Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function

J Optim Theory Appl (2012) 153:436–460DOI 10.1007/s10957-011-9962-8

Differential Properties of the Symmetric Matrix-ValuedFischer-Burmeister Function

Liwei Zhang · Ning Zhang · Liping Pang

Received: 3 April 2011 / Accepted: 2 November 2011 / Published online: 22 November 2011© Springer Science+Business Media, LLC 2011

Abstract This paper focuses on the study of differential properties of the symmetricmatrix-valued Fischer–Burmeister (FB) function. As the main results, the formulasfor the directional derivative, the B-subdifferential and the generalized Jacobian ofthe symmetric matrix-valued Fischer–Burmeister function are established, which canbe utilized in designing implementable Newton-type algorithms for nonsmooth equa-tions involving the symmetric matrix-valued FB function.

Keywords Fischer–Burmeister function · Directional derivative · B-subdifferential ·Generalized Jacobian

1 Introduction

It is well known that NCP-functions serve to transform a complementarity probleminto a nonsmooth system of equations or into a unconstrained minimization problem.Among which, one of the most popular NCP-functions is the Fischer–Burmeister(FB) function. Active research has been done on the applications of the FB-functionin solving the nonlinear complementarity problems and the second-order cone com-plementarity problems. The symmetric matrix-valued FB-function also plays impor-tant roles in optimization problems over the cone of positive and definite symmetric

Communicated by Liqun Qi.

The authors are grateful to the anonymous referees for their helpful suggestions and commentsand thank Professor Shaohua Pan from South China University of Technology for helping us to proveProposition 4.1.

The work is Supported by the National Natural Science Foundation of China under projects No.11071029 and No. 11171049 and the Fundamental Research Funds for the Central Universities.

L. Zhang (�) · N. Zhang · L. PangSchool of Mathematical Sciences, Dalian University of Technology, Dalian, Chinae-mail: [email protected]

mailto:[email protected]

J Optim Theory Appl (2012) 153:436–460 437

matrices. It has many nice properties that could be utilized in algorithmic develop-ment for semidefinite complementarity problems and nonsmooth equations involvingthe semidefinite conic complementarity condition. It is known from [1] that the func-tion ΦFB is strongly semismooth; it plays a fundamental role in the analysis of thequadratic convergence of Newton-type methods for solving systems of nonsmoothequations involving the semidefinite conic complementarity condition; see [2] for in-stance. It is proved by [3] that the gradient mapping of the squared norm of ΦFB isLipschitz continuous. However, these properties are generally not enough to facilitatethe computations in Newton-type methods, and the study of the differential proper-ties of the symmetric matrix-valued FB-function is becoming the key in designingimplementable Newton-type algorithms.

This paper investigates the differential properties of the matrix-valued FB-function, including the formulas for the directional derivative, the B-subdifferential,and the generalized Jacobian.

The organization of the paper is as follows. In Sect. 2, we give some preliminariesabout first-order derivatives of symmetric matrix-valued mappings and perturbationtheory for eigenvalues and eigenvectors of symmetric matrices from [4] and [5], re-spectively. The formula for the directional derivative of ΦFB is established in Sect. 3.In Sect. 4, we develop the formulas for the B-subdifferential and the generalized Ja-cobian of ΦFB. Finally conclusions are given in Sect. 5.

2 Preliminaries

Let Sn denote the linear space of n × n real symmetric matrices endowed with the

inner product

〈X,Y 〉 := tr(XY), for any X,Y ∈ Sn,

where “tr” denotes the trace, i.e., the sum of the diagonal entries.The symmetric matrix-valued FB-function is the function ΦFB : S

n × Sn → S

n

defined by

ΦFB(X,Y ) = X + Y −√

X2 + Y 2, X,Y ∈ Sn, (1)

which is an extension to the scalar-valued FB-function [6] as follows:

φ(a, b) = a + b −√

a2 + b2, a, b ∈ R × R.

In [7], Tseng proves that

ΦFB(X,Y ) = 0 ⇐⇒ 0 � X ⊥ Y 0,

where the symbol ⊥ means “perpendicular under the inner product." For this rela-tionship, Kanzow and Nagel [8] suggest a proof somewhat different from the onegiven by Tseng [7].

Throughout this paper, we employ the following notations, which are rather stan-dard in matrix analysis. Concretely, for X ∈ S

n, X 0 (X � 0) means that X is asymmetric and positive semidefinite matrix (positive definite matrix). For any m × n

438 J Optim Theory Appl (2012) 153:436–460

matrix A and index sets I ⊆ {1,2, . . . ,m} and J ⊆ {1,2, . . . , n}, AIJ denotes thesubmatrix of A with rows and columns specified by I , J , respectively. Particularly,Aij is the entry of A at (i, j) position. We use “◦” to denote the Hardamard prod-uct between matrices, i.e., for any two m × n matrices A and B , the (i, j)-th entry ofC := A◦B is Cij = AijBij . For X,H ∈ S

n, define LX(H) by LX(H) := XH +HX.For any X ∈ S

n, we use λ1(X) ≥ λ2(X) ≥ · · · ≥ λn(X) to denote the real eigen-values of X (counting multiplicity) being arranged in the non-increasing order. Letλ(X) := (λ1(X),λ2(X), . . . , λn(X))T ∈ R

n. Let Λ(X) ∈ Sn be the diagonal matrix

whose i-th diagonal entry is given by λi(X), i = 1, . . . , n, i.e., Λ(X) = diag(λ(X)).Denote by On the set of all n × n orthogonal matrices in R

n×n and On(X) a subsetof On by

On(X) := {P ∈ On : X = PΛ(X)P T

}.

Let μ1 > · · · > μr be the distinct eigenvalues of X and denote αk := {i : λi(X) = μk}for k = 1, . . . , r . For each i ∈ {1, . . . , n}, we define li to be the number of eigenval-ues that are equal to λi(X) but are ranked before i (including i) and mi to be thenumber of eigenvalues that are equal to λi(X) but are ranked after i (excluding i),respectively, i.e., we define li and mi such that

λ1(X) ≥ · · · ≥ λi−li (X) > λi−li+1(X) = · · · = λi(X) = · · · = λi+mi(X)

> λi+mi+1(X) ≥ · · · ≥ λn(X).

Now we list some useful results about the symmetric matrices, which are neededin the subsequent discussions.

Lemma 2.1 [9, Lemma 2.1] For any n × n real matrices A,B and any Z ∈ Sn+, it

holds that

(A + B)T Z(A + B) � 2(AT ZA + BT ZB

),

(A − B)T Z(A − B) � 2(AT ZA + BT ZB

).

Lemma 2.2 [9, Lemma 2.2] Let X,Y ∈ Sn satisfy X2 +Y 2 � 0. Then, for any n×m

real matrices A,B ,

AT A + BT B − (AT X + BT Y

)(X2 + Y 2)−1

(XA + YB) 0.

For the analysis in next section, we apply Stewart theorem [5] to matrix per-turbation G + �G and zero eigenvalues of G, where G ∈ S

n+ and �G ∈ Sn are

n × n symmetric matrices. Let G = PDP T with D := Λ(G) and P ∈ On(G). Letα := {i : λi(G) > 0} and β := {i : λi(G) = 0}. For any �G ∈ S

n, denote

[Pα Pβ ]T �G[Pα Pβ ] :=[�Gαα �Gαβ

�GTαβ �Gββ

]

.

