1Introduction NonsingularityconditionsforFBsystemofnonlinearSDPs

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

Nonsingularity conditions for FB system of nonlinear SDPs

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

LetX 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)∈ Sⁿ+,

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 → Sⁿ are twice continuously differentiable functions,Sⁿ is the linear space of all n× n real symmetric matrices, and Sⁿ₊ is the cone of all n× n positive semidefinite matrices. By introducing a slack variable X ∈ Sⁿ+ for the conic constraint g(x)∈ Sⁿ+, we can rewrite the NLSDP (1) as follows:

min

(x,X)∈X×Sⁿf (x)

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

g(x)− X = 0, X∈ Sⁿ+.

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 Jx,XL(x, X, µ, S, Y ) = 0, h(x) = 0, g(x)− X = 0, −Y ∈ N_Sⁿ₊(X), (3) where the Lagrangian function L :X × Sⁿ× IR^m× Sⁿ× Sⁿ → IR is defined by

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

Jx,XL(x, X, µ, S, Y ) is the derivative of L at (x, X, µ, S, Y ) with respect to (x, X), and N_Sⁿ₊(X) denotes the normal cone of Sⁿ₊ at X in the sense of convex analysis [16]:

NSⁿ+(X) =

{ {Z ∈ Sⁿ: ⟨Z, W − X⟩ ≤ 0 ∀W ∈ Sⁿ} if X ∈ Sⁿ+,

∅ if X /∈ Sⁿ₊.

Recall that Φ :Sⁿ× Sⁿ→ Sⁿ is a semidefinite cone (SDC) complementarity function if Φ(X, Y ) = 0 ⇐⇒ X ∈ Sⁿ+, Y ∈ Sⁿ+, ⟨X, Y ⟩ = 0 ⇐⇒ −Y ∈ NSⁿ+(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 ) :=







Jx,XL(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− ΠSⁿ+(X − Y ) ∀X, Y ∈ Sⁿ and

Φ_FB(X, Y ) := (X + Y )−√

X²+ Y² ∀X, Y ∈ Sⁿ, (5)

respectively, where Π_Sⁿ

+(·) denotes the projection operator onto Sⁿ+. 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, Jzf (z) and Jzz²f (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 ∈ Sⁿ, we write X ≽ 0 (respectively, X ≻ 0) to mean X ∈ Sⁿ+

(respectively, X ∈ Sⁿ++). For a linear operatorA, we denote by A^∗ the adjoint ofA, and by ∥A∥2 the operator norm of A. For a linear operator A : Sⁿ → Sⁿ, we write A ≽ 0 (respectively, A ≻ 0) if ⟨W, A(W )⟩ ≥ 0 for any W ∈ Sⁿ (respectively, ⟨W, A(W )⟩ > 0 for any nonzero W ∈ Sⁿ). For any given sets of indices α and β, we designate by Aαβ

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 DΞ the set of points inO where Ξ is F-differentiable. Then Clarke’s Jacobian of Ξ at x is well defined [6]:

∂Ξ(x) := conv{∂BΞ(x)},

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

∂_BΞ(x) :=

{

V : V = lim

k→∞JxΞ(x^k), x^k → x, x^k∈ DΞ

} .

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 ∈ Sⁿ₊, it holds that

(A + B)^TZ(A + B) ≼ 2(A^TZA + B^TZB), (6) (A− B)^TZ(A− B) ≼ 2(A^TZA + B^TZB). (7) Proof. Fix any Z ∈ Sⁿ+. Then, for any n× m real matrices A and B, we have that

0 ≼ (A − B)^TZ(A− B) = (A^TZA + B^TZB)− (A^TZB + B^TZA), 0 ≼ (A + B)^TZ(A + B) = (A^TZA + B^TZB) + (A^TZB + B^TZA).

