SIAM Journal on Optimization, vol. 21, pp. 1392-1417, 2011

**Nonsingularity conditions for FB system of nonlinear SDPs**

^{1}

Shujun Bi* ^{†}*, Shaohua Pan

*and Jein-Shan Chen*

^{‡}

^{§}February 14, 2011

(revised on July 29, 2011, August 21, 2011)

**Abstract. For a locally optimal solution to the nonlinear semideﬁnite programming,**
under Robinson’s constraint qualiﬁcation, we show that the nonsingularity of Clarke’s
Jacobian of the Fischer-Burmeister (FB) nonsmooth system is equivalent to the strong
regularity of the Karush-Kuhn-Tucker point. Consequently, from Sun’s paper (Mathe-
matics of Operations Research, vol. 31, pp. 761-776, 2006), the semismooth Newton
method applied to the FB system may attain the locally quadratic convergence under
the strong second order suﬃcient condition and constraint nondegeneracy.

**Key words: nonlinear semideﬁnite programming; the FB system; Clarke’s Jacobian;**

nonsingularity; strong regularity.

**AMS subject classiﬁcations. 90C22, 90C25, 90C31, 65K05**

**1** **Introduction**

Let*X be a ﬁnite dimensional real vector space endowed with an inner product ⟨·, ·⟩ and*
its induced norm *∥ · ∥. Consider the nonlinear semideﬁnite programming (NLSDP)*

min

*x**∈X* *f (x)*

*s.t.* *h(x) = 0,* (1)

*g(x)∈ S** ^{n}*+

*,*

1This work was supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001).

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

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

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

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

^{n}*is the linear space of all n× n real symmetric matrices, and S*

^{n}_{+}is the cone

*of all n× n positive semideﬁnite matrices. By introducing a slack variable X ∈ S*

*+ for*

^{n}*the conic constraint g(x)∈ S*

*+, we can rewrite the NLSDP (1) as follows:*

^{n}min

*(x,X)**∈X×S*^{n}*f (x)*

*s.t.* *h(x) = 0,* (2)

*g(x)− X = 0,*
*X∈ S** ^{n}*+

*.*

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

The Karush-Kuhn-Tucker (KKT) condition for the NLSDP (2) takes the form
*J**x,X**L(x, X, µ, S, Y ) = 0, h(x) = 0, g(x)− X = 0, −Y ∈ N*_{S}^{n}_{+}*(X),* (3)
*where the Lagrangian function L :X × S*^{n}*× IR*^{m}*× S*^{n}*× S*^{n}*→ IR is deﬁned by*

*L(x, X, µ, S, Y ) := f (x) +⟨µ, h(x)⟩ + ⟨S, g(x) −X⟩ − ⟨X, Y ⟩,*

*J**x,X**L(x, X, µ, S, Y ) is the derivative of L at (x, X, µ, S, Y ) with respect to (x, X), and*
*N*_{S}^{n}_{+}*(X) denotes the normal cone of* S^{n}_{+} *at X in the sense of convex analysis [16]:*

*N*S* ^{n}*+

*(X) =*

{ *{Z ∈ S** ^{n}*:

*⟨Z, W − X⟩ ≤ 0 ∀W ∈ S*

^{n}*} if X ∈ S*

*+*

^{n}*,*

*∅* *if X /∈ S*^{n}_{+}*.*

Recall that Φ :S^{n}*× S*^{n}*→ S** ^{n}* is a semideﬁnite cone (SDC) complementarity function if

*Φ(X, Y ) = 0*

*⇐⇒ X ∈ S*

*+*

^{n}*, Y*

*∈ S*

*+*

^{n}*,*

*⟨X, Y ⟩ = 0 ⇐⇒ −Y ∈ N*S

*+*

^{n}*(X).*

Then, with an SDC complementarity function Φ, the KKT optimality conditions in (3) can be reformulated as the following nonsmooth system:

*E(x, X, µ, S, Y ) :=*

*J**x,X**L(x, X, µ, S, Y )*
*h(x)*

*g(x)− X*
*Φ(X, Y )*

* = 0.* (4)

The most popular SDC complementarity functions include the matrix-valued natural residual (NR) function and Fischer-Burmeister (FB) function, which are deﬁned as

Φ_{NR}*(X, Y ) := X− Π*S* ^{n}*+

*(X*

*− Y )*

*∀X, Y ∈ S*

*and*

^{n}Φ_{FB}*(X, Y ) := (X + Y )−√*

*X*^{2}*+ Y*^{2} *∀X, Y ∈ S*^{n}*,* (5)

respectively, where Π_{S}^{n}

+(*·) denotes the projection operator onto S** ^{n}*+. It turns out that
Φ

_{FB}has almost all favorable properties of Φ

_{NR}(see [21]). Also, the squared norm of Φ

_{FB}induces a continuously diﬀerentiable merit function whose derivative is globally Lipschitz continuous [18, 24]. This greatly facilitates the globalization of the semismooth Newton method [14, 15] for solving the FB system of (2). The FB system and the NR system

*mean E*

_{FB}

*(x, X, µ, S, Y ) = 0 and E*

_{NR}

*(x, X, µ, S, Y ) = 0, respectively, with the mappings*

*E*

_{FB}

*and E*

_{NR}

*deﬁned as in E except that Φ is speciﬁed as Φ*

_{FB}and Φ

_{NR}, respectively.

The strong regularity is one of the important concepts in sensitivity and perturbation analysis introduced by Robinson in his seminal paper [17]. For the NLSDP (1), Sun [22]

oﬀered a characterization for the strong regularity via the study of the nonsingularity of Clarke’s Jacobian of the NR system under the strong second order suﬃcient condition and constraint nondegeneracy, and established its equivalence to other characterizations discussed in a wide range of literatures. Later, for the linear SDP, Chan and Sun [3]

gained more insightful characterizations for the strong regularity via the study of the nonsingularity of Clarke’s Jacobian of the NR system, too. Then, it is natural for us to ask: is it possible to give a characterization for the strong regularity of NLSDPs by studying the nonsingularity of Clarke’s Jacobian of the FB system? Note that up to now one even does not know whether the B-subdiﬀerential of FB system is nonsingular or not without strict complementarity of locally optimal solutions.

