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1.Introduction ShaohuaPan, ShujunBi, andJein-ShanChen NonsingularityConditionsforFBSystemofReformulatingNonlinearSecond-OrderConeProgramming ResearchArticle

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Volume 2013, Article ID 602735,21pages http://dx.doi.org/10.1155/2013/602735

Research Article

Nonsingularity Conditions for FB System of Reformulating Nonlinear Second-Order Cone Programming

Shaohua Pan,

1

Shujun Bi,

1

and Jein-Shan Chen

2

1Department of Mathematics, South China University of Technology, Guangzhou 510641, China

2Mathematics Division, National Center for Theoretical Sciences, Taipei 11677, Taiwan

Correspondence should be addressed to Jein-Shan Chen; jschen@math.ntnu.edu.tw Received 2 June 2012; Accepted 9 December 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2013 Shaohua Pan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions:

the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

1. Introduction

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

min𝜁∈R𝑛 𝑓 (𝜁) s.t. ℎ (𝜁) = 0,

𝑔 (𝜁) ∈ K,

(1)

where𝑓 : R𝑛 → R, ℎ : R𝑛 → R𝑚, and𝑔 : R𝑛 → R𝑛are given twice continuously differentiable functions, andK is the Cartesian product of some second-order cones, that is,

K := K𝑛1× K𝑛2× ⋅ ⋅ ⋅ × K𝑛𝑟, (2)

with𝑛1+ ⋅ ⋅ ⋅ + 𝑛𝑟 = 𝑛 and K𝑛𝑗 being the second-order cone (SOC) inR𝑛𝑗defined by

K𝑛𝑗 := {(𝑥𝑗1, 𝑥𝑗2) ∈ R × R𝑛𝑗−1 | 𝑥𝑗1≥ 󵄩󵄩󵄩󵄩󵄩𝑥𝑗2󵄩󵄩󵄩󵄩󵄩} . (3)

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

𝜁,𝑥∈Rmin𝑛 𝑓 (𝜁) s.t. ℎ (𝜁) = 0,

𝑔 (𝜁) − 𝑥 = 0, 𝑥 ∈ K.

(4)

In this paper, we will concentrate on this equivalent formula- tion of problem (1).

Let𝐿 : R𝑛× R𝑛× R𝑚× R𝑛× K → R be the Lagrangian function of problem (4)

𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) := 𝑓 (𝜁) + ⟨𝜇, ℎ (𝜁)⟩ + ⟨𝑔 (𝜁) − 𝑥, 𝑠⟩ − ⟨𝑥, 𝑦⟩, (5) and denote byNK(𝑥) the normal cone of K at 𝑥 in the sense of convex analysis [1]:

NK(𝑥) = { {𝑑 ∈ R𝑛: ⟨𝑑, 𝑧 − 𝑥⟩ ≤ 0 ∀𝑧 ∈ K} , if 𝑥 ∈ K,

0, if𝑥 ∉ K.

(6)

(2)

Then the Karush-Kuhn-Tucker (KKT) conditions for (4) take the following form:

J𝜁,𝑥𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) = 0, ℎ (𝜁) = 0,

𝑔 (𝜁) − 𝑥 = 0, −𝑦 ∈ NK(𝑥) , (7) whereJ𝜁,𝑥𝐿(𝜁, 𝑥, 𝜇, 𝑠, 𝑦) is the derivative of 𝐿 at (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) with respect to(𝜁, 𝑥). Recall that 𝜙socis an SOC complemen- tarity function associated with the coneK if

𝜙soc(𝑥, 𝑦) = 0 ⇐⇒ 𝑥 ∈ K, 𝑦 ∈ K,

⟨𝑥, 𝑦⟩ = 0 ⇐⇒ −𝑦 ∈ NK(𝑥) . (8) With an SOC complementarity function𝜙socassociated with K, we may reformulate the KKT optimality conditions in (7) as the following nonsmooth system:

𝐸 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) :=[[[ [

J𝜁,𝑥𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) ℎ (𝜁) 𝑔 (𝜁) − 𝑥 𝜙soc(𝑥, 𝑦)

]] ] ]

= 0. (9)