The first equation means that (A^TZB + B^TZA)≼ (A^TZA + B^TZB), which along with the second equality yields (6). The second equation implies that −(A^TZB + B^TZA) ≼ (A^TZA + B^TZB), which along with the first equality yields (7). 2

Lemma 2.2 Let X, Y ∈ Sⁿ with X²+ Y² ≻0. Then for any n × m real matrices A, B, A^TA + B^TB− (A^TX + B^TY )(X²+ Y²)⁻¹(XA + Y B)≽ 0.

Proof. Note that A^TA + B^TB − (A^TX + B^TY )(X²+ Y²)⁻¹(XA + Y B) is the Schur complement of X²+ Y² in the following block symmetric matrix

Σ =

[ X² + Y² XA + Y B (XA + Y B)^T A^TA + B^TB

] .

We only need to prove Σ≽ 0 (see [10, Theorem 7.7.6]). For any ζ = (ζ1, ζ₂)∈ IRⁿ×IR^m, ζ^TΣζ = ζ₁^T(X²+ Y²)ζ₁+ 2ζ₁^T(XA + Y B)ζ₂+ ζ₂^T(A^TA + B^TB)ζ₂

= ∥Xζ1+ Aζ2∥² +∥Y ζ1+ Bζ2∥² ≥ 0, which shows that Σ≽ 0. The proof is then complete. 2

For any given X∈ Sⁿ, letLX:Sⁿ→Sⁿbe the Lyapunov operator associated with X:

LX(Y ) := XY + Y X ∀Y ∈ Sⁿ.

We next study several properties of the Lyapunov operators associated with X, Y ∈ Sⁿ and Z ∈ Sⁿ₊ with Z² ≽ X²+ Y². To this end, we need to establish two trace inequalities.

Lemma 2.3 Let X, Y ∈ Sⁿ with X ≽ |Y |. Then, for any W ∈ Sⁿ, it holds that Trace(W XW X)≥ Trace(W Y W Y ).

Proof. Fix any W ∈ Sⁿ. 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 ∈ Sⁿ and Z ∈ Sⁿ+ satisfying Z ≽√

X²+ Y², we have Trace(W ZW Z)≥ Trace(W |X|W |X|) + Trace(W |Y |W |Y |) ∀W ∈ Sⁿ.

Proof. Fix any W ∈ Sⁿ. Applying Lemma 2.3, we readily obtain that Trace(W ZW Z)≥ Trace(

W√

X²+ Y²W√

X²+ Y² )

. (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 Sⁿ+× Sⁿ+, which means that for any A1, A2, B1, B2 ∈ Sⁿ+,

φ

(A₁+ A₂

2 ,B₁+ B₂ 2

)

≥ 1

2[φ(A₁, B₁) + φ(A₂, B₂)] .

Using this inequality with A₁ = B₁ = X² and A₂ = B₂ = Y², we obtain that 2φ

(X²+ Y²

2 ,X²+ Y² 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 ∈ Sⁿ and Z ∈ Sⁿ+, the following implication holds:

Z² ≽ X²+ Y² =⇒ L²Z ≽ L²X +L²Y.

Proof. Since Z² ≽ X²+ Y² and Z ∈ Sⁿ₊, from [1, Proposition V.1.8] it follows that Z ≽√

X² + Y².

Now choose a matrix W ∈ Sⁿ arbitrarily. Then, a simple computation yields that

⟨W, (L²Z− L²X − L²Y)W⟩ = 2[

Trace(W ZW Z) + Trace(W²Z²)− Trace(W XW X)

−Trace(W²X²)− Trace(W²Y²)− Trace(W Y W Y )]

= 2[

Trace(

W²(Z²− X²− Y²))

+ 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 Z² ≽ X² + Y², and the second one is using Z ≽

√X²+ Y² and Lemmas 2.4 and 2.3. Since W is arbitrary inSⁿ, the result follows. 2

Proposition 2.2 For any given X, Y ∈ Sⁿ and Z ∈ Sⁿ++, define A: Sⁿ× Sⁿ → Sⁿ by A(△U, △V ) := L⁻¹_Z LX(△U) + L⁻¹_Z LY(△V ) ∀△U, △V ∈ Sⁿ.