In this work, for a locally optimal solution to the NLSDP (2), we prove that under Robinson’s constraint qualiﬁcation, the nonsingularity of Clarke’s Jacobian of the FB system is equivalent to the strong regularity of the KKT point, which by [22, Theorem 4.1] is further equivalent to the strong second order suﬃcient condition and constraint nondegeneracy. This result is of interest since, on one hand, it relates the nonsingularity of Clarke’s Jacobian of the FB system to Robinson’s strong regularity condition and, on the other hand, it allows us to obtain the quadratic convergence of the semismooth Newton method [15, 14] for the FB system without strict complementarity assumption.

In addition, it also extends the result of [9, Corollary 3.7] for the variational inequality with the polyhedral cone constraints to the setting of semideﬁnite cones. It is worthwhile to point out that [22, Theorem 4.1] plays a key role in achieving this objective.

Throughout this paper, *J**z**f (z) and* *J**zz*^{2}*f (z) denote the derivative and the second*
*order derivative, respectively, of a twice diﬀerentiable function f with respect to z, and*
*I denotes an identity operator. For any n × m real matrices A and B, ⟨A, B⟩ means*
their Frobenius inner product, and*∥A∥ denotes the norm of A induced by the Frobenius*
*inner product. For X* *∈ S*^{n}*, we write X* *≽ 0 (respectively, X ≻ 0) to mean X ∈ S** ^{n}*+

*(respectively, X* *∈ S** ^{n}*++). For a linear operator

*A, we denote by A*

*the adjoint of*

^{∗}*A, and*by

*∥A∥*2 the operator norm of

*A. For a linear operator A : S*

^{n}*→ S*

*, we write*

^{n}*A ≽ 0*(respectively,

*A ≻ 0) if ⟨W, A(W )⟩ ≥ 0 for any W ∈ S*

*(respectively,*

^{n}*⟨W, A(W )⟩ > 0*

*for any nonzero W*

*∈ S*

^{n}*). For any given sets of indices α and β, we designate by A*

*αβ*

*the submatrix of A whose row indices belong to α and column indices belong to β, and*

use *|α| to denote the number of elements in the set α.*

**2** **Preliminary results**

LetX and Y be two arbitrary ﬁnite dimensional real vector spaces each equipped with a
scalar product*⟨·, ·⟩ and its induced norm ∥·∥. Let O be an open set in X and Ξ : O → Y*
be a locally Lipschitz continuous function on the set *O. By Rademacher’s theorem, Ξ*
is almost everywhere F(r´echet)-diﬀerentiable in *O. We denote by D*Ξ the set of points
in*O where Ξ is F-diﬀerentiable. Then Clarke’s Jacobian of Ξ at x is well deﬁned [6]:*

*∂Ξ(x) := conv{∂**B**Ξ(x)},*

*where “conv” means the convex hull, and ∂**B**Ξ(x) is the B-subdiﬀerential of Ξ at x:*

*∂*_{B}*Ξ(x) :=*

{

*V : V = lim*

*k**→∞**J**x**Ξ(x*^{k}*), x*^{k}*→ x, x*^{k}*∈ D*Ξ

}
*.*

For the concepts of (strong) semismoothness, please refer to the literature [15, 14, 20].

The following matrix inequalities are used in the proof of Lemma 3.3; see Appendix.

**Lemma 2.1 For any n**× m real matrices A, B and any Z ∈ S^{n}_{+}*, it holds that*

*(A + B)*^{T}*Z(A + B)* *≼ 2(A*^{T}*ZA + B*^{T}*ZB),* (6)
*(A− B)*^{T}*Z(A− B) ≼ 2(A*^{T}*ZA + B*^{T}*ZB).* (7)
**Proof. Fix any Z***∈ S** ^{n}*+

*. Then, for any n× m real matrices A and B, we have that*

0 *≼ (A − B)*^{T}*Z(A− B) = (A*^{T}*ZA + B*^{T}*ZB)− (A*^{T}*ZB + B*^{T}*ZA),*
0 *≼ (A + B)*^{T}*Z(A + B) = (A*^{T}*ZA + B*^{T}*ZB) + (A*^{T}*ZB + B*^{T}*ZA).*

*The ﬁrst equation means that (A*^{T}*ZB + B*^{T}*ZA)≼ (A*^{T}*ZA + B*^{T}*ZB), which along with*
the second equality yields (6). The second equation implies that *−(A*^{T}*ZB + B*^{T}*ZA)* *≼*
*(A*^{T}*ZA + B*^{T}*ZB), which along with the ﬁrst equality yields (7).* *2*

**Lemma 2.2 Let X, Y***∈ S*^{n}*with X*^{2}*+ Y*^{2} *≻0. Then for any n × m real matrices A, B,*
*A*^{T}*A + B*^{T}*B− (A*^{T}*X + B*^{T}*Y )(X*^{2}*+ Y*^{2})^{−1}*(XA + Y B)≽ 0.*

**Proof. Note that A**^{T}*A + B*^{T}*B* *− (A*^{T}*X + B*^{T}*Y )(X*^{2}*+ Y*^{2})^{−1}*(XA + Y B) is the Schur*
*complement of X*^{2}*+ Y*^{2} in the following block symmetric matrix

Σ =

[ *X*^{2} *+ Y*^{2} *XA + Y B*
*(XA + Y B)*^{T}*A*^{T}*A + B*^{T}*B*

]
*.*

We only need to prove Σ*≽ 0 (see [10, Theorem 7.7.6]). For any ζ = (ζ*1*, ζ*_{2})*∈ IR*^{n}*×IR*^{m}*,*
*ζ*^{T}*Σζ* *= ζ*_{1}^{T}*(X*^{2}*+ Y*^{2}*)ζ*_{1}*+ 2ζ*_{1}^{T}*(XA + Y B)ζ*_{2}*+ ζ*_{2}^{T}*(A*^{T}*A + B*^{T}*B)ζ*_{2}

= *∥Xζ*1*+ Aζ*2*∥*^{2} +*∥Y ζ*1*+ Bζ*2*∥*^{2} *≥ 0,*
which shows that Σ*≽ 0. The proof is then complete.* *2*

*For any given X∈ S** ^{n}*, let

*L*

*X*:S

^{n}*→S*

^{n}*be the Lyapunov operator associated with X:*

*L**X**(Y ) := XY + Y X* *∀Y ∈ S*^{n}*.*

*We next study several properties of the Lyapunov operators associated with X, Y* *∈ S*^{n}*and Z* *∈ S*^{n}_{+} *with Z*^{2} *≽ X*^{2}*+ Y*^{2}. To this end, we need to establish two trace inequalities.

