to appear in Abstract and Applied Analysis, 2013

**Nonsingularity conditions for FB system of reformulating** **nonlinear second-order cone programming**

^{1}

Shaohua Pan* ^{†}*, Shujun Bi

*and Jein-Shan Chen*

^{‡}

^{§}June 2, 2012

**Abstract. This paper is a counterpart of [2]. Speciﬁcally, for a locally optimal solution**
to the nonlinear second-order cone programming (SOCP), under Robinson’s constraint
qualiﬁcation, we establish the equivalence among the following three conditions: the
nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for
the Karush-Kuhn-Tucker conditions, the strong second-order suﬃcient condition and
constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

**Key words: nonlinear second-order cone programming; FB nonsmooth system; non-**
singularity; Clarke’s generalized Jacobian; strong regularity.

**1** **Introduction**

The nonlinear second-order cone programming (SOCP) problem can be stated as

*ζ*min*∈IR*^{n}*f (ζ)*

*s.t.* *h(ζ) = 0,* (1)

*g(ζ)∈ K,*

*where f : IR*^{n}*→ IR, h : IR*^{n}*→ IR*^{m}*and g : IR*^{n}*→ IR** ^{n}* are given twice continuously
diﬀerentiable functions, and

*K is the Cartesian product of some second-order cones, i.e.,*

*K := K*^{n}^{1} *× K*^{n}^{2} *× · · · × K*^{n}^{r}

1This work was supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001), the Fundamental Research Funds for the Central Universities, and Project of Liaoning Innovative Research Team in UniversityWT2010004

*†*Department of Mathematics, South China University of Technology, Guangzhou, China (shh-
pan@scut.edu.cn).

*‡*Department of Mathematics, South China University of Technology, Guangzhou, China (beami-
lan@163.com).

*§*Corresponding author. Member of Mathematics Division, National Center for Theoretical Sciences,
Taipei Oﬃce. The author’s work is supported by National Science Council of Taiwan, Department of
Mathematics, National Taiwan Normal University, Taipei, Taiwan 11677 (jschen@math.ntnu.edu.tw).

*with n*_{1} +*· · · + n**r* *= n and* *K*^{n}* ^{j}* being the second-order cone (SOC) in IR

^{n}*deﬁned by*

^{j}*K*

^{n}*:={*

^{j}*(x**j1**, x**j2*)*∈ IR × IR*^{n}^{j}^{−1}*| x**j1* *≥ ∥x**j2**∥*}
*.*

By introducing a slack variable to the second constraint, the SOCP (1) is equivalent to

*ζ,x*min*∈IR*^{n}*f (ζ)*

*s.t.* *h(ζ) = 0,* (2)

*g(ζ)− x = 0, x ∈ K.*

In this paper, we will concentrate on this equivalent formulation of problem (1).

*Let L : IR*^{n}*× IR*^{n}*× IR*^{m}*× IR*^{n}*× K → IR be the Lagrangian function of problem (2)*
*L(ζ, x, µ, s, y) := f (ζ) +⟨µ, h(ζ)⟩ + ⟨g(ζ) − x, s⟩ − ⟨x, y⟩,*

and denote by *N**K**(x) the normal cone of* *K at x in the sense of convex analysis [19]:*

*N**K**(x) =*

{ *{d ∈ IR** ^{n}*:

*⟨d, z − x⟩ ≤ 0 ∀z ∈ K} if x ∈ K,*

*∅* *if x /∈ K.*

Then the Karush-Kuhn-Tucker (KKT) conditions for (2) take the following form
*J**ζ,x**L(ζ, x, µ, s, y) = 0, h(ζ) = 0, g(ζ)− x = 0 and − y ∈ N*_{K}*(x),* (3)
where *J**ζ,x**L(ζ, x, µ, s, y) is the derivative of L at (ζ, x, µ, s, y) with respect to (ζ, x).*

*Recall that ϕ*^{soc} is an SOC complementarity function associated with the cone *K if*
*ϕ*^{soc}*(x, y) = 0* *⇐⇒ x ∈ K, y ∈ K, ⟨x, y⟩ = 0 ⇐⇒ −y ∈ N**K**(x).* (4)
*With an SOC complementarity function ϕ*^{soc} associated with*K, we may reformulate the*
KKT optimality conditions in (3) as the following nonsmooth system:

*E(ζ, x, µ, s, y) :=*

*J**ζ,x**L(ζ, x, µ, s, y)*
*h(ζ)*
*g(ζ)− x*
*ϕ*^{soc}*(x, y)*

* = 0.* (5)

The most popular SOC complementarity functions include the vector-valued natural residual (NR) function and Fischer-Burmeister (FB) function, respectively, deﬁned as

*ϕ*_{NR}*(x, y) := x− Π*_{K}*(x− y)* *∀x, y ∈ IR** ^{n}*
and

*ϕ*_{FB}*(x, y) := (x + y)−*√

*x*^{2} *+ y*^{2} *∀x, y ∈ IR*^{n}*,* (6)

where Π* _{K}*(

*·) is the projection operator onto the closed convex cone K, x*

^{2}

*= x◦ x means*

*the Jordan product of x and itself, and*

*√*

*x denotes the unique square root of x* *∈ K.*

*It turns out that the FB SOC complementarity function ϕ*_{FB} enjoys almost all favorable
*properties of the NR SOC complementarity function ϕ*_{NR} (see [22]). Also, the squared
*norm of ϕ*_{FB} induces a continuously diﬀerentiable merit function with globally Lipschitz
continuous derivative [6, 7]. This greatly facilitates the globalization of the semismooth
Newton method [16, 17] for solving the FB nonsmooth system of KKT conditions:

*E*_{FB}*(ζ, x, µ, s, y) :=*

*J**ζ,x**L(ζ, x, µ, s, y)*
*h(ζ)*
*g(ζ)− x*
*ϕ*_{FB}*(x, y)*

* = 0.* (7)

Recently, with the help of [3, Theorem 30] and [5, Lemma 11], Wang and Zhang [23]

gave a characterization for the strong regularity of the KKT point of the SOCP (1) via the nonsingularity study of Clarke’s Jacobian of the NR nonsmooth system

*E*_{NR}*(ζ, x, µ, s, y) :=*

*J**ζ,x**L(ζ, x, µ, s, y)*
*h(ζ)*

*g(ζ)− x*
*ϕ*_{NR}*(x, y)*

* = 0.* (8)

They showed that the strong regularity of the KKT point, the nonsingularity of Clarke’s
*Jacobian of E*_{NR} at the KKT point, and the strong second order suﬃcient condition and
constraint nondegeneracy [3], are all equivalent. These nonsingularity conditions are
better structured than those of [14] for the nonsingularity of the B-subdiﬀerential of the
NR system. Then, it is natural to ask: is it possible to obtain a characterization for the
strong regularity of the KKT point by studying the nonsingularity of Clarke’s Jacobian
*of E*_{FB}. Note that up to now one even does not know whether the B-subdiﬀerential of
the FB system is nonsingular or not without the strict complementarity assumption.

In this work, for a locally optimal solution to the nonlinear SOCP (2), under Robin-
son’s constraint qualiﬁcation, we show that the strong second-order suﬃcient condition
and constraint nondegeneracy introduced in [3], the nonsingularity of Clarke’s Jacobian
*of E*_{FB} at the KKT point, and the strong regularity of the KKT point are equivalent to
each other. This, on the one hand, gives a new characterization for the strong regularity
of the KKT point; and on the other hand, provides a mild condition to guarantee the
quadratic convergence rate of the semismooth Newton method [16, 17] for the FB sys-
tem. Note that parallel results are obtained recently for the FB system of the nonlinear
semideﬁnite programming (see [2]), however, we do not duplicate them. As will be seen
in Section 3 and Section 4, the analysis techniques here are totally diﬀerent from those in
[2]. It seems hard to put them together in a uniﬁed framework under the Euclidean Jor-
dan algebra. The main reason causing this is due to completely diﬀerent analysis when

dealing with the Clarke Jacobians associated with FB SOC complementarity function and FB semideﬁnite cone complementarity function.

*Throughout this paper, I denotes an identity matrix of appropriate dimension, IR*^{n}*(n > 1) denotes the space of n-dimensional real column vectors, and IR*^{n}^{1}*× · · · × IR*^{n}* ^{r}* is
identiﬁed with IR

^{n}^{1}

^{+}

^{···+n}

^{r}*. Thus, (x*

_{1}

*, . . . , x*

*)*

_{r}*∈ IR*

^{n}^{1}

*× · · · × IR*

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

^{r}

^{n}^{1}

^{+}

^{···+n}

^{r}**. The notations int**

*K*

^{n}**, bd**

*K*

^{n}**and bd**

^{+}

*K*

*denote the interior, the boundary, and the boundary excluding the origin of*

^{n}*K*

^{n}*, respectively. For any x*

*∈ IR*

*,*

^{n}*we write x*

*≽*

*K*

^{n}*0 (respectively, x≻*

*K*

^{n}*0) if x*

*∈ K*

^{n}*(respectively, x*

**∈ intK***). For any*

^{n}*given real symmetric matrix A, we write A*

*≽ 0 (respectively, A ≻ 0) if A is positive*semideﬁnite (respectively, positive deﬁnite). In addition,

*J*

*ω*

*f (ω) and*

*J*

*ωω*

^{2}

*f (ω) denote*the derivative and the second order derivative, respectively, of a twice diﬀerentiable

*function f with respect to the variable ω.*

**2** **Preliminary results**

First we recall from [11] the deﬁnition of Jordan product and spectral factorization.

**Deﬁnition 2.1 The Jordan product of x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR× IR*^{n}^{−1}*is given by*
*x◦ y := (⟨x, y⟩, x*1*y*_{2}*+ y*_{1}*x*_{2}*).* (9)
Unlike scalar or matrix multiplication, the Jordan product is not associative in general.

*The identity element under this product is e := (1, 0, . . . , 0)*^{T}*∈ IR*^{n}*, i.e., e◦ x = x for all*
*x∈ IR*^{n}*. For each x = (x*_{1}*, x*_{2})*∈ IR × IR*^{n}* ^{−1}*, we deﬁne the associated arrow matrix by

*L** _{x}*:=

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

]

*.* (10)

*Then it is easy to verify that L*_{x}*y = x◦ y for any x, y ∈ IR*^{n}*. Recall that each x =*
*(x*_{1}*, x*_{2})*∈ IR × IR*^{n}* ^{−1}* admits a spectral factorization, associated with

*K*

*, of the form*

^{n}*x = λ*_{1}*(x)u*^{(1)}_{x}*+ λ*_{2}*(x)u*^{(2)}_{x}*,* (11)
*where λ*_{1}*(x), λ*_{2}*(x)* *∈ IR and u*^{(1)}*x* *, u*^{(2)}*x* *∈ IR** ^{n}* are the spectral values and the associated

*spectral vectors of x, respectively, with respect to the Jordan product, deﬁned by*

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

( 1

(*−1)*^{i}*x*˜2

)

*for i = 1, 2,* (12)
with ˜*x*_{2} = _{∥x}^{x}^{2}

2*∥* *if x*_{2} *̸= 0 and otherwise being any vector in IR*^{n}* ^{−1}* satisfying