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

𝜙NR(𝑥, 𝑦) := 𝑥 − ΠK(𝑥 − 𝑦) , ∀𝑥, 𝑦 ∈ R𝑛,

𝜙FB(𝑥, 𝑦) := (𝑥 + 𝑦) − √𝑥2+ 𝑦2, ∀𝑥, 𝑦 ∈ R𝑛, (10) whereΠK(⋅) is the projection operator onto the closed convex coneK, 𝑥2= 𝑥 ∘ 𝑥 means the Jordan product of 𝑥 and itself, and√𝑥 denotes the unique square root of 𝑥 ∈ K. It turns out that the FB SOC complementarity function𝜙FBenjoys almost all favorable properties of the NR SOC complementarity function𝜙NR(see [2]). Also, the squared norm of𝜙FBinduces a continuously differentiable merit function with globally Lipschitz continuous derivative [3,4]. This greatly facilitates the globalization of the semismooth Newton method [5,6]

for solving the FB nonsmooth system of KKT conditions:

𝐸FB(𝜁, 𝑥, 𝜇, 𝑠, 𝑦) :=[[[ [

J𝜁,𝑥𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) ℎ (𝜁) 𝑔 (𝜁) − 𝑥 𝜙FB(𝑥, 𝑦)

]] ] ]

= 0. (11)

Recently, with the help of [7, Theorem 30] and [8, Lemma 11], Wang and Zhang [9] gave a characterization for the strong regularity of the KKT point of the SOCP (1) via the nonsingularity study of Clarke’s Jacobian of the NR nonsmooth system

𝐸NR(𝜁, 𝑥, 𝜇, 𝑠, 𝑦) :=[[[ [

J𝜁,𝑥𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) ℎ (𝜁) 𝑔 (𝜁) − 𝑥 𝜙NR(𝑥, 𝑦)

]] ] ]

= 0. (12)

They showed that the strong regularity of the KKT point, the nonsingularity of Clarke’s Jacobian of𝐸NR at the KKT

point, and the strong second-order sufficient condition and constraint nondegeneracy [7] are all equivalent. These non- singularity conditions are better structured than those of [10] for the nonsingularity of the𝐵-subdifferential of the NR system. Then, it is natural to ask the following: is it possible to obtain a characterization for the strong regularity of the KKT point by studying the nonsingularity of Clarke’s Jacobian of 𝐸FB. Note that up till now one even does not know whether the𝐵-subdifferential of the FB system is nonsingular or not without the strict complementarity assumption.

In this work, for a locally optimal solution to the nonlin- ear SOCP (4), under Robinson’s constraint qualification, we show that the strong second-order sufficient condition and constraint nondegeneracy introduced in [7], the nonsingular- ity of Clarke’s Jacobian of𝐸FBat the KKT point, and the strong regularity of the KKT point are equivalent to each other. This, on the one hand, gives a new characterization for the strong regularity of the KKT point and, on the other hand, provides a mild condition to guarantee the quadratic convergence rate of the semismooth Newton method [5,6] for the FB system.

Note that parallel results are obtained recently for the FB system of the nonlinear semidefinite programming (see [11]);

however, we do not duplicate them. As will be seen in Sections 3and4, the analysis techniques here are totally different from those in [11]. It seems hard to put them together in a unified framework under the Euclidean Jordan algebra. The main reason causing this is due to completely different analysis when dealing with the Clarke Jacobians associated with FB SOC complementarity function and FB semidefinite cone complementarity function.

Throughout this paper, 𝐼 denotes an identity matrix of appropriate dimension,R𝑛 (𝑛 > 1) denotes the space of 𝑛- dimensional real column vectors, and R𝑛1 × ⋅ ⋅ ⋅ × R𝑛𝑟 is identified withR𝑛1+⋅⋅⋅+𝑛𝑟. Thus,(𝑥1, . . . , 𝑥𝑟) ∈ R𝑛1×⋅ ⋅ ⋅×R𝑛𝑟is viewed as a column vector inR𝑛1+⋅⋅⋅+𝑛𝑟. The notations int K𝑛, bd K𝑛, and bd+K𝑛denote the interior, the boundary, and the boundary excluding the origin ofK𝑛, respectively. For any 𝑥 ∈ R𝑛, we write𝑥 ⪰K𝑛 0 (resp., 𝑥 ≻K𝑛 0) if 𝑥 ∈ K𝑛(resp., 𝑥 ∈ int K𝑛). For any given real symmetric matrix𝐴, we write 𝐴 ⪰ 0 (resp., 𝐴 ≻ 0) if 𝐴 is positive semidefinite (resp., positive definite). In addition,J𝜔𝑓(𝜔) and J2𝜔𝜔𝑓(𝜔) denote the derivative and the second-order derivative, respectively, of a twice differentiable function 𝑓 with respect to the variable𝜔.