**Lemma 2.3 Let X, Y***∈ S*^{n}*with X* *≽ |Y |. Then, for any W ∈ S*^{n}*, it holds that*
*Trace(W XW X)≥ Trace(W Y W Y ).*

**Proof. Fix any W***∈ S** ^{n}*. By the trace property of symmetric matrices, we have that

*Trace(W XW X)− Trace(W Y W Y )*

*= Trace [W XW (X− Y )] + Trace [W (X − Y )W Y ]*

*= Trace [W (X− Y )W X] + Trace [W (X − Y )W Y ]*

*= Trace [W (X− Y )W (X + Y )] .*

*Since X* *≽ |Y |, we have W (X − Y )W ≽ 0 and X + Y ≽ 0. From [10, Theorem 7.6.3],*
*it then follows that Trace [W (X− Y )W (X + Y )] ≥ 0. The result is thus proved.* *2*
**Lemma 2.4 For any given X, Y***∈ S*^{n}*and Z* *∈ S** ^{n}*+

*satisfying Z*

*≽√*

*X*^{2}*+ Y*^{2}*, we have*
*Trace(W ZW Z)≥ Trace(W |X|W |X|) + Trace(W |Y |W |Y |)* *∀W ∈ S*^{n}*.*

**Proof. Fix any W***∈ S** ^{n}*. Applying Lemma 2.3, we readily obtain that

*Trace(W ZW Z)≥ Trace*(

*W√*

*X*^{2}*+ Y*^{2}*W√*

*X*^{2}*+ Y*^{2}
)

*.* (8)

*In addition, from [1, Theorem IX.6.1], we know that φ(A, B) := Trace(W√*
*AW√*

*B) is*
a jointly concave function on S* ^{n}*+

*× S*

*+*

^{n}*, which means that for any A*1

*, A*2

*, B*1

*, B*2

*∈ S*

*+,*

^{n}*φ*

(*A*_{1}*+ A*_{2}

2 *,B*_{1}*+ B*_{2}
2

)

*≥* 1

2*[φ(A*_{1}*, B*_{1}*) + φ(A*_{2}*, B*_{2}*)] .*

*Using this inequality with A*_{1} *= B*_{1} *= X*^{2} *and A*_{2} *= B*_{2} *= Y*^{2}, we obtain that
*2φ*

(*X*^{2}*+ Y*^{2}

2 *,X*^{2}*+ Y*^{2}
2

)

*≥ Trace(W |X|W |X|) + Trace(W |Y |W |Y |).*

*This, together with the deﬁnition of φ and inequality (8), implies the result.* *2*
The following proposition, extending the result of [8, Proposition 3.4] associated with
second-order cones to SDCs, is used to prove Proposition 2.2. Among others, Proposition
2.2 is the key to characterize the properties of Clarke’s Jacobian of Φ_{FB}; see Section 4.

**Proposition 2.1 For any given X, Y***∈ S*^{n}*and Z* *∈ S** ^{n}*+

*, the following implication holds:*

*Z*^{2} *≽ X*^{2}*+ Y*^{2} =*⇒ L*^{2}*Z* *≽ L*^{2}*X* +*L*^{2}*Y**.*

**Proof. Since Z**^{2} *≽ X*^{2}*+ Y*^{2} *and Z* *∈ S*^{n}_{+}, from [1, Proposition V.1.8] it follows that
*Z* *≽√*

*X*^{2} *+ Y*^{2}*.*

*Now choose a matrix W* *∈ S** ^{n}* arbitrarily. Then, a simple computation yields that

*⟨W, (L*^{2}*Z**− L*^{2}*X* *− L*^{2}*Y**)W⟩ = 2*[

*Trace(W ZW Z) + Trace(W*^{2}*Z*^{2})*− Trace(W XW X)*

*−Trace(W*^{2}*X*^{2})*− Trace(W*^{2}*Y*^{2})*− Trace(W Y W Y )*]

= 2[

Trace(

*W*^{2}*(Z*^{2}*− X*^{2}*− Y*^{2}))

*+ Trace(W ZW Z)*

*−Trace(W XW X) − Trace(W Y W Y )]*

*≥ 2 [Trace(W ZW Z) − Trace(W XW X) − Trace(W Y W Y )]*

*≥ 0,*

*where the ﬁrst inequality is due to Z*^{2} *≽ X*^{2} *+ Y*^{2}*, and the second one is using Z* *≽*

*√X*^{2}*+ Y*^{2} *and Lemmas 2.4 and 2.3. Since W is arbitrary in*S* ^{n}*, the result follows.

*2*

**Proposition 2.2 For any given X, Y***∈ S*^{n}*and Z* *∈ S** ^{n}*++

*, deﬁne*

*A: S*

^{n}*× S*

^{n}*→ S*

^{n}*by*

*A(△U, △V ) := L*

^{−1}

_{Z}*L*

*X*(

*△U) + L*

^{−1}

_{Z}*L*

*Y*(

*△V )*

*∀△U, △V ∈ S*

^{n}*.*

*If Z*^{2} *≽ X*^{2}*+ Y*^{2}*, then the linear operator* *A satisﬁes ∥A∥*_{2} *≤ 1, and consequently*
*
L*^{−1}*Z* *L**X*(*△U) + L*^{−1}_{Z}*L**Y*(*△V )
≤ √∥△U∥*^{2}+*∥△V ∥*^{2} *∀△U, △V ∈ S*^{n}*.* (9)
**Proof. Assume that Z**^{2} *≽X*^{2}*+ Y*^{2}. By the deﬁnition of*A and Proposition 2.1, we have*

*AA** ^{∗}* =

*L*

^{−1}*Z*(

*L*

^{2}

*X*+

*L*

^{2}

*Y*)

*L*

^{−1}*Z*

*≼ L*

^{−1}*Z*

*L*

^{2}

*Z*

*L*

^{−1}*Z*=

*I.*

This means that the largest eigenvalue of *AA** ^{∗}* is less than 1, and consequently,

*∥A∥*2 =√

*∥A*^{∗}*A∥*2 =√

*λ*_{max}(*A*^{∗}*A) =*√

*λ*_{max}(*AA** ^{∗}*)

*≤ 1.*

This completes the proof of the ﬁrst part. By the deﬁnition of operator norm, we have

*∥L*^{−1}*Z* *L**X*(*△U) + L*^{−1}*Z* *L**Y*(*△V )∥ = ∥A(△U, △V )∥ ≤ ∥A∥*_{2}*∥(△U, △V )∥.*

Together with the ﬁrst part, we prove that the inequality (9) holds. *2*
*Let α, β and γ be disjoint index sets with α∪ β ∪ γ = {1, 2, . . . , n}. Deﬁne*

*Γ(X, Y ) := (X*_{ββ}^{2} *+ Y*_{ββ}^{2} *+ X*_{βγ}*X*_{γβ}*+ Y*_{βα}*Y** _{αβ}*)

^{1/2}*∀X, Y ∈ S*

^{n}*.*(10) The following property of the function Γ will be used in the subsequent sections.