If Z² ≽ X²+ Y², then the linear operator A satisfies ∥A∥₂ ≤ 1, and consequently L⁻¹Z LX(△U) + L⁻¹_Z LY(△V ) ≤ √∥△U∥²+∥△V ∥² ∀△U, △V ∈ Sⁿ. (9) Proof. Assume that Z² ≽X²+ Y². By the definition ofA and Proposition 2.1, we have

AA^∗ =L⁻¹Z (L²X +L²Y)L⁻¹Z ≼ L⁻¹Z L²ZL⁻¹Z =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⁻¹Z LX(△U) + L⁻¹Z LY(△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_ββ² + Y_ββ² + X_βγX_γβ + Y_βαY_αβ)^1/2 ∀X, Y ∈ Sⁿ. (10) The following property of the function Γ will be used in the subsequent sections.

Proposition 2.3 Let X, Y ∈ Sⁿ be such that Γ(X, Y )≻ 0. Then for any G, H ∈ Sⁿ,

∥L⁻¹_{Γ(X,Y )}(X_βγG_γβ+ G_βγX_γβ)∥ ≤ 2√

|β||γ| ∥Gγβ∥,

∥L⁻¹_{Γ(X,Y )}(Y_βαH_αβ+ H_βαY_αβ)∥ ≤ 2√

|β||α| ∥Hαβ∥.

Proof. Let Γ(X, Y ) = Qβdiag(λ1, . . . , 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 eX_βγ := Q^T_βX_βγQ_γ and eY_βα := Q^T_βY_βαQ_α. Then, from the expression of Γ(X, Y ) and its spectral decomposition, it is easy to get that

λ²_i ≥

∑|γ|

k=1

Xe_ik² +

∑|α|

l=1

Ye_il² 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 ∈ Sⁿ, with eG_βγ = Q^T_βG_βγQ_γ and eH_βα = Q^T_βH_βαQ_α, we calculate that

Q^T_βL⁻¹_{Γ(X,Y )}(X_βγG_γβ+ G_βγX_γβ)Q_β = [∑_|γ|

k=1( eXikGekj+ eGikXekj) λ_i+ λ_j

]

1≤i,j≤|β|

,

Q^T_βL⁻¹_{Γ(X,Y )}(Y_βαH_αβ + H_βαY_αβ)Q_β =

[∑_|α|

l=1( eY_ilHe_lj + eH_ilYe_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 :Sⁿ× Sⁿ → Sⁿ to denote the function C(X, Y ) :=√

X²+ Y² ∀X, Y ∈ Sⁿ, (12) and for any given X, Y ∈ Sⁿ assume that C(X, Y ) has the spectral decomposition

C(X, Y ) = P diag(λ₁, . . . , λ_n)P^T = P DP^T, (13)

where P is an n× n orthogonal matrix, and D = diag(λ1, . . . , 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 : λi > 0} and β := {i : λi = 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 inSⁿ× Sⁿ; 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 ∈ Sⁿ, 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)∈ Sⁿ× Sⁿ has the following expression

(G + H)− P [L⁻¹Dκ

(LXeκκ( eG_κκ) +LYeκκ( eH_κκ) )

D⁻¹_κ ( eX_κκGe_κβ+ eY_κκHe_κβ) ( eG_βκXe_κκ+ eH_βκYe_κκ)D⁻¹_κ Θ( eG, eH)

]

P^T (14)

where eX := P^TXP , eY := P^TY P , eG := P^TGP , eH := P^THP , and Θ is defined by Θ(U, V ) :=

[

U_ββ² + V_ββ² + U_βκU_κβ + V_βκV_κβ −

(U_βκXe_κκ+ V_βκYe_κκ)D_κ⁻²( eX_κκU_κβ + eY_κκV_κβ) ]1/2

∀U, V ∈ Sⁿ. (15) Proof. Fix any G, H ∈ Sⁿ. Assume that (X, Y )̸= (0, 0). Then, for any t > 0, we have Φ_FB(X + tG, Y + tH)− ΦFB(X, Y ) = t(G + H)− △(t) (16) with

△(t) ≡[

C²(X, Y ) + t(LX(G) +LY(H)) + t²(G²+ H²)]1/2

− C(X, Y ).