*∥˜x*2

*∥ = 1.*

**Deﬁnition 2.2 The determinant of a vector x**∈IR^{n}*is deﬁned as det(x) := λ*_{1}*(x)λ*_{2}*(x),*
*and a vector x is said to be invertible if its determinant det(x) is nonzero.*

By the formula of spectral factorization, it is easy to compute that the projection of
*x∈ IR** ^{n}* onto the closed convex cone

*K*

*, denoted by Π*

^{n}

_{K}

^{n}*(x), has the expression*

Π_{K}^{n}*(x) = max(0, λ*1*(x))u*^{(1)}_{x}*+ max(0, λ*2*(x))u*^{(2)}_{x}*.*

Deﬁne*|x| := 2Π**K*^{n}*(x)− x. Then, using the expression of Π**K*^{n}*(x), it follows that*

*|x| = |λ*1*(x)|u*^{(1)}*x* +*|λ*2*(x)|u*^{(2)}*x* *.*
*The spectral factorization of the vectors x, x*^{2}*,* *√*

*x and the matrix L** _{x}* have various
interesting properties (see [12]). We list several properties that we will use later.

**Property 2.1 For any x = (x**_{1}*, x*_{2})*∈IR×IR*^{n}^{−1}*with spectral factorization (11), we have*
**(a) x**^{2} *= λ*^{2}_{1}*(x)u*^{(1)}*x* *+ λ*^{2}_{2}*(x)u*^{(2)}*x* *∈ K*^{n}*.*

**(b) If x**∈ K^{n}*, then 0* *≤ λ*1*(x)≤ λ*2*(x) and* *√*

*x =*√

*λ*_{1}*(x)u*^{(1)}*x* +√

*λ*_{2}*(x)u*^{(2)}*x* *.*
**(c) If x****∈ intK**^{n}*, then 0 < λ*_{1}*(x)≤ λ*2*(x) and L*_{x}*is invertible with*

*L*^{−1}* _{x}* = 1

*det(x)*

*x*_{1} *−x*^{T}_{2}

*−x*2

*det(x)*

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

* .* (13)

**(d) L**_{x}*≽ 0 (respectively, L**x* *≻ 0) if and only if x ∈ K*^{n}*(respectively, x ∈ intK*

^{n}*).*

*The following lemma states a result for the arrow matrices associated with x, y* *∈ IR*^{n}*and z* *≽**K*^{n}

√*x*^{2}*+ y*^{2}, which will be used in the next section to characterize an important
*property for the elements of Clarke’s Jacobian of ϕ*_{FB} at a general point.

**Lemma 2.1 For any given x, y**∈ IR^{n}*and z* *≻**K*^{n}*0, if z*^{2} *≽**K*^{n}*x*^{2}*+ y*^{2}*, then*
*[L*^{−1}_{z}*L*_{x}*L*^{−1}_{z}*L** _{y}*]

2 *≤ 1,*

*where* *∥A∥*2 *means the spectral norm of a real matrix A. Consequently, it holds that*

*∥L*^{−1}*z* *L*_{x}*△u + L*^{−1}*z* *L*_{y}*△v∥ ≤*√

*∥△u∥*^{2}+*∥△v∥*^{2} *∀△u, △v ∈ IR*^{n}*.*
**Proof. Let A = [L**^{−1}_{z}*L*_{x}*L*^{−1}_{z}*L** _{y}*]. From [12, Proposition 3.4], it follows that

*AA*^{T}*= L*^{−1}_{z}*(L*^{2}_{x}*+ L*^{2}_{y}*)L*^{−1}_{z}*≼ L*^{−1}*z* *L*^{2}_{z}*L*^{−1}_{z}*= I.*

This shows that *∥A∥*2 *≤ 1, and the ﬁrst part follows. Note that for any ξ ∈ IR** ^{2n}*,

*∥Aξ∥*^{2} *= ξ*^{T}*A*^{T}*Aξ≤ λ*max*(A*^{T}*A)∥ξ∥*^{2} *≤ ∥ξ∥*^{2}*.*

*By letting ξ = (△u, △v) ∈ IR*^{n}*× IR** ^{n}*, we immediately obtain the second part.

*2*

*The following two lemmas state the properties of x, y with x*

^{2}

*+ y*

^{2}

**∈ bdK***which are often used in the subsequent sections. The proof of Lemma 2.2 is given in [6, Lemma 2].*

^{n}**Lemma 2.2 For any x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR ×IR*^{n}^{−1}*with x*^{2}*+ y*^{2}**∈bdK**^{n}*, we have*
*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}*.*

**Lemma 2.3 For any x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR × IR*^{n}^{−1}*, let w = (w*_{1}*, w*_{2}*) := x*^{2}*+ y*^{2}*.*
**(a) If w****∈ bdK**^{n}*, then for any g = (g*_{1}*, g*_{2}*), h = (h*_{1}*, h*_{2})*∈ IR × IR*^{n}^{−1}*, it holds that*

*(x*_{1}*x*_{2}*+ y*_{1}*y*_{2})^{T}*(x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}*) = (x*^{2}_{1}*+ y*_{1}^{2}*)(x*^{T}*g + y*^{T}*h).*

**(b) If w****∈ bd**^{+}*K*^{n}*, then the following four equalities hold*
*x*_{1}*w*_{2}

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

*∥w*2*∥* *= x*_{1}*,* *y*_{1}*w*_{2}

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

*∥w*2*∥* *= y*_{1};
*and consequently the expression of ϕ*_{FB}*(x, y) can be simpliﬁed as*