**Proposition 2.3 Let X, Y***∈ S*^{n}*be such that Γ(X, Y )≻ 0. Then for any G, H ∈ S*^{n}*,*

*∥L*^{−1}_{Γ(X,Y )}*(X*_{βγ}*G*_{γβ}*+ G*_{βγ}*X** _{γβ}*)

*∥ ≤ 2*√

*|β||γ| ∥G**γβ**∥,*

*∥L*^{−1}_{Γ(X,Y )}*(Y*_{βα}*H*_{αβ}*+ H*_{βα}*Y** _{αβ}*)

*∥ ≤ 2*√

*|β||α| ∥H**αβ**∥.*

**Proof. Let Γ(X, Y ) = Q***β**diag(λ*1*, . . . , λ*_{|β|}*)Q*^{T}_{β}*be the spectral decomposition of Γ(X, Y ),*
*where λ*_{i}*> 0 for each i. Let Q*_{γ}*and Q** _{α}* be arbitrary but ﬁxed

*|γ| × |γ| and |α| × |α|*

orthogonal matrix, respectively. Deﬁne e*X*_{βγ}*:= Q*^{T}_{β}*X*_{βγ}*Q** _{γ}* and e

*Y*

_{βα}*:= Q*

^{T}

_{β}*Y*

_{βα}*Q*

*. Then,*

_{α}*from the expression of Γ(X, Y ) and its spectral decomposition, it is easy to get that*

*λ*^{2}_{i}*≥*

∑*|γ|*

*k=1*

*X*e_{ik}^{2} +

∑*|α|*

*l=1*

*Y*e_{il}^{2} *for all i = 1, . . . ,|β|.*

This means that for 1*≤ k ≤ |γ|, 1 ≤ l ≤ |α|, 1 ≤ i ≤ |β| and 1 ≤ j ≤ |β|,*

*| eX*_{ik}*|*

*λ*_{i}*+ λ*_{j}*≤ 1,* *| eX*_{kj}*|*

*λ*_{i}*+ λ*_{j}*≤ 1,* *|eY*_{il}*|*

*λ*_{i}*+ λ*_{j}*≤ 1,* *|eY*_{lj}*|*

*λ*_{i}*+ λ*_{j}*≤ 1.* (11)
*For any G, H* *∈ S** ^{n}*, with e

*G*

_{βγ}*= Q*

^{T}

_{β}*G*

_{βγ}*Q*

*and e*

_{γ}*H*

_{βα}*= Q*

^{T}

_{β}*H*

_{βα}*Q*

*, we calculate that*

_{α}*Q*^{T}_{β}*L*^{−1}_{Γ(X,Y )}*(X*_{βγ}*G*_{γβ}*+ G*_{βγ}*X*_{γβ}*)Q** _{β}* =
[∑

_{|γ|}*k=1*( e*X**ik**G*e*kj*+ e*G**ik**X*e*kj*)
*λ*_{i}*+ λ*_{j}

]

1*≤i,j≤|β|*

*,*

*Q*^{T}_{β}*L*^{−1}_{Γ(X,Y )}*(Y*_{βα}*H*_{αβ}*+ H*_{βα}*Y*_{αβ}*)Q** _{β}* =

[∑_{|α|}

*l=1*( e*Y*_{il}*H*e* _{lj}* + e

*H*

_{il}*Y*e

*)*

_{lj}*λ*

_{i}*+ λ*

_{j}]

1≤i,j≤|β|

*.*

Using the inequalities in (11) and noting that Frobenius norm is orthogonally invariant,
from the last two equalities we obtain the desired result. *2*

*In the subsequent sections, we always use C :*S^{n}*× S*^{n}*→ S** ^{n}* to denote the function

*C(X, Y ) :=√*

*X*^{2}*+ Y*^{2} *∀X, Y ∈ S*^{n}*,* (12)
*and for any given X, Y* *∈ S*^{n}*assume that C(X, Y ) has the spectral decomposition*

*C(X, Y ) = P diag(λ*_{1}*, . . . , λ*_{n}*)P*^{T}*= P DP*^{T}*,* (13)

*where P is an n× n orthogonal matrix, and D = diag(λ*1*, . . . , λ*_{n}*) with λ*_{i}*≥ 0 for all i.*

*Deﬁne the index sets κ and β associated with the eigenvalues of C(X, Y ) by*
*κ :=* *{i : λ**i* *> 0} and β := {i : λ**i* = 0*} .*

*Then, by permuting the rows and columns of C(X, Y ) if necessary, we may assume that*
*D =*

[ *D** _{κ}* 0
0

*D*

_{β}]

=

[ *D** _{κ}* 0
0 0

]
*.*

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

The function Φ_{FB} is directionally diﬀerentiable everywhere inS^{n}*× S** ^{n}*; see [21, Corollary
2.3]. But, to our best knowledge, the expression of its directional derivative is not given
in the literature. Next we derive it and use it to show that the B-subdiﬀerential of Φ

_{FB}at a general point coincides with that of its directional derivative function at the origin.