Define γ := ‖�Gαβ‖ and δ := mini∈α λi(G) − ‖�Gαα‖ − ‖�Gαβ‖.


Lemma 2.3 If δ > 0 and γ ≤ δ/2, then there exists a matrix solution K of

K�Gαβ − (Dα + �Gββ)K = �GTαβ − K�GαβK (2)

such that ‖K‖ ≤ 2γ /δ; if, moreover, we set Pβ := (Pβ + PαK)(I + KT K)− 12 , then

the columns of Pβ form an orthonormal basis for Range Pβ which is an invariantsubspace of G + �G. The representation of G + �G on Pβ is

Dβ = (I + KT K

) 12 (�Gαα + �GαβK)

(I + KT K

)− 12 .

From the above lemma, we get a perturbation result about eigenvalues λi(G +�G) for i ∈ β , which plays a key role in developing the formula for the directionalderivatives whose proof is quite similar to that of [10, Proposition 1.4].

Proposition 2.1 Let i ∈ β . Then, for any small �G ∈ Sn, there exists a matrix K

satisfying (2) with ‖K‖ = O(‖�G‖), such that

λi(G + �G) = λli

(P T

β �GPβ − P Tβ �GPαD−1

α P Tα �GPβ

)+ O(‖�G‖3).

Lemma 2.4 [4, Proposition 2.2] Let Q be an orthogonal matrix in On such thatQT Λ(X)Q = Λ(X). Then we have

{Qαkαl

= 0, if k, l = 1, . . . , r, k �= l,

QαkαkQT

αk,αk= QT

αkαkQαk,αk

, if k = 1, . . . , r.(3)

From Lemma 2.4, we obtain that On(Λ(X)) = {Q ∈ On : Q satisfies (3)}.

Lemma 2.5 [4, Proposition 2.3] For any H ∈ Sn, let U be an orthogonal matrix such

that

UT(Λ(X) + H

)U = Λ

(Λ(X) + H

).

Then, for any H → 0, we have

⎧⎪⎪⎨

⎪⎪⎩

Uαkαl= O(‖H‖), if k, l = 1, . . . , r, k �= l,

UαkαkUT

αkαk= I|αk | + O(‖H‖2), if k = 1, . . . , r ,

dist(Uαkαk, O|αk |) = O(‖H‖2), if k = 1, . . . , r ,

and

λi

(Λ(X) + H

)− λi(X) − λli (Hαkαk) = O

(‖H‖2), i ∈ αk, k = 1, . . . , r.

Hence, for any given direction H ∈ Sn, the eigenvalue function λi(·) is directional

differentiable at X with λ′i (Λ(X);H) = λli (Hαkαk

), i ∈ αk , k = 1, . . . , r .


For any given X ∈ Sn and each k ∈ {1, . . . , r}, there exists δk > 0 such that |μl −

μk| > δk , ∀ l �= k. Define a continuously scalar function gk(·) : R → R by

gk(t) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

− 6δk

(t − μk − δk

2 ), if t ∈ (μk + δk

3 ,μk + δk

2 ],1, if t ∈ [μk − δk

3 ,μk + δk

3 ],6δk

(t − μk + δk

2 ), if t ∈ [μk − δk

2 ,μk − δk

3 ),

0, otherwise.

Let Pk(·) be the matrix-valued function with respect to gk(·), i.e., for any Y ∈ Sn,

Pk(Y ) = P diag(gk

(λ1(Y )

), . . . , gk

(λn(Y )

))P T ,

where P ∈ On(X).Define

F s(Y ) :=r∑

k=1

f (μk)Pk(Y ). (4)

Obviously, the function gk(·) is twice continuously differentiable at each μk , thefollowing result follows from [4].

Lemma 2.6 For each k = 1, . . . , r , Pk(·) is twice continuously differentiable at X.

For a real-valued function f : R → R, the associated matrix-valued function F :S

n → Sn is defined by

F(X) := P diag(f(λ1(X)

), . . . , f

(λn(X)

))P T . (5)

Let f [1](Λ(X)) ∈ Sn be the first divided difference matrix whose (i, j)-entry is given

by

f [1](Λ(X))ij

:=⎧⎨

⎩

f (λi (X))−f (λj (X))

λi (X)−λj (X), if i �= j = 1,2, . . . , n,

f ′(λi(X)), if i = j = 1, . . . , n.

The following result for the differentiability of the symmetric matrix functions F

defined in (5) can be found in [11] or [12].

Lemma 2.7 Let X ∈ Sn be given and have the eigenvalue decomposition X =

PΛ(X)P T . The symmetric matrix-valued function F is Fréchet differentiable at X,if and only if f is differentiable at λi(X). In this case, the Fréchet derivative of F(·)at X is given by

F ′(X)(H) = P[f [1](Λ(X)

) ◦ (P T HP)]

P T , ∀H ∈ Sn. (6)

Note that the formula (6) is independent of the choice of P and the ordering ofλ1(X), . . . , λn(X).


3 Directional Derivative of ΦFB

To simplify the notation, we define a matrix-valued function � : Sn × S

n → Sn by

�(X,Y ) := X2 + Y 2.

Let G := �(X,Y ) and σ1 ≥ σ2 ≥ · · · ≥ σn be eigenvalues of G. Choose P ∈ On(G)

such that

G = PDP T , (7)

where D := diag(σ1, σ2, . . . , σn). Let f : R → R be the scalar function:

f (t) :={√

t, if t ≥ 0,√−t, if t < 0.

Define a mapping F : Sn → S

n+ as the matrix-valued function associated with thescalar function f , i.e.,

F(G) = P diag(f (σ1), f (σ2), . . . , f (σn)

)P T ,

it is obvious that F(G) = √G for G ∈ S

n+. Then we can rewrite ΦFB as

ΦFB(X,Y ) = X + Y − Ψ (X,Y ),

where Ψ is the compound function of F and �, i.e.,

Ψ (X,Y ) := (F • �)(X,Y ).

Let

D :=[Dα 00 0β

], (8)

where

α := {i : σi > 0} and β := {i : σi = 0}. (9)

We assume that μ1 > μ2 > · · · > μr > μr+1 = 0 are the distinct eigenvalues of G.Define the following subsets of {1, . . . , n}:

αk := {i : σi = μk}, k = 1, . . . , r + 1. (10)

Then {α1, . . . , αr+1} and {α1, . . . , αr} are partitions of {1, . . . , n} and α, respectively.Due to the strong semismoothness [1], ΦFB is directionally differentiable every-

where in Sn. Now, we characterize the directional derivative of ΦFB.

Proposition 3.1 For (X,Y ) ∈ Sn × S

n, let Z := [X Y ] ∈ Rn×2n, the matrix-valued

function ΦFB is directionally differentiable at (X,Y ) and the directional derivative,for any H := (HX,HY ) ∈ S

n × Sn is given by


Φ ′FB

((X,Y ); (HX,HY )

)

= HX + HY − P

[f [1](D)αα ◦ P T

α LZ(H)Pα f [1](D)αβ ◦ P Tα LZ(H)Pβ

f [1](D)βα ◦ P Tβ LZ(H)Pα K(HX,HY )

]

P T ,

where

LZ(H) = LX(HX) + LY (HY ),

K(HX,HY ) = {P T

β

[(HX)2 + (HY )2]Pβ − P T

β LZ(H)PαD−1α P T

α LZ(H)Pβ

} 12 .