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

△(t) := Pe ^T△(t)P = (D²+ fW )^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 = tf

(X eeG + eG eX + eY eH + eH eY )

+ t²

(Ge²+ eH² )

.

Since eX²+ eY² = D² and Dβ = 0, we have eX = diag( eXκκ, 0) and eY = diag( eYκκ, 0). So, fW = t

[ LXeκκ( eG_κκ) +LYeκκ( eH_κκ) Xe_κκGe_κβ + eY_κκHe_κβ GeβκXeκκ+ eHβκYeκκ 0

]

+

[ o(t) o(t)

o(t) t²

(Ge²_ββ+ eH_ββ² + eG_βκGe_κβ+ eH_βκHe_κβ ) ]

.

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







△(t)e κκ =L⁻¹_Dκ(fWκκ) + o(∥fW∥),

△(t)e κβ = D_κ⁻¹fW_κβ + o(∥fW∥), fW_ββ = e△(t)^Tκβ△(t)e κβ + e△(t)²ββ.

(18)

From the second equality of (18) and the expression of fW_κβ, it follows that

△(t)e κβ = tD⁻¹_κ

(Xe_κκGe_κβ + eY_κκHe_κβ )

+ o(t), (19)

and consequently,

△(t)e ^Tκβ△(t)e κβ = t²

(Ge_βκXe_κκ+ eH_βκYe_κκ )

D_κ⁻²

(Xe_κκGe_κβ + eY_κκHe_κβ )

+ o(t²).

This, together with the third equation of (18) and the expression of fW_ββ, implies that

△(t)e ²ββ = t²

(Ge_βκGe_κβ + eH_βκHe_κβ + eG²_ββ+ eH_ββ² )

−t²(

Ge_βκXe_κκ+ eH_βκYe_κκ )

D_κ⁻²

(Xe_κκGe_κβ + eY_κκHe_κβ )

+ o(t²).

Since D_β = 0, the expression of e△(t) in (17) implies that e△(t)ββ ≽ 0. Therefore,

limt↓0

△(t)e ββ

t = lim

t↓0

[ e△(t)²_ββ]^1/2

t = Θ( eG, eH).

In addition, from the first equation in (18) and the expression of fW_κκ, we have

△(t)e κκ = tL⁻¹_Dκ(

LXeκκ( eG_κκ) +LYeκκ( eH_κκ) )

+ o(t).

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

lim

t↓0

△(t)e t =

[ L⁻¹Dκ

(LXeκκ( eG_κκ) +LYeκκ( eH_κκ) )

D_κ⁻¹( eX_κκGe_κβ + eY_κκHe_κβ) ( eG_βκXe_κκ+ eH_βκYe_κκ)D_κ⁻¹ Θ( eG, 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)−√

G²+ H² = 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_βκU_κβ + V_βκV_κβ− (UβκXe_κκ+ V_βκYe_κκ)D⁻²_κ ( eX_κκU_κβ + eY_κκV_κβ)≽ 0

for all U, V ∈ Sⁿ. 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) ∈ Sⁿ× Sⁿ,

J ΦFB(X, Y )(G, H) = (G + H)− L⁻¹_{C(X,Y )}(LX(G) +LY(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 ΦFB is F-differentiable at (X, Y ), Φ^′

FB((X, Y ); (·, ·)) is a linear operator. But, letting (G1, H₁), (G₂, H₂)∈ Sⁿ× Sⁿ be such that G1 = G2 = 0, H1 = diag(0, I_|β|) and H2 =−H1, we obtain that