*ϕ*_{FB}*(x, y) =*

*x*_{1}*+ y*_{1}*−*√

*x*^{2}_{1}*+ y*_{1}^{2}
*x*_{2}*+ y*_{2}*−* *x*_{1}*x*_{2}*+ y*_{1}*y*_{2}

√*x*^{2}_{1}*+ y*_{1}^{2}

* .* (14)

**Proof. (a) The result is direct by the equalities of Lemma 2.2 since x**^{2}*+ y*^{2} **∈ bdK*** ^{n}*.

*(b) Since w*

**∈ bd**^{+}

*K*

^{n}*, we must have w*2

*= 2(x*1

*x*2

*+ y*1

*y*2)

*̸= 0. Using Lemma 2.2,*

*w*

_{2}

*= 2(x*

_{1}

*x*

_{2}

*+ y*

_{1}

*y*

_{2}) and

*∥w*2

*∥ = w*1

*= 2(x*

^{2}

_{1}

*+ y*

^{2}

_{1}), we easily obtain the ﬁrst part. Note

*that ϕ*

_{FB}

*(x, y) = (x + y)−√*

*w. Using Property 2.1(b) and Lemma 2.2 yields (14).* *2*
*When x, y* **∈ bdK*** ^{n}*satisfy the complementary condition, we have the following result.

**Lemma 2.4 For any given x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2}) *∈ IR × IR*^{n}^{−1}*, if x, y* **∈ bdK**^{n}*and*

*⟨x, y⟩ = 0, then there exists a constant α > 0 such that x*1 *= αy*_{1} *and x*_{2} =*−αy*2*.*
**Proof. Since x, y****∈ bdK**^{n}*, we have that x*_{1} =*∥x*2*∥ and y*1 =*∥y*2*∥, and consequently,*

0 = *⟨x, y⟩ = x*1*y*_{1}*+ x*^{T}_{2}*y*_{2} =*∥x*2*∥∥y*2*∥ + x** ^{T}*2

*y*

_{2}

*.*

*This means that there exists α > 0 such that x*2 =*−αy*2*, and then x*1 *= αy*1. *2*
Next we recall from [21] the strong regularity for a solution of generalized equation

0*∈ ϕ(z) + N**D**(z),* (15)

*where ϕ is a continuously diﬀerentiable mapping from a ﬁnite dimensional real vector*
space*Z to itself, D is a closed convex set in Z, and N**D**(z) is the normal cone of D at z.*

As will be shown in Sec. 4, the KKT condition (3) can be written in the form of (15).

**Deﬁnition 2.3 We say that ¯**z is a strongly regular solution of the generalized equation*(15) if there exist neighborhood* *B of the origin 0 ∈ Z and V of ¯z such that for every*
*δ∈ B, the linearized generalized equation δ ∈ ϕ(¯z) + J**z**ϕ(¯z)(z− ¯z) + N**D**(z) has a unique*
*solution in* *V, denoted by z**V**(δ), and the mapping z** _{V}* :

*B → V is Lipschitz continuous.*

To close this section, we recall from [8] Clarke’s (generalized) Jacobian of a locally
*Lipschitz mapping. Let S* *⊂ IR*^{n}*be an open set and Ξ : S* *→ IR** ^{n}* be a locally Lipschitz

*continuous function on S. By Rademacher’s theorem, Ξ is almost everywhere F(r´*echet)-

*diﬀerentiable in S. We denote by S*

_{Ξ}

*the set of points in S where Ξ is F-diﬀerentiable.*

*Then Clarke’s Jacobian of Ξ at y is deﬁned by ∂Ξ(y) := conv{∂**B**Ξ(y)}, where “conv”*

*means the convex hull, and B-subdiﬀerential ∂*_{B}*Ξ(y), a name coined in [18], has the form*

*∂*_{B}*Ξ(y) :=*

{

*V : V = lim*

*k**→∞**J**y**Ξ(y*^{k}*), y*^{k}*→ y, y*^{k}*∈ S*Ξ

}
*.*

For the concept of (strong) semismoothness, please refer to the literature [16, 17].

*Unless otherwise stated, in the rest of this paper, for any x∈ IR*^{n}*(n > 1), we write*
*x = (x*1*, x*2*) where x*1 *is the ﬁrst component of x, and x*2 is a column vector consisting
*of the remaining n−1 entries of x. For any x = (x*1*, x*_{2}*), y = (y*_{1}*, y*_{2})*∈ IR × IR*^{n}* ^{−1}*, let

*w = w(x, y) := x*^{2}*+ y*^{2}*,* *w*˜_{2} := *w*_{2}

*∥w*2*∥* *if w*_{2} *̸= 0 and z = z(x, y) =*√

*w(x, y).* (16)

**3** **Directional derivative and B-subdiﬀerential**

*The function ϕ*_{FB} is directionally diﬀerentiable everywhere by [22, Corollary 3.3]. But,
to the best of our knowledge, the expression of its directional derivative is not given
in the literature. In this section, we derive its expression, and then prove that the B-
*subdiﬀerential of ϕ*_{FB} at a general point coincides with that of its directional derivative
function at the origin. Throughout this section, we assume that *K = K** ^{n}*.