Proof For any (HX,HY ) ∈ Sn × S

n, let

�(t) : = ΦFB(X + tHX,Y + tHY ) − ΦFB(X,Y )

= t (HX + HY ) − [Ψ (X + tHX,Y + tHY ) − Ψ (X,Y )

]

= t (HX + HY ) − P(F(D + W(t)

)− F(D))P T ,

where

W(t) := t(

LX(HX) + LY (HY ))+ t2((HX)2 + (HY )2),

with

X := P T XP, Y := P T YP, HX := P T HXP and HY := P T HY P.

Let D(t) := D + W(t) admit the following spectral decomposition:

D(t) = U(t)diag(λ(D(t)

))U(t)T , U(t) ∈ On

(D(t)

).

Let Fn(D(t)) := F(D(t)) − F s(D(t)). Since F s(D) = F(D), we have

F(D(t)

)− F(D) = Fn(D(t)

)+ F s(D(t)

)− F s(D).

By Lemma 2.6, the mappings Pk(·), k = 1, . . . , r + 1 are twice continuously differ-entiable at D. Then we have

limt↓0

F s(D(t)) − F s(D)

t= lim

t↓0

∑r+1k=1 f (μk)(Pk(D(t)) − Pk(D))

t

=r+1∑

k=1

f (μk)P ′k(D)P T

(LX(HX) + LY (HY )

)P

=Ω1, (11)

where for k, l = 1, . . . , r + 1,

(Ω1)αkαl:=⎧⎨

⎩

1√μk+√

μlP T

αkLZ(H)Pαl

, if k �= l,

0αkαk, if k = l.


For t sufficiently close to 0, we have Pk(D(t)) = ∑i∈αk

ui(t)ui(t)T , where ui(t)

is the ith column of the matrix U(t), k = 1, . . . , r + 1. Therefore, we can writeFn(D(t)) as

Fn(D(t)

)=r+1∑

k=1

∑

i∈αk

[f(λi

(D(t)

))− f (μk)]ui(t)ui(t)

T .

For each k = 1, . . . , r + 1, let

�k(t) :=∑

i∈αk

[f(λi

(D(t)


T .

Noting that{

μk > 0, if k = 1, . . . , r ,

μk = 0, if k = r + 1,

we need consider the following two cases.

Case 1. Let k = 1, . . . , r and i ∈ αk . Since f is directionally differentiable at μk

and f ′(μk)(·) is positive homogeneous, we have

f(λi

(D(t)

))− f (μk) = f ′(μk)(λli

(W(t)αkαk

))+ O(t2).

Since ui(t)ui(t)T is uniformly bounded, we obtain that

�k(t) =∑

i∈αk

f ′(μk)(λli

(W(t)αkαk

))ui(t)ui(t)

T + O(t2).

By Lemma 2.5, we know that there exists Q(t)αkαk∈ O|αk | such that for any i ∈ αk ,

ui(t) =⎡

⎢⎣

O(‖W(t)‖)qi(t)

O(‖W(t)‖)

⎤

⎥⎦+ O

(‖W(t)‖2)=⎡

⎢⎣

O(t)

qi(t)

O(t)

⎤

⎥⎦+ O

(t2),

where qi(t) is the ith column of Q(t)αkαk. Since the eigenvalue function λi(·) is

locally Lipschitz continuous, we obtain that

λli

(W(t)αkαk

) = λli

((LX(tHX) + LY (tHY ) + (tHX)2 + (tHY )2)

αkαk

)

= λli

((LX(tHX) + LY (tHY )

)αkαk

)+ O(t2)

= tλli

(P T

αkLZ(H)Pαk

)+ O(t2),

which implies

�k(t) =⎡

⎢⎣

0 0 0

0∑

i∈αkf ′(μk)(λli (tP

Tαk

LZ(H)Pαk))qi(t)qi(t)

T 0

0 0 0

⎤

⎥⎦+ O

(t2).


On the other hand, by the spectral decomposition of D(t) and Lemma 2.5, we get

Λαk

(D(t)

) = U(t)Tαk

(D + W(t)

)U(t)αk

= μkI|αk | + U(t)TαkαkW(t)αkαk

U(t)αkαk+ O

(t2)

= μkI|αk | +(Q(t)Tαkαk

+ O(‖W(t)‖2))W(t)αkαk

(Q(t)αkαk

+ O(‖W(t)‖2))+ O

(t2)

= μkI|αk | + tQ(t)Tαkαk

(P T

αkLZ(H)Pαk

)Q(t)αkαk

+ O(t2).

This, together with the fact that

Λαk

(D(t)

)− μkI|αk | = tΛ(P T

αkLZ(H)Pαk

)+ O(t2),

implies that

tΛ(P T

αkLZ(H)Pαk

)= tQ(t)Tαkαk

(P T

αkLZ(H)Pαk

)Q(t)αkαk

+ O(‖W(t)‖2).

Therefore,∑

i∈αk

tλli

(P T

αkLZ(H)Pαk

)qi(t)qi(t)

T

= tQ(t)αkαkΛ(P T

αkLZ(H)Pαk

)Q(t)Tαkαk

+ O(t2)

= tP Tαk

LZ(H)Pαk+ O

(t2),

and in turn,

�k(t) = 1

2√

μk

⎡

⎢⎣

0 0 0

0 tP Tαk

LZ(H)Pαk+ O(t2) 0

0 0 0

⎤

⎥⎦+ O

(t2)

= 1

2√

μk

⎡

⎢⎣

0 0 0

0 tP Tαk

LZ(H)Pαk0

0 0 0

⎤

⎥⎦+ O

(t2). (12)

Then, for any k ∈ {1,2, . . . , r}, we have

limt↓0

�k(t)

t= 1

2√

μk

⎡

⎢⎣

0 0 0

0 P Tαk

LZ(H)Pαk0

0 0 0

⎤

⎥⎦ (13)

Case 2. Suppose that k = r + 1, or equivalently αk = β , then

�r+1(W(t)

) =∑

i∈β

[f(λi

(D(t)


T


=∑

i∈β

f(λi

(D(t)

))ui(t)ui(t)

T .

By Proposition 2.1, we have

λi

(D(t)

)= λli

(W(t)ββ − W(t)βαD−1

α W(t)αβ

)+ O(‖W(t)‖3).

Define

Eββ(t) := W(t)ββ − W(t)βαD−1α W(t)αβ

and

Eββ(HX,HY ) := P Tβ

[(HX)2 + (HY )2]Pβ − P T


α LZ(H)Pβ,

then

Eββ(t) = t2Eββ(HX,HY ) + O(‖W(t)‖3).

In the remainder of the proof, we use Eββ instead of Eββ(HX,HY ), for notationsimplicity.

