*_{2})*∈ IR×IR*^{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})*∥*

*≤ 2∥u∥*^{2}+ 2*∥v∥*^{2}+ 3*|τ − 2|∥u∥∥v∥ ≤ 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 [4, 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}

*)*

^{′}*∈ IR × IR*

^{n}*, it is easy to verify that*

^{−1}*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 ∈ IR** ^{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 diﬀerentiable at (x*

^{′}*, y*

^{′}*) = (x, y). By Lemma 3.1,*

*w*2

*= 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*

*)*

^{′}*∈ IR*

^{n}*× IR*

^{n}*suﬃciently 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, notice 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** ^{T}*2

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

√

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

where the equality follows from Lemma 3.1. 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*^{2}_{1}*+ 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*^{2}_{1}*+ y*_{1}^{2}*+ (τ* *− 2)x*1*y*_{1}
0

]

*−* 2

√

*x*^{2}_{1}*+ 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).* *2*

*Proposition 3.2 shows that ψ** _{τ}* is diﬀerentiable with a computable gradient. To estab-

*lish 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 appendix.

**Lemma 3.2 For any x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR × IR*^{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 IR*^{n}*× IR*^{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)∈ IR*^{n}*× IR*^{n}*. If (a− b)*^{2}*+ τ (a◦ b) ∈ int(K** ^{n}*), the
conclusion has been shown in Proposition 3.2. 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*

*), then*

^{n}*∇*

*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*^{2}_{1}*+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*

*). In subcase (2a),*

^{n}*∇*

*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** ^{T}*2

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

(

*(a− b)*^{2}*+ τ (a◦ b)*^{)}^{1/2}*= a +τ* *− 2*
2 *b.*