^{2}

**Proposition 3.1 For any given X, Y***∈ S*

^{n}*, let C(X, Y ) have the spectral decomposition*

*as in (13). Then, the directional derivative Φ*

^{′}FB*((X, Y ); (G, H)) of Φ*_{FB} *at (X, Y ) with*
*the direction (G, H)∈ S*^{n}*× S*^{n}*has the following expression*

*(G + H)− P*
[*L*^{−1}*D**κ*

(*L**X*e*κκ*( e*G** _{κκ}*) +

*L*

*Y*e

*κκ*( e

*H*

*) )*

_{κκ}*D*^{−1}* _{κ}* ( e

*X*

_{κκ}*G*e

*+ e*

_{κβ}*Y*

_{κκ}*H*e

*) ( e*

_{κβ}*G*

_{βκ}*X*e

*+ e*

_{κκ}*H*

_{βκ}*Y*e

_{κκ}*)D*

^{−1}*Θ( e*

_{κ}*G, eH)*

]

*P** ^{T}* (14)

*where eX := P*^{T}*XP , eY := P*^{T}*Y P , eG := P*^{T}*GP , eH := P*^{T}*HP , and Θ is deﬁned by*
*Θ(U, V ) :=*

[

*U*_{ββ}^{2} *+ V*_{ββ}^{2} *+ U*_{βκ}*U*_{κβ}*+ V*_{βκ}*V*_{κβ}*−*

*(U*_{βκ}*X*e_{κκ}*+ V*_{βκ}*Y*e_{κκ}*)D*_{κ}* ^{−2}*( e

*X*

_{κκ}*U*

*+ e*

_{κβ}*Y*

_{κκ}*V*

*) ]*

_{κβ}*1/2*

*∀U, V ∈ S*^{n}*.* (15)
**Proof. Fix any G, H***∈ S*^{n}*. Assume that (X, Y )̸= (0, 0). Then, for any t > 0, we have*
Φ_{FB}*(X + tG, Y + tH)− Φ*FB*(X, Y ) = t(G + H)− △(t)* (16)
with

*△(t) ≡*[

*C*^{2}*(X, Y ) + t(L**X**(G) +L**Y**(H)) + t*^{2}*(G*^{2}*+ H*^{2})]*1/2*

*− C(X, Y ).*

Let e*X, eY , eG and eH be deﬁned as in the proposition. It is easy to see that*

*△(t) := P*e ^{T}*△(t)P = (D*^{2}+ f*W )*^{1/2}*− D,* (17)

2When we are preparing this manuscript, we learn that these results are obtained by Zhang, Zhang and Pang (see [26]) via the singular value decomposition. To the contrast, we achieve them indepen- dently by the eigenvalue decomposition in order to obtain Proposition 3.2.

where

*W = t*f

(*X e*e*G + eG eX + eY eH + eH eY*
)

*+ t*^{2}

(*G*e^{2}+ e*H*^{2}
)

*.*

Since e*X*^{2}+ e*Y*^{2} *= D*^{2} *and D**β* = 0, we have e*X = diag( eX**κκ**, 0) and eY = diag( eY**κκ**, 0). So,*
f*W = t*

[ *L**X*e*κκ*( e*G** _{κκ}*) +

*L*

*Y*e

*κκ*( e

*H*

*)*

_{κκ}*X*e

_{κκ}*G*e

*+ e*

_{κβ}*Y*

_{κκ}*H*e

_{κβ}*G*e

*βκ*

*X*e

*κκ*+ e

*H*

*βκ*

*Y*e

*κκ*0

]

+

[ *o(t)* *o(t)*

*o(t) t*^{2}

(*G*e^{2}* _{ββ}*+ e

*H*

_{ββ}^{2}+ e

*G*

_{βκ}*G*e

*+ e*

_{κβ}*H*

_{βκ}*H*e

*) ]*

_{κβ}*.*

By equation (17) and [24, Lemma 6.2], we know that

*△(t)*e *κκ* =*L*^{−1}_{D}* _{κ}*(f

*W*

*κκ*

*) + o(∥fW∥),*

*△(t)*e *κβ* *= D*_{κ}* ^{−1}*f

*W*

_{κβ}*+ o(∥fW∥),*f

*W*

*= e*

_{ββ}*△(t)*

^{T}*κβ*

*△(t)*e

*κβ*+ e

*△(t)*

^{2}

*ββ*

*.*

(18)

From the second equality of (18) and the expression of f*W** _{κβ}*, it follows that

*△(t)*e *κβ* *= tD*^{−1}_{κ}

(*X*e_{κκ}*G*e* _{κβ}* + e

*Y*

_{κκ}*H*e

*)*

_{κβ}*+ o(t),* (19)

and consequently,

*△(t)*e ^{T}*κβ**△(t)*e *κβ* *= t*^{2}

(*G*e_{βκ}*X*e* _{κκ}*+ e

*H*

_{βκ}*Y*e

*)*

_{κκ}*D*_{κ}^{−2}

(*X*e_{κκ}*G*e* _{κβ}* + e

*Y*

_{κκ}*H*e

*)*

_{κβ}*+ o(t*^{2}*).*

This, together with the third equation of (18) and the expression of f*W** _{ββ}*, implies that

*△(t)*e ^{2}*ββ* *= t*^{2}

(*G*e_{βκ}*G*e* _{κβ}* + e

*H*

_{βκ}*H*e

*+ e*

_{κβ}*G*

^{2}

*+ e*

_{ββ}*H*

_{ββ}^{2})

*−t*^{2}(

*G*e_{βκ}*X*e* _{κκ}*+ e

*H*

_{βκ}*Y*e

*)*

_{κκ}*D*_{κ}^{−2}

(*X*e_{κκ}*G*e* _{κβ}* + e

*Y*

_{κκ}*H*e

*)*

_{κβ}*+ o(t*^{2}*).*

*Since D** _{β}* = 0, the expression of e

*△(t) in (17) implies that e△(t)*

*ββ*

*≽ 0. Therefore,*

lim*t**↓0*

*△(t)*e *ββ*

*t* = lim

*t**↓0*

[ e*△(t)*^{2}* _{ββ}*]

^{1/2}*t* = Θ( e*G, eH).*

In addition, from the ﬁrst equation in (18) and the expression of f*W** _{κκ}*, we have