Partition ui(t) as ui(t) = [uαi (t) u

βi (t)]T . The spectral decomposition of D(t) im-

plies(D + W(t)

)ui(t) = λi

(D(t)

)ui(t),

from which we get

[W(t)ββ W(t)βα

W(t)αβ Dα + W(t)αα

][u

βi (t)

uαi (t)

]

= λi

(D(t)

)[u

βi (t)

uαi (t)

]

,

i.e.,⎧⎨

⎩

W(t)ββuβi (t) + W(t)βαuα

i (t) = λi(D(t))uβi (t),

W(t)αβuβi (t) + (Dα + W(t)αα)uα

i (t) = λi(D(t))uαi (t).

(14)

Noticing the nonsingularity of Dα , we have

uαi (t) = D−1

α

(λi

(D(t)

)uα

i (t) − W(t)ααuαi (t)

)− D−1α W(t)αβu

βi (t).

Then, by substituting the above equation into the first formula in (14), we have

W(t)ββuβi (t) − W(t)βαD−1

α W(t)αβuβi (t)

= λi

(D(t)

)u

βi (t) − W(t)βαD−1

α

(λi

(D(t)

)uα

i (t) − W(t)ααuαi (t)

).

Together with the fact U(t)αβ = O(‖W(t)‖), we obtain that

U(t)TββE(t)ββU(t)ββ = Λβ

(D(t)

)+ O(‖W(t)‖3).


From Lemma 2.5, we know that there exists Q(t)ββ ∈ O|β| such that

U(t)ββ = Q(t)ββ + O(‖W(t)‖2),

then(Q(t)Tββ + O

(‖W(t)‖2))Eββ(t)(Q(t)ββ + O

(‖W(t)‖2))

= Λβ

(D(t)

)+ O(‖W(t)‖3). (15)

Simple calculation yields that

Q(t)ββΛβ

(D(t)

)Q(t)Tββ = Eββ(t) + L

(W(t)

)with L

(W(t)

) := O(‖W(t)‖3).

Hence,

�r+1(W(t)

) =∑

i∈β

[f(λi

(D(t)

))]ui(t)ui(t)

T

=∑

i∈β

√λi

(D(t)

)qi(t)qi(t)

T + O(‖W(t)‖2)

=√

Eββ(t) + L(W(t)

)+ O(‖W(t)‖2).

Let P ∈ O|β|(Eββ), then

Eββ = P diag(λ(Eββ)

)P T ,

with λ1(Eββ) ≥ · · · ≥ λ|β|(Eββ). Let μ1 > μ2 > · · · > μr ′ > μr ′+1 = 0 be the dis-tinct eigenvalues of Eββ . Denote

D :=(

Dα′ 0

0 0|β ′|×|β ′|

),

where α′ :=⋃r ′k=1 α′

k and β ′ := α′r ′+1 with

α′k := {

i : λi(Eββ) = μk

}, k = 1, . . . , r ′ + 1. (16)

Then√

Mββ(t) −√

t2Eββ = P T(F(t2D(t) + O

(‖W(t)‖3))− F(t2D(t)

))P ,

where Mββ(t) := Eββ(t)+L(W(t)). Similar to the above estimating precess, we get

√Mββ(t) −

√t2Eββ =

[O(‖W(t)‖6) O(‖W(t)‖3)

O(‖W(t)‖3) O(‖W(t)‖3/2)

]

ββ

+ O(‖W(t)‖3)

=[

0α′α′ 0

0 O(t3/2)β ′β ′

]+ O

(t3),


i.e.,

√M(t)ββ =

√t2Eββ + O

(t3/2). (17)

Then

�r+1(t) = t

√Eββ + O

(‖W(t)‖2).

This together with (17) shows that

limt↓0

�r+1(t)

t= K(HX,HY ), (18)

where

K(HX,HY ) = {P T

β

[(HX)2 + (HY )2]Pβ − P T


α LZ(H)Pβ

} 12 .

Combining (11), (13), with (18), we obtain the result. �

Now we give an explanation for the well-definedness of K(HX,HY ). Let Z =[X Y ] ∈ R

n×2n admit the following singular value decomposition:

Z = P[�(Z) 0

]QT = P

[�(Z) 0

][Q1 Q2]T = P�(Z)QT1

= P[�(Z) 0

][Qα Qβ ]T , (19)

where Q ∈ O2n, Q1,Q2 ∈ R2n×n, Qα ∈ R

2n×|α|, Qβ ∈ R2n×|β| and Q = [Q1 Q2],

Q1 = [Qα Qβ ], and �(Z) := diag(√

σ1,√

σ2, . . . ,√

σn). The set of such matrices(P,Q) in the singular value decomposition (19) is denoted by On,2n(Z), i.e.,

On,2n(Z) := {(P,Q) ∈ R

n×n × R2n×2n : Z = P

[�(Z) 0

]QT

}.

Obviously, we can rewrite LZ(H) as

LZ(H) = Z

[HX

HY

]+ [HX HY ]ZT . (20)

By the fact P Tβ XP = 0 and P T

β YP = 0, we obtain that

Eββ = P Tβ

(H 2

X + H 2Y

)Pβ − P T


α LZ(H)Pβ

= P Tβ [HX HY ]

(

I2n − Q

[�(Z)

0

]P T PαD−1

α P Tα P

[�(Z) 0

]QT

)[HX

HY

]Pβ

= P Tβ [HX HY ](I2n − QαQT

α

)[HX

HY

]Pβ

= P Tβ

([HX HY ](QαQT

α

)[HX

HY

])Pβ


= Tβ(HX,HY )Tβ(HX,HY )T 0, (21)

where Qα := [Qβ Q2] ∈ R2n×(|β|+n) and Tβ(HX,HY ) := P T

β [HX HY ]Qα . There-fore, we obtain that

K(HX,HY ) =√

Tβ(HX,HY )Tβ(HX,HY )T

is well defined.

4 B-subdifferential and Generalized Jacobian of ΦFB

In this section, we characterize the B-subdifferential and the generalized Jacobianof ΦFB.

Let A ∈ Sn+ be given and have the eigenvalue decomposition (8). Let

Wβ(A) :={Θ ∈ R

|β|×|β| :√

λi+|α|(Am)√

λi+|α|(Am) +√λj+|α|(Am)

→ Θij ,

Am → A with Am � 0

},

where α and β are the corresponding subsets given by (9).The following result plays an important role in establishing the formula for the

B-subdifferential of the FB-function ΦFB.1

Proposition 4.1 For any given X,Y ∈ Sn, let Υ (·, ·) := Φ ′

FB((X,Y ); (·, ·)). Then

∂BΦFB(X,Y ) = ∂BΥ (0,0). (22)

Proof Let W ∈ ∂BΦFB(X,Y ). It follows the definition of ∂BΦFB(X,Y ) that thereexists a sequence {(Xm,Ym)} in S

n × Sn converging to (X,Y ) such that ΦFB(·, ·) is

F-differentiable at every (Xm,Ym) and W = limm→∞ Φ ′FB(Xm,Ym). Let P m ∈ On,

Qm ∈ O2n such that

�(Xm,Ym

)= P mΛ(Gm

)(P m

)T, Zm := [

Xm Ym]= P m

[�(Zm)

0](

Qm)T

,

where Gm := (Xm)2 + (Ym)2 and �(Zm) = Λ(Gm)12 .