2. Preliminary Results

First we recall from [12] the definition of Jordan product and spectral factorization.

Definition 1. The Jordan product of𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2) ∈ R × R𝑛−1is given by

𝑥 ∘ 𝑦 := (⟨𝑥, 𝑦⟩ , 𝑥1𝑦2+ 𝑦1𝑥2) . (13)

Unlike scalar or matrix multiplication, the Jordan product is not associative in general. The identity element under this product is𝑒 := (1, 0, . . . , 0)𝑇 ∈ R𝑛, that is,𝑒 ∘ 𝑥 = 𝑥 for all

(3)

𝑥 ∈ R𝑛. For each𝑥 = (𝑥1, 𝑥2) ∈ R × R𝑛−1, we define the associated arrow matrix by

𝐿𝑥:= [𝑥1 𝑥2𝑇

𝑥2 𝑥1𝐼] . (14)

Then it is easy to verify that𝐿𝑥𝑦 = 𝑥 ∘ 𝑦 for any 𝑥, 𝑦 ∈ R𝑛. Recall that each𝑥 = (𝑥1, 𝑥2) ∈ R × R𝑛−1admits a spectral factorization, associated withK𝑛, of the form

𝑥 = 𝜆1(𝑥) 𝑢(1)𝑥 + 𝜆2(𝑥) 𝑢(2)𝑥 , (15) where𝜆1(𝑥), 𝜆2(𝑥) ∈ R and 𝑢(1)𝑥 , 𝑢(2)𝑥 ∈ R𝑛are the spectral values and the associated spectral vectors of𝑥, respectively, with respect to the Jordan product, defined by

𝜆𝑖(𝑥) := 𝑥1+ (−1)𝑖󵄩󵄩󵄩󵄩𝑥2󵄩󵄩󵄩󵄩, 𝑢(𝑖)𝑥 := 1

2( 1

(−1)𝑖̃𝑥2) , for 𝑖 = 1, 2, (16) with̃𝑥2= 𝑥2/‖𝑥2‖ if 𝑥2 ̸= 0 and otherwise being any vector in R𝑛−1satisfying‖̃𝑥2‖ = 1.

Definition 2. The determinant of a vector𝑥 ∈ R𝑛is defined as det(𝑥) := 𝜆1(𝑥)𝜆2(𝑥), and a vector 𝑥 is said to be invertible if its determinant det(𝑥) is nonzero.

By the formula of spectral factorization, it is easy to compute that the projection of𝑥 ∈ R𝑛onto the closed convex coneK𝑛, denoted byΠK𝑛(𝑥), has the expression

ΠK𝑛(𝑥) = max (0, 𝜆1(𝑥)) 𝑢(1)𝑥 + max (0, 𝜆2(𝑥)) 𝑢(2)𝑥 . (17) Define|𝑥| := 2ΠK𝑛(𝑥) − 𝑥. Then, using the expression of ΠK𝑛(𝑥), it follows that

|𝑥| = 󵄨󵄨󵄨󵄨𝜆1(𝑥)󵄨󵄨󵄨󵄨 𝑢(1)𝑥 + 󵄨󵄨󵄨󵄨𝜆2(𝑥)󵄨󵄨󵄨󵄨 𝑢(2)𝑥 . (18) The spectral factorization of the vectors𝑥, 𝑥2,√𝑥 and the matrix𝐿𝑥have various interesting properties (see [13]). We list several properties that we will use later.

Property 3. For any𝑥 = (𝑥1, 𝑥2) ∈ R × R𝑛−1with spectral factorization (15), we have the following.