**Proposition 3.1 For any given x = (x**_{1}*, x*_{2}*), y = (y*_{1}*, y*_{2}) *∈ IR × IR*^{n}^{−1}*, the directional*
*derivative ϕ*^{′}

FB*((x,y); (g,h)) of ϕ*_{FB} *at (x, y) with the direction (g,h) has the following form.*

**(a) If (x, y) = (0, 0), then ϕ**^{′}

FB*((x, y); (g, h)) = ϕ*_{FB}*(g, h).*

**(b) If x**^{2}*+ y*^{2} **∈ intK**^{n}*, then ϕ*^{′}

FB*((x, y); (g, h)) = (I− L*^{−1}_{z}*L*_{x}*) g + (I− L*^{−1}_{z}*L*_{y}*) h.*

**(c) If x**^{2}*+ y*^{2} **∈ bd**^{+}*K*^{n}*, then*
*ϕ*^{′}

FB*((x, y); (g, h)) = (g + h)−* *φ(g, h)*
2

( 1

*− ˜w*_{2}
)

+*x*^{T}_{2}*g*_{2}*+ y*^{T}_{2}*h*_{2}
2√

*x*^{2}_{1}*+ y*^{2}_{1}
( 0

˜
*w*_{2}

)

*−* 1

2√

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

( *x*^{T}*g + y*^{T}*h*

*2x*_{1}*g*_{2} *+ g*_{1}*x*_{2}*+ 2y*_{1}*h*_{2}*+ h*_{1}*y*_{2}
)

(17)

*where g = (g*_{1}*, g*_{2}*), h = (h*_{1}*, h*_{2})*∈ IR × IR*^{n}^{−1}*, and φ : IR*^{n}*× IR*^{n}*→ IR is deﬁned by*
*φ(g, h) :=*

√*(x*_{1}*g*_{1}*− x** ^{T}*2

*g*

_{2}

*+ y*

_{1}

*h*

_{1}

*−y*

*2√*

^{T}*h*

_{2})

^{2}+

*∥x*1

*h*

_{2}

*− h*1

*x*

_{2}

*+g*

_{1}

*y*

_{2}

*− y*1

*g*

_{2}

*∥*

^{2}

*x*^{2}_{1} *+ y*^{2}_{1} *.* (18)

**Proof. Part (a) is immediate by noting that ϕ**_{FB} is a positively homogeneous function.

Part (b) is due to [12, Proposition 5.2]. We next prove part (c) by two subcases as shown
*below. In the rest of proof, we let λ*_{1}*, λ*_{2} *with λ*_{1} *≤ λ*2 *denote the spectral values of w.*

*Since w = x*^{2} *+ y*^{2} **∈ bd**^{+}*K*^{n}*, we have w*2 *̸= 0, and from Lemma 2.3(b) it follows that*
*w*_{1} =*∥w*2*∥ = 2∥x*1*x*_{2}*+ y*_{1}*y*_{2}*∥ = 2∥x*^{2}1*w*e_{2}*+ y*_{1}^{2}*w*e_{2}*∥ = 2(x*^{2}1*+ y*_{1}^{2}*),*

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

**(c.1): (x + tg)**^{2} *+ (y + th)*^{2} **∈ bd**^{+}*K*^{n}*for suﬃciently small t > 0. In this case, from*
*Lemma 2.3(b), we know that ϕ*_{FB}*(x + tg, y + th) has the following expression*

*(x*_{1}*+ y*_{1}*) + t(g*_{1}*+ h*_{1})*−*√

*(x*_{1}*+ tg*_{1})^{2}*+ (y*_{1}*+ th*_{1})^{2}

*(x*_{2} *+ y*_{2}*) + t(g*_{2}*+ h*_{2})*−(x*_{1}*+ tg*_{1}*)(x*_{2}*+ tg*_{2}*) + (y*_{1}*+ th*_{1}*)(y*_{2}*+ th*_{2})

√*(x*_{1}*+ tg*_{1})^{2}*+ (y*_{1}*+ th*_{1})^{2}

* .*

*Let [ϕ*_{FB}*(x, y)]*1 *be the ﬁrst element of ϕ*_{FB}*(x, y) and [ϕ*_{FB}*(x, y)]*2 be the vector consisting
*of the rest n− 1 components of ϕ*FB*(x, y). By the above expression of ϕ*_{FB}*(x + tg, y + th),*

lim*t**↓0*

*[ϕ*_{FB}*(x + tg, y + th)]*1*− [ϕ*FB*(x, y)]*1

*t*

*= (g*1*+ h*1)*− lim*

*t**↓0*

√*(x*_{1}*+ tg*_{1})^{2}*+ (y*_{1}*+ th*_{1})^{2}*−*√

*x*^{2}_{1}*+ y*_{1}^{2}
*t*

*= (g*_{1}*+ h*_{1})*−* *x*_{1}*g*_{1}*+ y*_{1}*h*_{1}

√*x*^{2}_{1}*+ y*_{1}^{2}
and

lim*t**↓0*

*[ϕ*_{FB}*(x + tg, y + th)]*2*− [ϕ*FB*(x, y)]*2

*t*

*= (g*2*+ h*2)*− lim*

*t**↓0*

[

*(x*_{1}*+ tg*_{1}*)(x*_{2}*+ tg*_{2}*) + (y*_{1}*+ th*_{1}*)(y*_{2}*+ th*_{2})
*t*√

*(x*_{1}*+ tg*_{1})^{2}*+ (y*_{1}*+ th*_{1})^{2} *−x*_{1}*x*_{2} *+ y*_{1}*y*_{2}
*t*√

*x*^{2}_{1}*+ y*_{1}^{2}
]

*= (g*2*+ h*2)*−* *g*_{1}*x*_{2}*+ x*_{1}*g*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}

√*x*^{2}_{1}*+ y*^{2}_{1}

*− lim*

*t**↓0*

[

*x*_{1}*x*_{2}*+ y*_{1}*y*_{2}
*t*√

*(x*_{1}*+ tg*_{1})^{2}*+ (y*_{1}*+ th*_{1})^{2} *−x*_{1}*x*_{2}*+ y*_{1}*y*_{2}
*t*√

*x*^{2}_{1}*+ y*^{2}_{1}
]

*= (g*_{2}*+ h*_{2})*−* *g*_{1}*x*_{2}*+ x*_{1}*g*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}