0 = Φ^′_FB((X, Y ); (G1, H1) + (G2, H2))

= Φ^′_FB((X, Y ); (G₁, H₁)) + Φ^′_FB((X, Y ); (G₂, H₂))

= −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) ∈ Sⁿ× Sⁿ, which is used to characterize the F-differentiable points of Θ in Lemma 3.2 below. Define Ω₁ :Sⁿ× Sⁿ→ IR^|β|×|κ| and Ω₂ :Sⁿ× Sⁿ → IR^|β|×|κ| by

Ω₁(U, V ) := U_βκ− (UβκXe_κκ+ V_βκYe_κκ)D⁻²_κ Xe_κκ ∀U, V ∈ Sⁿ and

Ω2(U, V ) := Vβκ− (UβκXeκκ+ VβκYeκκ)D⁻²_κ Yeκκ ∀U, V ∈ Sⁿ,

respectively. Noting that eX_κκ² + eY_κκ² = D_κ², we can rewrite the function Θ in (15) as Θ(U, V ) =[

U_ββ² + V_ββ² + Ω₁(U, V )Ω₁(U, V )^T+ Ω₂(U, V )Ω₂(U, V )^T]1/2

∀U, V ∈ Sⁿ. (21)

For any given U, V ∈ Sⁿ, assume that Θ(U, V ) has the following spectral decomposition Θ(U, V ) = RΛR^T = Rdiag(ϑ₁, . . . , ϑ_|β|)R^T,

where Λ = diag(ϑ₁, . . . , 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: ϑi > 0} and J := {i: ϑi = 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^TUββ]J β = 0, [R^TVββ]J β = 0, [R^TΩ1(U, V )]J κ = 0, [R^TΩ2(U, V )]J κ = 0. (22) Lemma 3.1 For any given (U, V )∈ Sⁿ× Sⁿ, 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)∈ Sⁿ× Sⁿ has the following expression

R

[ L⁻¹_ΛI[fW_II] Λ⁻¹_I Wf_IJ

Wf_IJ^TΛ⁻¹_I ( eΘJJ − fW_IJ^TΛ⁻²_I WfIJ)^1/2 ]

R^T, (23)

where eΘ := R^TΘ²(G, H)R, and fW := R^TW (G, H)R with W (G, H) given by W (G, H) := Ω₁(U, V )Ω₁(G, H)^T + Ω₁(G, H)Ω₁(U, V )^T +LUββ(G_ββ)

+LV_ββ(Hββ) + Ω2(U, V )Ω2(G, H)^T + Ω2(G, H)Ω2(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 )

= [

Θ²(U, V ) + tW (G, H) + t²Θ²(G, H)]1/2

− Θ(U, V ).

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

∆(t) := Re ^T∆(t)R = (

Λ²+ tfW + t²Θe )1/2

− Λ, (24)

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







∆(t)e _II = tL⁻¹ΛI[fW_II] + o(t),

∆(t)e _IJ = tΛ⁻¹_I fW_IJ + o(t),

tfW_JJ+ t²Θe_JJ = e∆(t)^T_IJ∆(t)e _IJ + e∆(t)²_JJ.

(25)

By equation (22) and the definition of fW , we have fW_JJ = 0. Then, from the last two equalities of (25), it follows that

∆(t)e ²_JJ = t²ΘeJJ− e∆(t)^T_IJ∆(t)e IJ = t²

(ΘeJJ − fW_IJ^TΛ⁻²_I fWIJ

)

+ o(t²).

Since Λ_J = 0, the expression of e∆(t) in (24) implies that e∆(t)_JJ ≽ 0. Therefore,

limt↓0

∆(t)e _JJ

t = lim

t↓0

√∆(t)e ²_JJ

t =

(Θe_JJ − fW_IJ^TΛ⁻²_I fW_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⁻¹ΛI[fW_II] Λ⁻¹_I Wf_IJ

Wf_IJ^TΛ⁻¹_I ( eΘ_JJ− fW_IJ^TΛ⁻²_I Wf_IJ)^1/2 ]

R^T.