(a)𝑥2= 𝜆21(𝑥)𝑢(1)𝑥 + 𝜆22(𝑥)𝑢(2)𝑥 ∈ K𝑛.

(b) If 𝑥 ∈ K𝑛, then 0 ≤ 𝜆1(𝑥) ≤ 𝜆2(𝑥) and √𝑥 =

√𝜆1(𝑥)𝑢(1)𝑥 + √𝜆2(𝑥)𝑢(2)𝑥 .

(c) If𝑥 ∈ int K𝑛, then0 < 𝜆1(𝑥) ≤ 𝜆2(𝑥) and 𝐿𝑥 is invertible with

𝐿−1𝑥 = 1 det(𝑥)[[

[

𝑥1 −𝑥𝑇2

−𝑥2 det(𝑥)

𝑥1 𝐼 + 𝑥2𝑥𝑇2 𝑥1

]] ]

. (19)

(d)𝐿𝑥 ⪰ 0 (resp., 𝐿𝑥 ≻ 0) if and only if 𝑥 ∈ K𝑛 (resp., 𝑥 ∈ int K𝑛).

The following lemma states a result for the arrow matrices associated with𝑥, 𝑦 ∈ R𝑛and𝑧 ⪰K𝑛√𝑥2+ 𝑦2, which will be used in the next section to characterize an important property for the elements of Clarke’s Jacobian of𝜙FBat a general point.

Lemma 4. For any given 𝑥, 𝑦 ∈ R𝑛and𝑧 ≻K𝑛0, if 𝑧2K𝑛𝑥2+ 𝑦2, then

󵄩󵄩󵄩󵄩󵄩[𝐿−1𝑧 𝐿𝑥 𝐿−1𝑧 𝐿𝑦]󵄩󵄩󵄩󵄩󵄩2≤ 1, (20) where ‖𝐴‖2 means the spectral norm of a real matrix 𝐴.

Consequently, it holds that

󵄩󵄩󵄩󵄩󵄩𝐿−1𝑧 𝐿𝑥Δ𝑢 + 𝐿−1𝑧 𝐿𝑦Δ𝑣󵄩󵄩󵄩󵄩󵄩 ≤ √‖Δ𝑢‖2+ ‖Δ𝑣‖2, ∀Δ𝑢, Δ𝑣 ∈ R𝑛. (21) Proof. Let𝐴 = [𝐿−1𝑧 𝐿𝑥 𝐿−1𝑧 𝐿𝑦]. From [13, Proposition 3.4], it follows that

𝐴𝐴𝑇= 𝐿−1𝑧 (𝐿2𝑥+ 𝐿2𝑦) 𝐿−1𝑧 ⪯ 𝐿−1𝑧 𝐿2𝑧𝐿−1𝑧 = 𝐼. (22) This shows that‖𝐴‖2≤ 1, and the first part follows. Note that, for any𝜉 ∈ R2𝑛,

󵄩󵄩󵄩󵄩𝐴𝜉󵄩󵄩󵄩󵄩2= 𝜉𝑇𝐴𝑇𝐴𝜉 ≤ 𝜆max(𝐴𝑇𝐴) 󵄩󵄩󵄩󵄩𝜉󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝜉󵄩󵄩󵄩󵄩2. (23) By letting𝜉 = (Δ𝑢, Δ𝑣) ∈ R𝑛×R𝑛, we immediately obtain the second part.

The following two lemmas state the properties of𝑥, 𝑦 with 𝑥2 + 𝑦2 ∈ bd K𝑛 which are often used in the subsequent sections. The proof ofLemma 5is given in [3, Lemma 2].

Lemma 5. For any 𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2) ∈ R × R𝑛−1 with𝑥2+ 𝑦2∈ bd K𝑛, one has

𝑥21= 󵄩󵄩󵄩󵄩𝑥2󵄩󵄩󵄩󵄩2, 𝑦12= 󵄩󵄩󵄩󵄩𝑦2󵄩󵄩󵄩󵄩2,

𝑥1𝑦1= 𝑥𝑇2𝑦2, 𝑥1𝑦2= 𝑦1𝑥2. (24) Lemma 6. For any 𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2) ∈ R × R𝑛−1, let 𝑤 = (𝑤1, 𝑤2) := 𝑥2+ 𝑦2.