*△(t)*e *κκ* *= tL*^{−1}_{D}* _{κ}*(

*L**X*e*κκ*( e*G** _{κκ}*) +

*L*

*Y*e

*κκ*( e

*H*

*) )*

_{κκ}*+ o(t).*

Combining the last two equations with (19), we immediately obtain that

lim

*t**↓0*

*△(t)*e
*t* =

[ *L*^{−1}*D**κ*

(*L**X*e*κκ*( e*G** _{κκ}*) +

*L*

*Y*e

*κκ*( e

*H*

*) )*

_{κκ}*D*_{κ}* ^{−1}*( e

*X*

_{κκ}*G*e

*+ e*

_{κβ}*Y*

_{κκ}*H*e

*) ( e*

_{κβ}*G*

_{βκ}*X*e

*+ e*

_{κκ}*H*

_{βκ}*Y*e

_{κκ}*)D*

_{κ}*Θ( e*

^{−1}*G, eH)*

]
*.*

This, along with (16), shows that Φ^{′}

FB*((X, Y ); (G, H)) has the expression given by (14).*

*When (X, Y ) = (0, 0), by the positive homogeneity of Φ*_{FB}, we immediately have
Φ^{′}

FB*((X, Y ); (G, H)) = (G + H)−√*

*G*^{2}*+ H*^{2} = Φ_{FB}*(G, H).*

*Note that this is a special case of (14) with κ =∅. The result then follows.* *2*
Note that the function Θ in (15) is always well deﬁned since, by Lemma 2.2,

*U*_{βκ}*U*_{κβ}*+ V*_{βκ}*V*_{κβ}*− (U**βκ**X*e_{κκ}*+ V*_{βκ}*Y*e_{κκ}*)D*^{−2}* _{κ}* ( e

*X*

_{κκ}*U*

*+ e*

_{κβ}*Y*

_{κκ}*V*

*)*

_{κβ}*≽ 0*

*for all U, V* *∈ S** ^{n}*. As a consequence of Proposition 3.1, we readily obtain the following
necessary and suﬃcient characterization for the diﬀerentiable points of the function Φ

_{FB}.

**Corollary 3.1 The function Φ**_{FB}

*is F-diﬀerentiable at (X, Y ) if and only if C(X, Y )≻ 0.*

*Furthermore, when C(X, Y )≻ 0, we have for any (G, H) ∈ S*^{n}*× S*^{n}*,*

*J Φ*FB*(X, Y )(G, H) = (G + H)− L*^{−1}* _{C(X,Y )}*(

*L*

*X*

*(G) +L*

*Y*

*(H)) .*(20)

**Proof. The “if” part is direct by [1, Theorem V.3.3] or [5, Proposition 4.3]. We next**prove the “only if” part by contradiction. Suppose that Φ

_{FB}

*is F-diﬀerentiable at (X, Y ),*

*but C(X, Y )*

*≻ 0 does not hold. Then |β| ̸= ∅. Since Φ*FB

*is F-diﬀerentiable at (X, Y ),*Φ

^{′}FB*((X, Y ); (·, ·)) is a linear operator. But, letting (G*1*, H*_{1}*), (G*_{2}*, H*_{2})*∈ S*^{n}*× S** ^{n}* be such

*that G*1

*= G*2

*= 0, H*1

*= diag(0, I*

_{|β|}*) and H*2 =

*−H*1, we obtain that

0 = Φ^{′}_{FB}*((X, Y ); (G*1*, H*1*) + (G*2*, H*2))

= Φ^{′}_{FB}*((X, Y ); (G*_{1}*, H*_{1})) + Φ^{′}_{FB}*((X, Y ); (G*_{2}*, H*_{2}))

= *−2P*

( 0 0
*0 I*_{|β|}

)
*P*^{T}*,*

which is a contradiction. This contradiction shows that the “only if” part holds. The
formula in (20) follows by [4, Lemma 2] or [11, Theorem 3.4]. *2*

*Next we derive the expression of the directional derivative of Θ at (U, V ) with the*
*direction (G, H)* *∈ S*^{n}*× S** ^{n}*, which is used to characterize the F-diﬀerentiable points of
Θ in Lemma 3.2 below. Deﬁne Ω

_{1}:S

^{n}*× S*

^{n}*→ IR*

*and Ω*

^{|β|×|κ|}_{2}:S

^{n}*× S*

^{n}*→ IR*

*by*

^{|β|×|κ|}Ω_{1}*(U, V ) := U*_{βκ}*− (U**βκ**X*e_{κκ}*+ V*_{βκ}*Y*e_{κκ}*)D*^{−2}_{κ}*X*e_{κκ}*∀U, V ∈ S** ^{n}*
and

Ω2*(U, V ) := V**βκ**− (U**βκ**X*e*κκ**+ V**βκ**Y*e*κκ**)D*^{−2}_{κ}*Y*e*κκ* *∀U, V ∈ S*^{n}*,*

respectively. Noting that e*X*_{κκ}^{2} + e*Y*_{κκ}^{2} *= D*_{κ}^{2}, we can rewrite the function Θ in (15) as
*Θ(U, V ) =*[

*U*_{ββ}^{2} *+ V*_{ββ}^{2} + Ω_{1}*(U, V )Ω*_{1}*(U, V )** ^{T}*+ Ω

_{2}

*(U, V )Ω*

_{2}

*(U, V )*

*]*

^{T}*1/2*

*∀U, V ∈ S*^{n}*. (21)*

*For any given U, V* *∈ S*^{n}*, assume that Θ(U, V ) has the following spectral decomposition*
*Θ(U, V ) = RΛR*^{T}*= Rdiag(ϑ*_{1}*, . . . , ϑ*_{|β|}*)R*^{T}*,*

*where Λ = diag(ϑ*_{1}*, . . . , ϑ*_{|β|}*) is the diagonal matrix of eigenvalues of Θ(U, V ) and R is*
*a corresponding matrix of orthonormal eigenvectors. Deﬁne the index sets I and J*
*associated with the eigenvalues of Θ(U, V ) by*

*I :=* *{i: ϑ**i* *> 0} and J := {i: ϑ**i* = 0*} .*

*Then, by permuting the rows and columns of Θ(U, V ) if necessary, we may assume that*
Λ =

[ Λ* _{I}* 0
0 Λ

_{J}]

=

[ Λ* _{I}* 0
0 0

]
*.*

*From (21) and the spectral decomposition of Θ(U, V ), it is easy to obtain that*

*[R*^{T}*U**ββ*]*J β* *= 0, [R*^{T}*V**ββ*]*J β* *= 0, [R** ^{T}*Ω1

*(U, V )]*

*J κ*

*= 0, [R*

*Ω2*

^{T}*(U, V )]*

*J κ*

*= 0.*(22)