Define Mm := (MmX ,Mm

Y ) by2

MmX := P

[0 Xm

αβ

Xmβα Xm

ββ

]

P T and MmY := P

[0 Y m

αβ

Ymβα Ym

ββ

]

P T , (23)

1When we were preparing this manuscript, we learned that the following result was obtained by Bi, Pan,and Chen (see [9]) via the eigenvalue decomposition. Our proof is different from that in [9]; here, we adoptthe singular value decomposition technique.2In the first version of this paper, we constructed Mm and (Xm,Ym) different from those in (23) and (32),Professor Shaohua Pan found mistakes in our proof, She suggested that we should adopt the constructionsin (23) and (32). We are grateful to her for the contributions.


where Xm := (P m)T XmP m, Y m := (P m)T YmP m. Then (MmX ,Mm

Y ) → (0,0) asm → ∞. An elementary calculation shows that

Eββ

(Mm

X ,MmY

)=P Tβ

((Mm

X

)2 + (Mm

Y

)2)Pβ

− P Tβ

(Mm

X X + MmY Y

)PαD−1

α P Tα

(XMm

X + YMmY

)Pβ

=Dmβ − (

XmβαXαα + Y m

βαYαα

)D−1

α

(XααXm

αβ + YααYmαβ

)

=Dmβ + O

(‖Mm‖2).

Consequently, there exists Um ∈ O|β|(Eββ(MmX ,Mm

Y )) such that

Um = I|β| + O(‖M‖2),

Eββ

(Mm

X ,MmY

)= Um(Ξm

)2(Um

)T with(Ξm

)2 = Dmβ + O

(‖Mm‖2).

Then, by singular decomposition theorem [11, Theorem 7.3.5], we may assume thatV m ∈ O|β|+n such that

Tβ

(Mm

X ,MmY

)= Um[Ξm 0

](V m

)T,

and by taking a subsequence if necessary, (Um,V m) → (U,V ) ∈ O|β|,|β|+n, as m →∞.

We now demonstrate that Eββ is positive at (MmX ,Mm

Y ).Since (Xm)2 + (Y m)2 = Λ(Gm), we have

XmααXm

αβ + XmαβXm

ββ + Y mααYm

αβ + Y mαβYm

ββ = 0, (24)

and consequently,

XααXmαβ + YααYm

αβ = (Xm

ααXmαβ + Y m

ααYmαβ

)− (Xm

αβXmββ + Y m

αβYmββ

), (25)

where Xmαα := Xαα − Xm

αα and Y mαα := Yαα − Y m

αα .This together with Lemma 2.1 yields that

(Xm

βαXαα + Y mβαYαα

)D−1

α

(XααXm

αβ + YααYmαβ

)

= [(Xm

ααXmαβ + Y m

ααYmαβ

)− (Xm

αβXmββ + Y m

αβYmββ

)]TD−1

α

× [(Xm

ααXmαβ + Y m

ααYmαβ

)− (Xm

αβXmββ + Y m

αβYmββ

)]

� 4(Xm

ααXmαβ

)TD−1

α

(Xm

ααXmαβ

)+ 4(Y m

ααYmαβ

)TD−1

α

(Y m

ααYmαβ

)

+ 4(Xm

αβXmββ

)TD−1

α

(Xm

αβXmββ

)+ 4(Y m

αβYmββ

)TD−1

α

(Y m

αβYmββ

). (26)

And then, by [11, Theorem 6.1.10], for sufficiently large m,

P Tβ

((Mm

X

)2 + (Mm

Y

)2)Pβ − P T

β

(Mm

XX + MmY Y

)PαD−1

α P Tα

(XMm

X + YMmY

)Pβ


Xmβα

(I − 4Xm

ααD−1α Xm

αα

)Xm

αβ + Y mβα

(I − 4Y m

ααD−1α Ym

αα

)Xm

αβ + 1

2

(Dm

β

)

+ 1

2Xm

ββ

(I − 8Xm

βαD−1α Xm

αβ

)Xm

ββ + 1

2Y m

ββ

(I − 8Y m

βαD−2α Ym

αβ

)Y m

ββ

1

2

(Dm

β

)� 0.

The definition of Eββ(MmX ,Mm

Y ) shows that, for m large enough, Eββ is well definedat (Mm

X ,MmY ) and Eββ(Mm

X ,MmY )� 0, i.e., Υ (·, ·) is F-differentiable at (Mm

X ,MmY ).

The Fréchet derivative of K at (MmX ,Mm

Y ) is

K ′(MmX,Mm

Y

)(HX,HY )

= Um[f [1](Ξm2) ◦ (Um

)T (Tβ

(Mm

X,MmY

)Tβ(HX,HY )T

+ Tβ(HX,HY )Tβ

(Mm

X,MmY

)T )Um

](Um

)T

= Um[Θm ◦ (V m

β

)TTβ(HX,HY )T Um

+ (1|β|1T|β| − Θm

) ◦ (Um)T

Tβ(HX,HY )V mβ

](Um

)T,

where

(Θm

)ij

:= Ξmi

Ξmi + Ξm

j

and 1|β| is a |β|-dimensional vector whose entries are all ones.Then, for any (HX,HY ) ∈ S

n × Sn, we obtain that

limm→∞Υ ′(Mm

X ,MmY )(HX,HY )

= HX + HY

− limm→∞P

[f [1](D)αα ◦ P T

α LZ(H)Pα f [1](D)αβ ◦ P Tα LZ(H)ZPβ

f [1](D)βα ◦ P Tβ LZ(H)Pα K ′(Mm

X ,MmY )(HX,HY )

]

P T

= HX + HY

− P

[f [1](D)αα ◦ P T


f [1](D)βα ◦ P Tβ LZ(H)Pα limm→∞ K ′(Mm

X ,MmY )(HX,HY )

]

P T

= HX + HY − P

[f [1](D)αα ◦ P T


f [1](D)βα ◦ P Tβ LZ(H)Pα S(HX,HY )

]

P T ,

(27)

where

S(HX,HY ) = Θ ◦ (Vβ)T Tβ(HX,HY )T + (1|β|1T|β| − Θ

) ◦ Tβ(HX,HY )Vβ (28)


and Θ ∈ Wβ(G).

Again by singular decomposition theorem, we know that Qm := [Qmα Qm

α V m] satis-fies

[Xm Ym

]= P m[�(Zm)

0]Qm.

Define Q := [Qα QαV ], then

limm→∞

(P m,Qm

)= (P, Q).

For any H := (HX,HY ) ∈ Sn × S

n, we have

limm→∞Φ ′

FB(Xm,Ym)(HX,HY )

= HX + HY − limm→∞P m

[f [1](Λ(Gm)

) ◦ (P m)T (LXm(HX)

+ LYm(HY ))P m

](P m

)T

= HX + HY

− P

[f [1](Λ(G))αα ◦ P T

α LZ(H)Pα f [1](Λ(G))αβ ◦ P Tα LZ(H)Pβ

f [1](Λ(G))βα ◦ P Tβ LZ(H)Pα lim

m→∞f [1](Λ(Gm))ββ ◦ (P mβ )T LZm(H)P m

β

]

P T

(29)

and

limm→∞f [1](Λ

(Gm

))ββ

◦ (P mβ

)TLZm(H)P m

β

= limm→∞f [1](Λ

(Gm

))ββ

◦ (P mβ

)T (LXm(HX) + LYm(HY ))P m

β


(Gm

))ββ

◦ (P mβ

)T(

P m[f(Λ(Gm

))0](

Qm)T[HX

HY

]

+ [HX HY ]Qm

[f (Λ(Gm))

0

](P m

)T)

P mβ

= Θ ◦ QTβ

[HX

HY

]Pβ + (

1|β|1T|β| − Θ) ◦ P T

β [HX HY ]Qβ,

where LZm(H) := LXm(HX) + LYm(HY ), and Θ ∈ Wβ(G) which is coincide withthe one in (28).