(a) If𝑤 ∈ bd K𝑛, then for any𝑔 = (𝑔1, 𝑔2), ℎ = (ℎ1, ℎ2) ∈ R × R𝑛−1, it holds that

(𝑥1𝑥2+ 𝑦1𝑦2)𝑇(𝑥1𝑔2+ 𝑔1𝑥2+ 𝑦12+ ℎ1𝑦2)

= (𝑥21+ 𝑦21) (𝑥𝑇𝑔 + 𝑦𝑇ℎ) . (25)

(b) If𝑤 ∈ bd+K𝑛, then the following four equalities hold 𝑥1𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 𝑥2, 𝑥𝑇2𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 𝑥1, 𝑦1𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 𝑦2, 𝑦2𝑇𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 𝑦1;

(26)

(4)

and consequently the expression of𝜙FB(𝑥, 𝑦) can be simplified as

𝜙FB(𝑥, 𝑦) = (

𝑥1+ 𝑦1− √𝑥21+ 𝑦12 𝑥2+ 𝑦2−𝑥1𝑥2+ 𝑦1𝑦2

√𝑥21+ 𝑦12

) . (27)

Proof. (a) The result is direct by the equalities ofLemma 5 since𝑥2+ 𝑦2∈ bd K𝑛.

(b) Since 𝑤 ∈ bd+K𝑛, we must have𝑤2 = 2(𝑥1𝑥2 + 𝑦1𝑦2) ̸= 0. UsingLemma 5,𝑤2 = 2(𝑥1𝑥2+ 𝑦1𝑦2) and ‖𝑤2‖ = 𝑤1 = 2(𝑥21 + 𝑦12), we easily obtain the first part. Note that 𝜙FB(𝑥, 𝑦) = (𝑥 + 𝑦) − √𝑤. UsingProperty 3(b) andLemma 5 yields (27).

When𝑥, 𝑦 ∈ bd K𝑛satisfies the complementary condi- tion, we have the following result.

Lemma 7. For any given 𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2) ∈ R × R𝑛−1, if𝑥, 𝑦 ∈ bd K𝑛 and⟨𝑥, 𝑦⟩ = 0, then there exists a constant𝛼 > 0 such that 𝑥1= 𝛼𝑦1and𝑥2= −𝛼𝑦2.

Proof. Since𝑥, 𝑦 ∈ bd K𝑛, we have that𝑥1 = ‖𝑥2‖ and 𝑦1 =

‖𝑦2‖, and consequently,

0 = ⟨𝑥, 𝑦⟩ = 𝑥1𝑦1+ 𝑥𝑇2𝑦2= 󵄩󵄩󵄩󵄩𝑥2󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑦2󵄩󵄩󵄩󵄩 + 𝑥𝑇2𝑦2. (28) This means that there exists𝛼 > 0 such that 𝑥2 = −𝛼𝑦2, and then𝑥1= 𝛼𝑦1.

Next we recall from [14] the strong regularity for a solu- tion of generalized equation

0 ∈ 𝜙 (𝑧) + N𝐷(𝑧) , (29) where𝜙 is a continuously differentiable mapping from a finite dimensional real vector spaceZ to itself, 𝐷 is a closed convex set inZ, and N𝐷(𝑧) is the normal cone of 𝐷 at 𝑧. As will be shown inSection 4, the KKT condition (7) can be written in the form of (29).

Definition 8. We say that𝑧 is a strongly regular solution of the generalized equation (29) if there exist neighborhoodB of the origin0 ∈ Z and V of 𝑧 such that for every 𝛿 ∈ B, the linearized generalized equation𝛿 ∈ 𝜙(𝑧) + J𝑧𝜙(𝑧)(𝑧 − 𝑧) + N𝐷(𝑧) has a unique solution in V, denoted by 𝑧V(𝛿), and the mapping𝑧V: B → V is Lipschitz continuous.

To close this section, we recall from [15] Clarke’s (gen- eralized) Jacobian of a locally Lipschitz mapping. Let 𝑆 ⊂ R𝑛 be an open set and Ξ : 𝑆 → R𝑛 a locally Lipschitz continuous function on 𝑆. By Rademacher’s theorem, Ξ is almost everywhere𝐹(r´echet)-differentiable in 𝑆. We denote by 𝑆Ξ the set of points in 𝑆 where Ξ is 𝐹-differentiable.