**Lemma 3.1 For any given (U, V )**∈ S

^{n}*× S*

^{n}*, assume that Θ(U, V ) has the spectral de-*

*composition as above. Then, the directional derivative Θ*

^{′}*((U, V ); (G, H)) of Θ at (U, V )*

*with the direction (G, H)∈ S*

^{n}*× S*

^{n}*has the following expression*

*R*

[ *L*^{−1}_{Λ}* _{I}*[f

*W*

*] Λ*

_{II}

^{−1}

_{I}*W*f

_{IJ}*W*f_{IJ}* ^{T}*Λ

^{−1}*( eΘ*

_{I}*JJ*

*− fW*

_{IJ}*Λ*

^{T}

^{−2}

_{I}*W*f

*IJ*)

*]*

^{1/2}*R*^{T}*,* (23)

*where eΘ := R** ^{T}*Θ

^{2}

*(G, H)R, and fW := R*

^{T}*W (G, H)R with W (G, H) given by*

*W (G, H) := Ω*

_{1}

*(U, V )Ω*

_{1}

*(G, H)*

*+ Ω*

^{T}_{1}

*(G, H)Ω*

_{1}

*(U, V )*

*+*

^{T}*L*

*U*

*ββ*

*(G*

*)*

_{ββ}+*L**V*_{ββ}*(H**ββ*) + Ω2*(U, V )Ω*2*(G, H)** ^{T}* + Ω2

*(G, H)Ω*2

*(U, V )*

^{T}*.*

**Proof. Assume that Θ(U, V )**̸= 0. For any t > 0, we calculate that*∆(t) := Θ(U + tG, V + tH)− Θ(U, V )*

= [

Θ^{2}*(U, V ) + tW (G, H) + t*^{2}Θ^{2}*(G, H)*]*1/2*

*− Θ(U, V ).*

*From the spectral decomposition of Θ(U, V ), it then follows that*

*∆(t) := R*e ^{T}*∆(t)R =*
(

Λ^{2}*+ tfW + t*^{2}Θe
)*1/2*

*− Λ,* (24)

where eΘ and f*W are deﬁned as in the lemma. From (24) and [24, Lemma 6.2], we have*

*∆(t)*e _{II}*= tL** ^{−1}*Λ

*I*[f

*W*

_{II}*] + o(t),*

*∆(t)*e _{IJ}*= tΛ*^{−1}* _{I}* f

*W*

_{IJ}*+ o(t),*

*tfW*_{JJ}*+ t*^{2}Θe* _{JJ}* = e

*∆(t)*

^{T}

_{IJ}*∆(t)*e

*+ e*

_{IJ}*∆(t)*

^{2}

_{JJ}*.*

(25)

By equation (22) and the deﬁnition of f*W , we have fW** _{JJ}* = 0. Then, from the last two
equalities of (25), it follows that

*∆(t)*e ^{2}_{JJ}*= t*^{2}Θe*JJ**− e∆(t)*^{T}_{IJ}*∆(t)*e *IJ* *= t*^{2}

(Θe*JJ* *− fW*_{IJ}* ^{T}*Λ

^{−2}*f*

_{I}*W*

*IJ*

)

*+ o(t*^{2}*).*

Since Λ* _{J}* = 0, the expression of e

*∆(t) in (24) implies that e∆(t)*

_{JJ}*≽ 0. Therefore,*

lim*t**↓0*

*∆(t)*e _{JJ}

*t* = lim

*t**↓0*

√*∆(t)*e ^{2}_{JJ}

*t* =

(Θe_{JJ}*− fW*_{IJ}* ^{T}*Λ

^{−2}*f*

_{I}*W*

*)*

_{IJ}*1/2*

*.*
This, together with the ﬁrst two equalities of (25), yields that

Θ^{′}*((U, V ); (G, H)) = lim*

*t**↓0*

*R e∆(t)R*^{T}

*t* *= R*

[ *L** ^{−1}*Λ

*I*[f

*W*

*] Λ*

_{II}

^{−1}

_{I}*W*f

_{IJ}*W*f_{IJ}* ^{T}*Λ

^{−1}*( eΘ*

_{I}

_{JJ}*− fW*

_{IJ}*Λ*

^{T}

^{−2}

_{I}*W*f

*)*

_{IJ}*]*

^{1/2}*R*^{T}*.*

*If Θ(U, V ) = 0, then U*_{ββ}*= 0, V** _{ββ}* = 0, Ω

_{1}

*(U, V ) = 0 and Ω*

_{2}

*(U, V ) = 0. By this, it*is easy to compute that Θ

^{′}*((U, V ); (G, H)) = Θ(G, H). Note that Θ(G, H) is a special*

*case of (23) with I =∅. The result then follows.*

*2*

**Remark 3.1 Lemma 3.1 shows that the function Θ deﬁned by (15) is directionally dif-***ferentiable everywhere in* S^{n}*× S*^{n}*. In fact, Θ is also globally Lipschitz continuous and*
*strongly semismooth in* S^{n}*× S*^{n}*. Let Ψ(U, V ) := [U*_{ββ}*V** _{ββ}* Ω

_{1}

*(U, V ) Ω*

_{2}

*(U, V )] for*

*U, V*

*∈ S*

^{n}*, and G*

^{mat}

*(A) :=*

*√*

*AA*^{T}*for A∈ IR*^{|β|×2n}*. Comparing with (21), we have that*
*Θ(U, V )* *≡ G*^{mat}*(Ψ(U, V )). By [21, Theorem 2.2], G*^{mat} *is globally Lipschitz continuous*
*and strongly semismooth everywhere in IR*^{|β|×2n}*. Since Ψ is a linear function, the compo-*
*sition of G*^{mat} *and Ψ, i.e. the function Θ, is globally Lipschitz continuous, and strongly*
*semismooth everywhere in* S^{n}*× S*^{n}*by [7, Theorem 19].*

By the expression of the directional derivative of Θ, we may present the necessary and suﬃcient characterization for the diﬀerentiable points of Θ.