From the definition of Q and comparing (27) with (29), we conclude that

W(HX,HY ) = limm→∞Υ ′(Mm

X ,MmY

)(HX,HY ).

Because (HX,HY ) is arbitrary in Sn × S

n, one has W ∈ ∂BΦFB(X,Y ).Now we prove the reverse inclusion. As before, we use the notation

Tβ(MX,MY ) = P Tβ [MX MY ]Qα , then Eββ(MX,MY ) = Tβ(MX,MY )Tβ(MX,MY )T.

For any given W ∈ ∂BΥ (0,0), there exists a sequence of matrices {(MmX ,Mm

Y )} ⊂


Sn × S

n converging to (0, 0) such that Eββ(MmX ,Mm

Y ) is nonsingular for every m andW = limm→∞ Υ ′(Mm

X ,MmY ). Let (Um,V m) ∈ O|β|,(|β|+n)(Tβ(Mm

X ,MmY )), such that

Tβ

(Mm

X ,MmY

)= Um[Ξm 0

](V m

)T,

then, for any (HX,HY ) ∈ Sn × S

n,

K ′(MmX,Mm

Y

)(HX,HY )

= Um[f [1](Ξm2) ◦ (Um

)T (Tβ

(Mm

X,MmY

)Tβ(HX,HY )T

+ Tβ(HX,HY )Tβ

(Mm

X,MmY

)T )Um

](Um

)T

= Um[Θm ◦ (V m

β

)TTβ(HX,HY )T Um

+ (1|β|1T|β| − Θm

) ◦ (Um)T

Tβ(HX,HY )V mβ

](Um

)T, (30)

where (Θm)ij = Ξmi

Ξmi +Ξm

j.

Therefore, we have that, for any (HX,HY ) ∈ Sn × S

n,

W(HX,HY ) − HX − HY

= limm→∞Υ ′(Mm

X ,MmY

)(HX,HY ) − HX − HY

= − limm→∞P m

[f [1](D)αα ◦ P T


f [1](D)βα ◦ P Tβ LZ(H)Pα K ′(Mm

X ,MmY )(HX,HY )

]

(P m)T

= −P

[f [1](D)αα ◦ P T


f [1](D)βα ◦ P Tβ LZ(H)Pα lim

m→∞K ′(MmX ,Mm

Y )(HX,HY )

]

P T

= −P

[f [1](D)αα ◦ P T



]

P T , (31)

where

S(HX,HY ) =U[Θ ◦ (Vβ)T Tβ(HX,HY )T U

+ (1|β|1T|β| − Θ

) ◦ UT Tβ(HX,HY )Vβ

]UT

and Θ ∈ Wβ(G).Let

Xm := P

[Xαα Sm

αβ

Smβα (Mm

X )ββ

]

P T and Ym := P

[Yαα T m

αβ

T mβα (Mm

Y )ββ

]

P T , (32)

where

Smαβ := (

MmX

)αβ

− XααD−1α

(Xαα

(Mm

X

)αβ

+ Yαα

(Mm

Y

)αβ

),


T mαβ := (

MmY

)αβ

− YααD−1α

(Xαα

(Mm

X

)αβ

+ Yαα

(Mm

Y

)αβ

),

with MmX := P T Mm

XP , MmY := P T Mm

Y P , X := P T XP and Y := P T YP .A simple calculation yields that

(Xm

)2 + (Ym)2

= P

[Am Sm

αβ(MmX )ββ + T m

αβ(MmY )ββ

(Smαβ(Mm

X )ββ + T mαβ(Mm

Y )ββ)T Eββ(MmX ,Mm

Y )

]

P T

= P

[Dα

Eββ(MmX ,Mm

Y )

]P T + O‖Mm‖2

= P m

[Dα

(Ξm)2

](P m

)T + O‖Mm‖2, (33)

where Am := Dα + SmαβSm

βα + T mαβT m

βα and P m =: [Pα PβUm].Consequently, there exists P m ∈ On(Gm) satisfying P m = P m +O(‖Mm‖2) such

that

Gm = P mΛm(P m

)T with Λm =[Dα

Dmβ

]

+ O(‖Mm‖2).

Define Zm := [Xm Ym], then we there exists Qm ∈ Om such that

[Xm Ym

]= P m[Λm 0

](Qm

)T. (34)

By the singular decomposition theorem, the orthogonal matrix Qm := [Qmα Qm

α V m]also satisfies (34). And then (P m, Qm) → (P , Q) as m → ∞, where P :=[Pα PβU ], Q := [Qα QαV ].

Next, we deduce the positive definiteness of (Xm)2 + (Ym)2 from the positivedefiniteness of its corresponding Schur complementary matrix. Indeed,

Eββ

(Mm

X,MmY

)− (Sm

αβ

(Mm

X

)ββ

+ T mαβ

(Mm

Y

)ββ

)T (Am)−1

× (Sm

αβ

(Mm

X

)ββ

+ T mαβ

(Mm

Y

)ββ

)

Eββ

(Mm

X ,MmY

)− (Sm

αβ

(Mm

X

)ββ

+ T mαβ

(Mm

Y

)ββ

)TD−1

α

× (Sm

αβ

(Mm

X

)ββ

+ T mαβ

(Mm

Y

)ββ

)

Eββ

(Mm

X ,MmY

)− 2(Mm

X

)ββ

SmβαD−1

α Smαβ

(Mm

X

)ββ

− 2(Mm

Y

)ββ

T mβαD−1

α T mαβ

(Mm

Y

)ββ

1

2Eββ

(Mm

X,MmY

)+ 1

2

((Mm

X

)βα

(Mm

X

)αβ

+ (Mm

Y

)βα

(Mm

Y

)αβ

)

− 1

2

((Mm

X

)ββ

Xαα + (Mm

Y

)ββ

Yαα

)D−1

α

(Xαα

(Mm

X

)αβ

+ Yαα

(Mm

Y

)αβ

)� 0,


where the second and the third inequalities are due to Lemma 2.1 and [11, Theo-rem 6.1.10], respectively, and we get the last inequality by Lemma 2.2. Then ΦFB(·, ·)is F-differentiable at (Xm,Ym). Therefore,

limm→∞Φ ′

FB

(Xm,Ym

)(HX,HY ) − HX − HY

= P

[f [1](Λ(G))αα ◦ P T

α LZ(H)Pα f [1](Λ(G))αβ ◦ P Tα LZ(H)Pβ

f [1](Λ(G))βα ◦ P Tβ LZ(H)Pα lim

m→∞f [1](Λ(Gm))ββ ◦ (P mβ )T LZm(H)P m

β

]

P T ,

(35)

where

limm→∞f [1](Λ

(Gm

))ββ

◦ (P mβ

)TLZm(H)P m

β


(Gm

))ββ

◦ (P mβ

)T (LXm(HX) + LYm(HY ))P m

β


(Gm

))ββ

◦ (P mβ

)T (P m

[Λ(Gm

) 12 0](

Qm)T[HX

HY

]

+ [HXHY ]Qm

[Λ(Gm)

12

0

](P m

)T )P m

β

= Θ ◦ QTβ

[HX

HY

]Pβ + (

1|β|1T|β| − Θ) ◦ P T

β [HX HY ]Qβ.