Then Clarke’s Jacobian of Ξ at 𝑦 is defined by 𝜕Ξ(𝑦) :=

conv{𝜕𝐵Ξ(𝑦)}, where “conv” means the convex hull, and 𝐵- subdifferential𝜕𝐵Ξ(𝑦), a name coined in [16], has the form

𝜕𝐵Ξ (𝑦) := {𝑉 : 𝑉 = lim

𝑘 → ∞J𝑦Ξ (𝑦𝑘) , 𝑦𝑘󳨀→ 𝑦, 𝑦𝑘∈ 𝑆Ξ} . (30)

For the concept of (strong) semismoothness, please refer to the literature [5,6].

Unless otherwise stated, in the rest of this paper, for any 𝑥 ∈ R𝑛 (𝑛 > 1), we write 𝑥 = (𝑥1, 𝑥2), where 𝑥1 is the first component of𝑥 and 𝑥2is a column vector consisting of the remaining𝑛 − 1 entries of 𝑥. For any 𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2) ∈ R × R𝑛−1, let

𝑤 = 𝑤 (𝑥, 𝑦) := 𝑥2+ 𝑦2, 𝑤̃2:= 𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩, if 𝑤2 ̸= 0, 𝑧 = 𝑧 (𝑥, 𝑦) = √𝑤 (𝑥, 𝑦).

(31)

3. Directional Derivative and 𝐵-Subdifferential

The function𝜙FB is directionally differentiable everywhere by [2, Corollary3.3]. But, to the best of our knowledge, the expression of its directional derivative is not given in the literature. In this section, we derive its expression and then prove that the 𝐵-subdifferential of 𝜙FB at a general point coincides with that of its directional derivative function at the origin. Throughout this section, we assume thatK = K𝑛. Proposition 9. For any given 𝑥 = (𝑥1, 𝑥2), 𝑦 = (𝑦1, 𝑦2) ∈ R×R𝑛−1, the directional derivative𝜙FB󸀠 ((𝑥, 𝑦); (𝑔, ℎ)) of 𝜙FBat (𝑥, 𝑦) with the direction (𝑔, ℎ) has the following form.

(a) If(𝑥, 𝑦) = (0, 0), then 𝜙FB󸀠 ((𝑥, 𝑦); (𝑔, ℎ)) = 𝜙FB(𝑔, ℎ).

(b) If𝑥2+ 𝑦2 ∈ int K𝑛, then𝜙󸀠FB((𝑥, 𝑦); (𝑔, ℎ)) = (𝐼 − 𝐿−1𝑧 𝐿𝑥)𝑔 + (𝐼 − 𝐿−1𝑧 𝐿𝑦)ℎ.

(c) If𝑥2+ 𝑦2∈ bd+K𝑛, then 𝜙󸀠FB((𝑥, 𝑦); (𝑔, ℎ))

= (𝑔 + ℎ) −𝜑 (𝑔, ℎ)

2 ( 1−̃𝑤2) + 𝑥𝑇2𝑔2+ 𝑦2𝑇2 2√𝑥21+ 𝑦12 ( 0𝑤̃2)

− 1

2√𝑥21+ 𝑦12( 𝑥𝑇𝑔 + 𝑦𝑇

2𝑥1𝑔2+ 𝑔1𝑥2+ 2𝑦12+ ℎ1𝑦2) , (32) where𝑔 = (𝑔1, 𝑔2), ℎ = (ℎ1, ℎ2) ∈ R × R𝑛−1, and 𝜑 : R𝑛× R𝑛 → R is defined by

𝜑 (𝑔, ℎ)

:=√(𝑥1𝑔1−𝑥𝑇2𝑔2+𝑦11−𝑦2𝑇2)2+󵄩󵄩󵄩󵄩𝑥12−ℎ1𝑥2+𝑔1𝑦2−𝑦1𝑔2󵄩󵄩󵄩󵄩2

√𝑥21+ 𝑦12 .