**Lemma 3.2 The function Θ is F-diﬀerentiable at (U, V ) if and only if Θ(U, V )***≻ 0.*

*Furthermore, when Θ(U, V )≻ 0, we have for any (G, H) ∈ S*^{n}*× S*^{n}*,*
*J Θ(U, V )(G, H) = L*^{−1}* _{Θ(U,V )}*[

*(U*_{βκ}*G*_{κβ}*+ G*_{βκ}*U*_{κβ}*) + (V*_{βκ}*H*_{κβ}*+ H*_{βκ}*V** _{κβ}*)

*−*(

*G*_{βκ}*X*e_{κκ}*+ H*_{βκ}*Y*e* _{κκ}*
)

*D*^{−2}_{κ}

(*X*e_{κκ}*U** _{κβ}*+ e

*Y*

_{κκ}*V*

*)*

_{κβ}*−*(

*U**βκ**X*e*κκ**+ V**βκ**Y*e*κκ*

)
*D*_{κ}^{−2}

(*X*e*κκ**G**κβ* + e*Y**κκ**H**κβ*

)

+*L**U**ββ**(G** _{ββ}*) +

*L*

*V*

*ββ*

*(H*

*) ]*

_{ββ}*.*

* Proof. We only need to prove the “only if” part. If Θ is F-diﬀerentiable at (U, V ), then*
Θ

^{′}*((U, V ); (G, H)) is a linear function of (G, H) which, by equation (23) implies that*( eΘ

_{JJ}*− fW*

_{IJ}*Λ*

^{T}

^{−2}*f*

_{I}*W*

*)*

_{IJ}

^{1/2}*is a linear function of (G, H). We next argue that this holds true*

*only if J =∅. Indeed, if J ̸= ∅, by taking G =*

[ 0 0
*0 G*_{ββ}

]

*and H =*

[ 0 0
*0 H*_{ββ}

]
with
*G*_{ββ}*≻ 0 and H**ββ* *≻ 0, we have Ω*1*(G, H) = 0 and Ω*_{2}*(G, H) = 0 which, together with*
*[R*^{T}*U** _{ββ}*]

_{J β}*= 0 and [R*

^{T}*V*

*]*

_{ββ}*= 0, implies that f*

_{J β}*W*

_{J I}*= [R*

^{T}*W (G, H)R]*

*= 0. Note that Θ*

_{J I}^{2}

*(G, H) = G*

^{2}

_{ββ}*+ H*

_{ββ}^{2}. Then, ( eΘ

_{JJ}*− fW*

_{IJ}*Λ*

^{T}

^{−2}*f*

_{I}*W*

*)*

_{IJ}*=*

^{1/2}√

*[R*^{T}*(G*^{2}_{ββ}*+ H*_{ββ}^{2} *)R]*_{JJ}*, which*
is clearly nonlinear. The Jacobian formula of Θ is direct by a simple computation. *2*
* Remark 3.2 Combining Proposition 3.1 with Lemma 3.2, we immediately obtain that*
Φ

^{′}FB*((X, Y ); (·, ·)) is F-diﬀerentiable at (G, H) if and only if Θ(P*^{T}*GP, P*^{T}*HP )≻ 0.*

By the deﬁnition of Θ and Lemma 3.2, we can prove the following result (see the
Appendix for the proof) which corresponds to the property of Φ_{NR} in [13, Lemma 11].

**Lemma 3.3 For any given X, Y***∈ S*^{n}*, let Ψ*_{FB}(*·, ·) ≡ Φ*^{′}_{FB}*((X, Y ); (·, ·)). Then,*

*∂** _{B}*Φ

_{FB}

*(X, Y ) = ∂*

*Ψ*

_{B}_{FB}

*(0, 0).*

Now Lemma 3.3 and Proposition 3.1 allow us to obtain the main result of this section.

**Proposition 3.2 For any given X, Y***∈ S*^{n}*, let C(X, Y ) have the spectral decomposition*
*as in (13). Then, a (U, V) ∈ ∂**B*Φ_{FB}*(X, Y ) (respectively, ∂Φ*_{FB}*(X, Y )) if and only if there*
*exists a (G, H) ∈ ∂**B**Θ(0, 0) (respectively, ∂Θ(0, 0)) such that for any G, H* *∈ S*^{n}*,*

(*I − U) (G) + (I − V) (H)*

*= P*

*L*^{−1}*D**κ*

(*L**X*e*κκ*( e*G** _{κκ}*) +

*L*

*Y*e

*κκ*( e

*H*

*) )*

_{κκ}*D*_{κ}^{−1}

(*X*e_{κκ}*G*e* _{κβ}* + e

*Y*

_{κκ}*H*e

*) (*

_{κβ}*G*e

_{βκ}*X*e

*+ e*

_{κκ}*H*

_{βκ}*Y*e

_{κκ})

*D*_{κ}^{−1}*G( eG) +H( eH)*

* P*^{T}*,* (26)

*where eX := P*^{T}*XP , eY := P*^{T}*Y P , eG := P*^{T}*GP , and eH := P*^{T}*HP .*

**Proof. For any G, H***∈ S*^{n}*, let Ψ(G, H) := (P*^{T}*GP, P*^{T}*HP ). Deﬁne Ξ :*S^{n}*× S*^{n}*→ S** ^{n}* by

*Ξ(S, T ) := P*

[*L*^{−1}_{D}* _{κ}*(

*L*_{X}_{e}_{κκ}*(S** _{κκ}*)+

*L*

_{Y}_{e}

_{κκ}*(T*

*))*

_{κκ}*D*_{κ}* ^{−1}*( e

*X*

_{κκ}*S*

*+ e*

_{κβ}*Y*

_{κκ}*T*

*)*

_{κβ}*(S*

*βκ*

*X*e

*κκ*

*+ T*

*βκ*

*Y*e

*κκ*

*)D*

_{κ}

^{−1}*Θ(S, T )*

]
*P*^{T}*.*

By Proposition 3.1, clearly, Ψ_{FB}*(G, H) = (G + H)− Ξ(Ψ(G, H)) for any G, H ∈ S** ^{n}*.
Note that Ξ is globally Lipschitz continuous in S

^{n}*× S*

*by the remarks after (21), and*

^{n}*J Ψ(G, H) for any G, H ∈ S*

*is onto. Applying [3, Lemma 2.1] to the composite mapping Ξ*

^{n}*◦ Ψ at (0, 0), we have that ∂*

*B*(Ξ

*◦ Ψ)(0, 0) = ∂*

*B*

*Ξ(Ψ(0, 0))J Ψ(0, 0) = ∂*

*B*

*Ξ(0, 0)Ψ. So,*

*∂** _{B}*Ψ

_{FB}

*(0, 0) = (I, I) − ∂*

*B*

*Ξ(0, 0)Ψ.*

This, together with Lemma 3.3 and the expression of Ξ, completes the proof. *2*