√*x*^{2}_{1}*+ y*^{2}_{1} +*(x*_{1}*x*_{2}*+ y*_{1}*y*_{2}*)(x*_{1}*g*_{1}*+ y*_{1}*h*_{1})
*(x*^{2}_{1}*+ y*_{1}^{2})√

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

*= (g*_{2}*+ h*_{2})*−* *x*_{1}*g*_{2}*+ y*_{1}*h*_{2}

√*x*^{2}_{1}*+ y*_{1}^{2}

*where the last equality is using x*_{1}*y*_{2} *= y*_{1}*x*_{2} by Lemma 2.2. The above two limits imply
*ϕ*^{′}_{FB}*((x, y); (g, h)) = (g + h)−* *x*1

√*x*^{2}_{1}*+ y*_{1}^{2}*g−* *y*1

√*x*^{2}_{1}*+ y*^{2}_{1}*h.* (19)

**(c.2): (x + tg)**^{2} *+ (y + th)*^{2} **∈ intK**^{n}*for suﬃciently small t > 0. Let u = (u*_{1}*, u*_{2}) :=

*(x + tg)*^{2}*+ (y + th)*^{2} *with the spectral values µ*1*, µ*2. An elementary calculation gives
*u*_{1} = *∥x + tg∥*^{2}+*∥y + th∥*^{2} *= w*_{1}*+ 2t(x*^{T}*g + y*^{T}*h) + t*^{2}(*∥g∥*^{2}+*∥h∥*^{2}*),* (20)
*u*_{2} *= 2(x*_{1}*+ tg*_{1}*)(x*_{2} *+ tg*_{2}*) + 2(y*_{1}*+ th*_{1}*)(y*_{2}*+ th*_{2})

*= w*_{2}*+ 2t(x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}*) + 2t*^{2}*(g*_{1}*g*_{2}*+ h*_{1}*h*_{2}*).* (21)
*Also, since w*_{2} *̸= 0, applying the Taylor formula of ∥ · ∥ at w*2 and Lemma 2.3(a) yields

*∥u*2*∥ = ∥w*2*∥ +w*^{T}_{2}*(u*_{2}*− w*2)

*∥w*2*∥* *+ o(t) =∥w*2*∥ + 2t(x*^{T}*g + y*^{T}*h) + o(t).* (22)
*Now using the deﬁnition of ϕ*_{FB} *and noting that λ*_{1} *= 0 and w*_{2} *̸= 0, we have that*

*ϕ*_{FB}*(x + tg, y + th)− ϕ*FB*(x, y)*

*= (x + tg + y + th)−√*

*u− (x + y) +√*
*w*

*= t(g + h)−*

*√µ*_{1}+*√*

*µ*_{2}*−√*
*λ*_{2}

*√* 2

*µ*_{2}*− √µ*1

2

*u*_{2}

*∥u*2*∥* *−*

*√λ*_{2}
2

*w*_{2}

*∥w*2*∥*

* ,*

which in turn implies that

*ϕ*^{′}

FB*((x, y); (g, h)) = (g + h)−*

lim*t**↓0*

*√µ*_{1}+*√*

*µ*_{2}*−√*
*λ*_{2}
*2t*

lim*t**↓0*

*(√µ*_{2} *− √µ*1

*2t*

*u*_{2}

*∥u*2*∥* *−*

*√λ*_{2}
*2t*

*w*_{2}

*∥w*2*∥*
)

* .* (23)

We ﬁrst calculate lim_{t}_{↓0}^{√}^{µ}^{2}^{−}

*√**λ*2

*t* . Using equations (20) and (22), it is easy to see that
*µ*2*− λ*2 *= (u*1*− w*1) + (*∥u*2*∥ − ∥w*2*∥) = 4t(x*^{T}*g + y*^{T}*h) + o(t),*

and consequently, lim

*t**↓0*

*√µ*_{2}*−√*
*λ*_{2}

*t* = lim

*t**↓0*

*µ*_{2}*− λ*2

*t* *·* 1

*√µ*2+*√*
*λ*2

= *x*^{T}*g + y*^{T}*h*
2*√*

*λ*2

= *x*^{T}*g + y*^{T}*h*

√*x*^{2}_{1}*+ y*_{1}^{2} *.* (24)

We next calculate lim_{t↓0}^{√}_{t}^{µ}^{1}*. Since w*_{1} *− ∥w*2*∥ = 0, using (20)-(21) and Lemma 2.3(a),*
*µ*_{1} *= (u*_{1}*− w*1)*− (∥u*2*∥ − ∥w*2*∥) = (u*1*− w*1)*−* *∥u*2*∥*^{2}*− ∥w*2*∥*^{2}

*∥u*2*∥ + ∥w*2*∥*

*= 2t(x*^{T}*g + y*^{T}*h)− 4tw*_{2}^{T}*(x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ y*_{1}*h*_{2} *+ h*_{1}*y*_{2})

*∥u*2*∥ + ∥w*2*∥* *+ t*^{2}(*∥g∥*^{2}+*∥h∥*^{2})

*−4t*^{2}*∥g*1*x*_{2}*+ x*_{1}*g*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}*∥*^{2}

*∥u*2*∥ + ∥w*2*∥* *− 4t*^{2}*w*_{2}^{T}*(g*_{1}*g*_{2}*+ h*_{1}*h*_{2})

*∥u*2*∥ + ∥w*2*∥* *+ o(t*^{2})

*= 2t(x*^{T}*g + y*^{T}*h)− 8t(x*^{2}_{1}*+ y*^{2}_{1}*)(x*^{T}*g + y*^{T}*h)*

*∥u*2*∥ + ∥w*2*∥* *+ t*^{2}(*∥g∥*^{2}+*∥h∥*^{2}*) + o(t*^{2})