Comparing (31) with (35), we conclude that W(HX,HY ) = limm→∞ Φ ′FB(Xm,Ym)×

(HX,HY ). This proves that W ∈ ∂BΦFB(X,Y ). The proof is completed. �

From Proposition 4.1, we obtain the following result which plays an importantrole in many nonsmooth Newton-type methods.

Theorem 4.1 For any given X, Y ∈ Sn, let G := X2 + Y 2 have the spectral decom-

position as in (7) and Z := [X,Y ] have the singular value decomposition as in (19).Then W ∈ ∂BΦFB(X,Y ), if and only if there exists S ∈ ∂BK(0,0) such that for any(HX,HY ) ∈ S

n × Sn,

W(HX,HY ) =HX + HY

− P

[f [1](D)αα ◦ P T



]

P T .

(36)

For S ∈ ∂BK(0,0), there exist (U,V ) ∈ O|β| × On+|β| and Θ ∈ Wβ(G) such that forany (HX,HY ) ∈ S

n × Sn,

S(HX,HY ) = U

(

Θ ◦ QTβ

[HX

HY

]Pβ + (

1|β|1T|β| − Θ) ◦ P T

β [HX HY ]Qβ

)

UT , (37)


where P = [Pα PβU ], Q = [Qα QαV ].

Proof The first part of the theorem comes from Proposition 4.1, namely W ∈∂BΦFB(X,Y ) if and only if there exists S ∈ ∂BK(0,0) such that for any (HX,HY ) ∈S

n × Sn, W(HX,HY ) can be expressed as (36).

For S ∈ ∂BK(0,0), there exists a sequence {(MmX ,Mm

Y )} converging to (0,0) suchthat K(·, ·) is F-differentiable at (Mm

X ,MmY ) and S(HX,HY ) = limm→∞ K ′(Mm

X ,

MmY )(HX,HY ). Let Tβ(Mm

X ,MmY ) admit the following singular value decomposition

Tβ

(Mm

X ,MmY

)= Um[Ξm 0

](V m

)T, (38)

where Um ∈ O|β|, V m ∈ On+|β|. By taking a subsequence if necessary, we assumethat limm→∞(Um,V m) = (U,V ), let P m := [Pα PβUm], Qm := [Qα QαV m], thenlimm→∞(P m, Qm) = (P , Q).

It is easy to check the inclusion (P m, Qm) ∈ On,2n([X Y ]) and so that (P , Q) ∈On,2n([X Y ]). Therefore, S(HX,HY ) can be expressed as

S(HX,HY ) = limm→∞K ′(Mm

X,MmY

)(HX,HY )

= limm→∞Um

([Θm ◦ (V m

β

)TTβ(HX,HY )T Um

]

+ (1|β|1T|β| − Θm

) ◦ (Um)T

Tβ(HX,HY )T V mβ

)(Um

)T

=U[Θ ◦ V T

β Tβ(HX,HY )T + (1|β|1T|β| − Θ

) ◦ UT Tβ(HX,HY )Vβ

]UT ,

(39)

where

(Θm

)ij

= Ξmi

Ξmi + Ξm

j

, Θij = limm→∞

(Θm

)ij, i, j = 1, . . . , |β|.

Let Xm, Ym have the form as (32), define Gm := (Xm)2 + (Ym)2, then limm→∞(Xm,

Ym) = (X,Y ) and Gm → G = X2 +Y 2. Similar to the analysis in the proof of Propo-sition 4.1, we can demonstrate that

limm→∞

(Θm

)ij

= limm→∞

√λi+|α|(Gm)

√λi+|α|(Gm) +√

λj+|α|(Gm), i, j = 1, . . . , |β|,

which implies Θ ∈ Wβ(G). Since Tβ(HX,HY ) = P Tβ [HX HY ]Qα and Pβ = PβU ,

Qβ = QαVβ , we obtain (37) from (39). The proof is completed. �

Remark 4.1 From a simple inspection of the proof of Theorem 4.1, we draw that theorthogonal matrices U and V are the clusters of the sequences {Um} and {V m} whichsatisfy (38), respectively.

Theorem 4.2 For any given X,Y ∈ Sn, let G := X2 + Y 2 have the spectral decom-

position as in (7) and Z := [X Y ] have the singular value decomposition as in (19).


Then W ∈ ∂ΦFB(X,Y ), if and only if there exists S ∈ ∂K(0,0) such that for any(HX,HY ) ∈ S

n × Sn,

W(H) =HX + HY

− P

[f [1](D)αα ◦ P T



]

P T . (40)

Proof For any W ∈ ∂ΦFB(X,Y ), by Carathéodory theorem and the definition of∂ΦFB(X,Y ), there exists a positive integer l, such that W =∑l

i=1 ηiWi , where

ηi ∈ [0,1],l∑

i=1

ηi = 1

and Wi ∈ ∂BΦFB(X,Y ), i = 1, . . . , l.For any H := (HX,HY ) ∈ S

n × Sn, there exist P i ∈ On(G), Ui ∈ O|β| and V i ∈

On+|β| (i = 1, . . . , l) such that

W(H) =l∑

i=1

ηiWi(H)

= HX + HY

−l∑

i=1

ηiPi

[f [1](D)αα ◦ (P i

α)T LZ(H)P iα f [1](D)αβ ◦ (P i

α)T LZ(H)P iβ

f [1](D)βα ◦ (P iβ)T LZ(H)P i

α Si(HX,HY )

](P i)T

,

where

Si(HX,HY ) =Ui[Θi ◦ (V i

β

)TT i

β(HX,HY )T Ui

+ (1|β|1T|β| − Θi

) ◦ (Ui)T

T iβ(HX,HY )V i

β

](Ui)T

,

with Θi ∈ W (G).

By Lemma 2.4, we know that there exist Ri ∈ On(Λ(G)) and Ri ∈ O2n(Λ(G))

such that

P i = P

[Ri

α 0

0 Riβ

]

and Qi = Q

[Ri

α 0

0 Riα

]

, (41)

where G := ZT Z.Then simple calculation yields that

W(H) = HX + HY

− P

[f [1](D)αα ◦ P T


f [1](D)βα ◦ P Tβ LZ(H)Pα

∑li=1 ηiR

iβSi(HX,HY )(Ri

β)T

]

P T .