If Θ(U, V ) = 0, then U_ββ = 0, V_ββ = 0, Ω₁(U, V ) = 0 and Ω₂(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 Sⁿ× Sⁿ. In fact, Θ is also globally Lipschitz continuous and strongly semismooth in Sⁿ× Sⁿ. Let Ψ(U, V ) := [U_ββ V_ββ Ω₁(U, V ) Ω₂(U, V )] for U, V ∈ Sⁿ, 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 Sⁿ× Sⁿ 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) ∈ Sⁿ× Sⁿ, J Θ(U, V )(G, H) = L⁻¹_{Θ(U,V )}[

(U_βκG_κβ + G_βκU_κβ) + (V_βκH_κβ + H_βκV_κβ)

−(

G_βκXe_κκ+ H_βκYe_κκ )

D⁻²_κ

(Xe_κκU_κβ+ eY_κκV_κβ )

−(

UβκXeκκ+ VβκYeκκ

) D_κ⁻²

(XeκκGκβ + eYκκHκβ

)

+LUββ(G_ββ) +LVββ(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^TΛ⁻²_I fW_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 Hββ ≻ 0, we have Ω1(G, H) = 0 and Ω₂(G, H) = 0 which, together with [R^TU_ββ]_{J β} = 0 and [R^TV_ββ]_{J β} = 0, implies that fW_{J I} = [R^TW (G, H)R]_{J I} = 0. Note that Θ²(G, H) = G²_ββ+ H_ββ² . Then, ( eΘ_JJ − fW_IJ^TΛ⁻²_I fW_IJ)^1/2 =

√

[R^T(G²_ββ+ H_ββ² )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^TGP, P^THP )≻ 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 ∈ Sⁿ, let Ψ_FB(·, ·) ≡ Φ^′_FB((X, Y ); (·, ·)). Then,

∂_BΦ_FB(X, Y ) = ∂_BΨ_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 ∈ Sⁿ, let C(X, Y ) have the spectral decomposition as in (13). Then, a (U, V) ∈ ∂BΦ_FB(X, Y ) (respectively, ∂Φ_FB(X, Y )) if and only if there exists a (G, H) ∈ ∂BΘ(0, 0) (respectively, ∂Θ(0, 0)) such that for any G, H ∈ Sⁿ,

(I − U) (G) + (I − V) (H)

= P



L⁻¹Dκ

(LXeκκ( eG_κκ) +LYeκκ( eH_κκ) )

D_κ⁻¹

(Xe_κκGe_κβ + eY_κκHe_κβ ) (Ge_βκXe_κκ+ eH_βκYe_κκ

)

D_κ⁻¹ G( eG) +H( eH)



 P^T, (26)

where eX := P^TXP , eY := P^TY P , eG := P^TGP , and eH := P^THP .

Proof. For any G, H ∈ Sⁿ, let Ψ(G, H) := (P^TGP, P^THP ). Define Ξ :Sⁿ× Sⁿ→ Sⁿ by Ξ(S, T ) := P

[L⁻¹_Dκ(

L_Xeκκ(S_κκ)+L_Yeκκ(T_κκ))

D_κ⁻¹( eX_κκS_κβ+ eY_κκT_κβ) (SβκXeκκ+ TβκYeκκ)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 ∈ Sⁿ. Note that Ξ is globally Lipschitz continuous in Sⁿ× Sⁿ by the remarks after (21), and J Ψ(G, H) for any G, H ∈ Sⁿis onto. Applying [3, Lemma 2.1] to the composite mapping Ξ◦ Ψ at (0, 0), we have that ∂B(Ξ◦ Ψ)(0, 0) = ∂BΞ(Ψ(0, 0))J Ψ(0, 0) = ∂BΞ(0, 0)Ψ. So,

∂_BΨ_FB(0, 0) = (I, I) − ∂BΞ(0, 0)Ψ.

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