*−4t*^{2}*∥g*1*x*2*+ x*1*g*2*+ y*1*h*2*+ h*1*y*2*∥*^{2}

*∥u*2*∥ + ∥w*2*∥* *− 8t*^{2}*(x*1*x*2*+ y*1*y*2)^{T}*(g*1*g*2*+ h*1*h*2)

*∥u*2*∥ + ∥w*2*∥* *. (25)*
Using*∥w*2*∥ = 2(x*^{2}_{1}*+ y*_{1}^{2}) and (22), we simplify the sum of the ﬁrst two terms in (25) as

*2t(x*^{T}*g + y*^{T}*h)−* *4t∥w*2*∥(x*^{T}*g + y*^{T}*h)*

*∥u*2*∥ + ∥w*2*∥* *= 2t(x*^{T}*g + y*^{T}*h)∥u*2*∥ − ∥w*2*∥*

*∥u*2*∥ + ∥w*2*∥*

= *4t*^{2}*(x*^{T}*g + y*^{T}*h)*^{2}

*∥u*2*∥ + ∥w*2*∥* *+ o(t*^{2}*).*

Then, from equation (25) and *∥w*2*∥ = 2(x*^{2}1*+ y*_{1}^{2}), we obtain that
lim

*t**↓0*

*µ*_{1}

*t*^{2} = *(x*^{2}_{1}*+ y*_{1}^{2})(*∥g∥*^{2} +*∥h∥*^{2})*− ∥g*1*x*_{2}*+ x*_{1}*g*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}*∥*^{2}
*x*^{2}_{1} *+ y*^{2}_{1}

+*(x*^{T}*g + y*^{T}*h)*^{2}*− 2(x*1*x*2*+ y*1*y*2)^{T}*(g*1*g*2*+ h*1*h*2)

*x*^{2}_{1}*+ y*^{2}_{1} *.* (26)

We next make simpliﬁcation for the numerator of the right hand side of (26). Note that
*(x*^{2}_{1}*+ y*_{1}^{2})(*∥g∥*^{2}+*∥h∥*^{2})*− ∥g*1*x*2*+ x*1*g*2*+ y*1*h*2*+ h*1*y*2*∥*^{2}

*= (x*^{2}_{1}*+ y*_{1}^{2})(*∥g∥*^{2}+*∥h∥*^{2})*− ∥g*1*x*_{2}*+ x*_{1}*g*_{2}*∥*^{2}*− ∥y*1*h*_{2}*+ h*_{1}*y*_{2}*∥*^{2}

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

*= x*^{2}_{1}*∥h∥*^{2}*+ y*^{2}_{1}*∥g∥*^{2}*− 2x*1*g*_{1}*x*^{T}_{2}*g*_{2}*− 2y*1*h*_{1}*y*_{2}^{T}*h*_{2}*− 2(g*1*x*_{2}*+ x*_{1}*g*_{2})^{T}*(y*_{1}*h*_{2}*+ h*_{1}*y*_{2})
and

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

*= (x*_{1}*g*_{1}*+ x*^{T}_{2}*g*_{2})^{2}*+ (y*_{1}*h*_{1}*+ y*_{2}^{T}*h*_{2})^{2}*+ 2x*^{T}*gy*^{T}*h− 2(x*1*x*_{2}*+ y*_{1}*y*_{2})^{T}*(g*_{1}*g*_{2}*+ h*_{1}*h*_{2})

*= (x*1*g*1)^{2}*+ (x*^{T}_{2}*g*2)^{2}*+ (y*1*h*1)^{2}*+ (y*_{2}^{T}*h*2)^{2}*+ 2x*^{T}*gy*^{T}*h− 2x*1*h*1*x*^{T}_{2}*h*2*− 2g*1*y*1*g*_{2}^{T}*y*2*.*
Therefore, adding the last two equalities and using Lemma 2.2 yields that

*(x*^{2}_{1}*+ y*_{1}^{2})(*∥g∥*^{2}+*∥h∥*^{2})*− ∥g*1*x*2*+ x*1*g*2*+ y*1*h*2*+ h*1*y*2*∥*^{2}
*+(x*^{T}*g + y*^{T}*h)*^{2}*− 2(x*1*x*_{2}*+ y*_{1}*y*_{2})^{T}*(g*_{1}*g*_{2}*+ h*_{1}*h*_{2})

*= (x*^{2}_{1}*∥h∥*^{2}*− 2x*1*h*_{1}*x*^{T}_{2}*h*_{2}*) + (y*_{1}^{2}*∥g∥*^{2}*− 2g*1*y*_{1}*g*_{2}^{T}*y*_{2}) +(

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

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

*+ 2x*^{T}*gy*^{T}*h− 2(g*1*x*_{2}*+ x*_{1}*g*_{2})^{T}*(y*_{1}*h*_{2}*+ h*_{1}*y*_{2})

= *∥x*1*h*_{2}*− h*1*x*_{2}*∥*^{2}+*∥g*1*y*_{2}*− y*1*g*_{2}*∥*^{2} *+ (x*_{1}*g*_{1} *− x** ^{T}*2

*g*

_{2})

^{2}

*+ (y*

_{1}

*h*

_{1}

*− y*

*2*

^{T}*h*

_{2})

^{2}

*+2(g*

_{1}

*x*

_{1}

*+ g*

_{2}

^{T}*x*

_{2}

*)(y*

_{1}

*h*

_{1}

*+ y*

_{2}

^{T}*h*

_{2})

*− 2(g*1

*x*

_{2}

*+ x*

_{1}

*g*

_{2})