Next, we only need to show that RiβSi(Ri

β)T ∈ ∂BK(0,0), i.e., there exists a sequence{(Mm

X ,MmY )} converging to (0,0) such that K(·, ·) is F-differentiable at (Mm

X ,MmY )

and RiβSi(HX,HY )(Ri

β)T = limm→∞ K ′(MmX ,Mm

Y )(HX,HY ).For any i ∈ {1, . . . , l}, Proposition 4.1 implies that there is a sequence {(Mm

X ,MmY )}

in Sn × S

n which convergent to (0,0) such that Wi(HX,HY ) = limm→∞ Υ ′(MmX ,

MmY )(HX,HY ) and K(Mm

X ,MmY ) � 0. For every m, let Xm and Ym have the follow-

ing form:

Xm := P i

[Xαα (Sm

αβ)i

(Smβα)i (Mm

X )iββ

](P i)T and Ym := P i

[Yαα (T m

αβ)i

(T mβα)i (Mm

Y )iββ

](P i)T

,

(42)

where

(Sm

αβ

)i := (Mm

X

)iαβ

− XααD−1α

(Xαα

(Mm

X

)iαβ

+ Yαα

(Mm

Y

)iαβ

),

(T m

αβ

)i := (Mm

Y

)iαβ

− YααD−1α

(Xαα

(Mm

X

)iαβ

+ Yαα

(Mm

Y

)iαβ

),

with (MmX )i := (P i)T Mm

X P i , (MmY )i := (P i)T Mm

Y P i , X := (P i)T XP i and Y :=(P i)T YP i .

Similar discussion to Proposition 4.1, we know there exist orthogonal matrices(P m)i and (Qm)i such that

[Xm Ym

]= (P m

)i[Λ(Gm

) 12 0]((

Qm)i)T

,

where

(P m

)i =[Iα

(Um)i

]+ O

(‖Mm‖2) and(Qm

)i =[Iα

(V m)i

]+ O

(‖Mm‖2),

with ((Um)i, (V m)i) ∈ O|β|,n+|β|(T iβ(Mm

X ,MmY )).

This, together with (41) shows that

(P m

)i := P i(P m

)i = [PαRi

α PβRiβ

(Um

)i],

(Qm

)i := Qi(Qm

)i = [QαRi

α QαRiα

(V m

)i].

Taking a subsequence if necessary, we may assume that (Um)i → Ui and (V m)i →V i as m → ∞, so that limm→∞((P m)i, (Qm)i) = (P i , Qi), with P i :=[PαRi

α PβRiβUi] and Qi := [QαRi

α QαRiαV i].

Consequently,

Wi(HX,HY )

= limm→∞Φ ′

FB

(Xm,Ym

)(HX,HY )

= HX + HY


+ P i

[f [1](Λ(G))αα ◦ (P i

α)T LZ(H)P iα f [1](Λ(G))αβ ◦ (P i

α)T LZ(H)P iβ

f [1](Λ(G))βα ◦ (P iβ )T LZ(H)P i

α limm→∞f [1](Λ(Gm))ββ ◦ ((P m

β )i )T LZm (H)(P mβ )i

](P i)T

,

where

limm→∞f [1]((Ξm

)2) ◦ ((P mβ

)i)TLZm(H)

(P m

β

)i

= limm→∞f [1]((Ξm

)2) ◦ ((P mβ

)i)T (LXm(HX) + LYm(HY ))(

P mβ

)i

= limm→∞f [1]((Ξm

)2) ◦ ((P mβ

)i)T((P m

)i[Λ(Gm

) 12 0]((

Qm)i)T

[HX

HY

]

+ [HX HY ](Qm)i[Λ(Gm)

12

0

](P m

)i)(P m

β

)i

= Θi ◦ (V iβ

)TT i

β(HX,HY )T Ui + (1|β|1T|β| − Θi

) ◦ (Ui)T

T iβ(HX,HY )V i

β,

which together with the definition of P and Proposition 4.1 implies RiβSi(Ri

β)T ∈∂BK(0,0).

Conversely, suppose that there exists S ∈ ∂K(0,0) such that (40) holds. The def-inition of Clark’s generalized Jacobian implies that there exist Si ∈ ∂BK(0,0) and∑l

i=1 ηi = 1 such that S = ∑li=1 ηiS

i . By Theorem 4.1, we know that there exist(Ui,V i) ∈ O|β| × On+|β| and Θi ∈ Wβ(G) such that for any (HX,HY ) ∈ S

n × Sn,

Si(HX,HY ) =Ui[Θi ◦ (V i

β

)TT i

β(HX,HY )T Ui

+ (1|β|1T|β| − Θi

) ◦ (Ui)T

T iβ(HX,HY )V i

β

](Ui)T

. (43)

Lemma 2.4 implies that, for any (P i,Qi) ∈ On,2n([X Y ]), there exist Ri ∈On(Λ(G)) and Ri ∈ O2n(Λ(G)) such that

P i = P

[Ri

α 0

0 Riβ

]

and Qi = Q

[Ri

α 0

0 Riα

]

.

Simple calculation shows that

W(H) =l∑

i=1

ηiWi(H),

where

Wi(H) =HX + HY

− P i

[f [1](D)αα ◦ (P i

α)T LZ(H)P iα f [1](D)αβ ◦ (P i

α)T LZ(H)P iβ

f [1](D)βα ◦ (P iβ)T LZ(H)P i

α (Riβ)T Si(HX,HY )Ri

β

](P i)T

.

Hence, we only need to prove that Wi ∈ ∂BΦFB(X,Y ).


Since Si ∈ ∂BK(0,0), we have a sequence {(MmX ,Mm

Y )} in Sn × S

n converging to(0,0) and Si(HX,HY ) = limm→∞ K ′(Mm

X ,MmY )(HX,HY ).

Let Xm and Ym have the same form of (42), and similarly, we get

[Xm Ym

]= P(P m

)i[Dm 0

]((Qm

)iQ)T

, (44)

where

(P m

)i =[Iα

(Um)i

]+ O

(‖Mm‖2) and(Qm

)i =[Iα

(V m)i

]+ O

(‖Mm‖2),

with ((Um)i, (V m)i) ∈ O|β|,n+|β|(T iβ(Mm

X ,MmY )).

This, together with (41) shows that

(P m

)i := P(P m

)i = [P i

α

(Ri

α

)TP i

β

(Ri

β

)T (Um

)i],

(Qm

)i := Q(Qm

)i = [Qi

α

(Ri

α

)TQi

α

(Ri

α

)T (V m

)i].

Then similar discussion to the first part of the proof implies that

Wi(HX,HY ) = limm→∞Φ ′

FB

(Xm,Ym

)(HX,HY ),

i.e., Wi ∈ ∂BΦFB(X,Y ). This completes the proof. �

5 Conclusions

In this paper, we have developed the formulas for the directional derivative, the B-subdifferential and the generalized Jacobian. The formulas can be used to analyzeconvergence properties of nonsmooth equation approach for solving semidefiniteconic complementarity problems, in which the nonsmooth equation is constructedby using ΦFB. For instance, to obtain the superlinear convergence of the nonsmoothequation approach, the strong BD-regularity of the mapping at the solution point isrequired and the formula of ∂BΦFB will be employed.

We can find applications of the differential properties of ΦFB to mathematicalprograms with semidefinite conic complementarity constraints, which is one of ourfuture concerns in the study of SDP related optimization problems.

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Differential Properties of the Symmetric Matrix-Valued Fischer-Burmeister Function