^{T}*(y*

_{1}

*h*

_{2}

*+ h*

_{1}

*y*

_{2})

= *∥x*1*h*2*− h*1*x*2*∥*^{2}+*∥g*1*y*2*− y*1*g*2*∥*^{2} *+ (x*1*g*1 *− x** ^{T}*2

*g*2)

^{2}

*+ (y*1

*h*1

*− y*

*2*

^{T}*h*2)

^{2}

*+2(x*

_{1}

*h*

_{2}

*− h*1

*x*

_{2})

^{T}*(g*

_{1}

*y*

_{2}

*− g*2

*y*

_{1}

*) + 2(x*

_{1}

*g*

_{1}

*− x*

*2*

^{T}*g*

_{2}

*)(y*

_{1}

*h*

_{1}

*− y*

*2*

^{T}*h*

_{2})

= *∥x*1*h*_{2}*− h*1*x*_{2} *+ g*_{1}*y*_{2}*− y*1*g*_{2}*∥*^{2}*+ (x*_{1}*g*_{1} *− x** ^{T}*2

*g*

_{2}

*+ y*

_{1}

*h*

_{1}

*− y*

*2*

^{T}*h*

_{2})

^{2}

*.*

*Combining this equality with (26) and using the deﬁnition of φ in (18), we readily get*
lim*t**↓0*

*√µ*_{1}

*t* *= φ(g, h).* (27)

We next calculate lim_{t}_{↓0}

[*√**µ*2*−√µ*1

*2t*
*u*2

*∥u*^{2}*∥* *−* ^{√}_{2t}^{λ}^{2}_{∥w}^{w}^{2}_{2}* _{∥}*]

*. To this end, we also need to take a*
look at *∥w*2*∥u*2*− ∥u*2*∥w*2. From equations (20)-(21) and (22), it follows that

*∥w*2*∥u*2*− ∥u*2*∥w*2 *= 2t∥w*2*∥*[

*(x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ y*_{1}*h*_{2} *+ h*_{1}*y*_{2})*− (x*^{T}*g + y*^{T}*h) ˜w*_{2}]

*+ o(t).*

Together with equations (24) and (27), we have that lim

*t**↓0*

*[√µ*_{2}*− √µ*1

*2t*

*u*_{2}

*∥u*2*∥−*

*√λ*_{2}
*2t*

*w*_{2}

*∥w*2*∥*
]

= *− lim*

*t**↓0*

*√µ*_{1}
*2t*

*u*_{2}

*∥u*2*∥* + lim

*t**↓0*

*[√µ*_{2}
*2t*

*u*_{2}

*∥u*2*∥−*

*√λ*_{2}
*2t*

*w*_{2}

*∥w*2*∥*
]

= *− lim*

*t**↓0*

*√µ*_{1}
*2t*

*u*_{2}

*∥u*2*∥* + lim

*t**↓0*

*√µ*_{2}*−√*
*λ*_{2}
*2t*

*u*_{2}

*∥u*2*∥* + lim

*t**↓0*

*√λ*_{2}(*∥w*2*∥u*2*− ∥u*2*∥w*2)
*2t∥u*2*∥∥w*2*∥*

= *−φ(g, h)*

2 *w*˜_{2}+ *x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2}

√*x*^{2}_{1}*+ y*^{2}_{1} *−* *x*^{T}*g + y*^{T}*h*
2√

*x*^{2}_{1}*+ y*_{1}^{2}*w*˜_{2}

= *−φ(g, h)*

2 *w*˜_{2}+ *2x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ 2y*_{1}*h*_{2}*+ h*_{1}*y*_{2}
2√

*x*^{2}_{1}*+ y*^{2}_{1} *−* *x*^{T}_{2}*g*_{2}*+ y*_{2}^{T}*h*_{2}
2√

*x*^{2}_{1}*+ y*^{2}_{1} *w*˜_{2}*,*

*where the last equality is using x*_{1}*w*˜_{2} *= x*_{2} *and y*_{1}*w*˜_{2} *= y*_{2}. Combining with (23), (24)
*and (27), a suitable rearrangement shows that ϕ*^{′}

FB*((x, y); (g, h)) has the expression (17).*

*Finally, we show that when (x + tg)*^{2}*+ (y + th)*^{2} **∈ bd**^{+}*K*^{n}*for suﬃciently small t > 0,*
the formula in (17) reduces to the one in (19). Indeed, an elementary calculation yields

*λ*_{1}(

*(x + tg)*^{2}*+ (y + th)*^{2})

= [

*∥x + tg∥*^{2}+*∥y + th∥*^{2}]2

*− 4 ∥(x*1*+ tg*_{1}*)(x*_{2}*+ tg*_{2}*) + (y*_{1}*+ th*_{1}*)(y*_{2}*+ th*_{2})*∥*^{2}

*= 4t*^{2}*φ(g, h)*

√

*x*^{2}_{1}*+ y*^{2}_{1} *+ 4t*^{3}*(x*^{T}*g + y*^{T}*h)(∥g∥*^{2}+*∥h∥*^{2})

*−8t*^{2}*(x*_{1}*g*_{2}*+ g*_{1}*x*_{2}*+ y*_{1}*h*_{2}*+ h*_{1}*y*_{2})^{T}*(g*_{1}*g*_{2} *+ h*_{1}*h*_{2})
*+t*^{4}[

(*∥g∥*^{2}+*∥h∥*^{2})^{2}*− 2∥g*1*g*_{2}*+ h*_{1}*h*_{2}*∥*^{2}]

*= 4t*^{2}*φ(g, h)*

√

*x*^{2}_{1}*+ y*^{2}_{1} *+ o(t*^{2}*).*