(33)

Proof. Part (a) is immediate by noting that𝜙FBis a positively homogeneous function. Part (b) is due to [13, Proposition 5.2]. We next prove part (c) by two subcases as shown in the following. In the rest of proof, we let𝜆1, 𝜆2 with𝜆1 ≤ 𝜆2

(5)

denote the spectral values of𝑤. Since 𝑤 = 𝑥2+ 𝑦2∈ bd+K𝑛, we have𝑤2 ̸= 0, and fromLemma 6(b) it follows that

𝑤1= 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 2󵄩󵄩󵄩󵄩𝑥1𝑥2+ 𝑦1𝑦2󵄩󵄩󵄩󵄩

= 2 󵄩󵄩󵄩󵄩󵄩𝑥21𝑤̃2+ 𝑦12𝑤̃2󵄩󵄩󵄩󵄩󵄩 = 2 (𝑥21+ 𝑦12) ,

𝜆1= 𝑤1− 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 0, 𝜆2= 𝑤1+ 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 = 4(𝑥21+ 𝑦12) . (34)

(c.1):(𝑥 + 𝑡𝑔)2+ (𝑦 + 𝑡ℎ)2∈ bd+K𝑛for sufficiently small 𝑡 > 0. In this case, fromLemma 6(b), we know that𝜙FB(𝑥 + 𝑡𝑔, 𝑦 + 𝑡ℎ) has the following expression:

(

(𝑥1+ 𝑦1) + 𝑡 (𝑔1+ ℎ1) − √(𝑥1+ 𝑡𝑔1)2+ (𝑦1+ 𝑡ℎ1)2 (𝑥2+ 𝑦2) + 𝑡 (𝑔2+ ℎ2)

−(𝑥1+ 𝑡𝑔1) (𝑥2+ 𝑡𝑔2) + (𝑦1+ 𝑡ℎ1) (𝑦2+ 𝑡ℎ2)

√(𝑥1+ 𝑡𝑔1)2+ (𝑦1+ 𝑡ℎ1)2

) .

(35)

Let [𝜙FB(𝑥, 𝑦)]1 be the first element of 𝜙FB(𝑥, 𝑦) and [𝜙FB(𝑥, 𝑦)]2the vector consisting of the rest𝑛−1 components of𝜙FB(𝑥, 𝑦). By the above expression of 𝜙FB(𝑥 + 𝑡𝑔, 𝑦 + 𝑡ℎ),

lim𝑡↓0

[𝜙FB(𝑥 + 𝑡𝑔, 𝑦 + 𝑡ℎ)]1− [𝜙FB(𝑥, 𝑦)]1 𝑡

= (𝑔1+ ℎ1) − lim

𝑡↓0

√(𝑥1+𝑡𝑔1)2+ (𝑦1+𝑡ℎ1)2− √𝑥21+𝑦21 𝑡

= (𝑔1+ ℎ1) −𝑥1𝑔1+ 𝑦11

√𝑥21+ 𝑦12 ,

lim𝑡↓0

[𝜙FB(𝑥 + 𝑡𝑔, 𝑦 + 𝑡ℎ)]2− [𝜙FB(𝑥, 𝑦)]2 𝑡

= (𝑔2+ ℎ2)

− lim

𝑡↓0

[[ [

(𝑥1+𝑡𝑔1) (𝑥2+𝑡𝑔2) + (𝑦1+𝑡ℎ1) (𝑦2+𝑡ℎ2) 𝑡√(𝑥1+ 𝑡𝑔1)2+ (𝑦1+ 𝑡ℎ1)2

−𝑥1𝑥2+ 𝑦1𝑦2 𝑡√𝑥21+ 𝑦12

]] ]

= (𝑔2+ ℎ2) −𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2

√𝑥21+ 𝑦12

− lim

𝑡↓0

[[ [

𝑥1𝑥2+𝑦1𝑦2

𝑡√(𝑥1+𝑡𝑔1)2+ (𝑦1+𝑡ℎ1)2 −𝑥1𝑥2+𝑦1𝑦2 𝑡√𝑥21+𝑦12

]] ]

= (𝑔2+ ℎ2) −𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2

√𝑥21+ 𝑦12

+(𝑥1𝑥2+ 𝑦1𝑦2) (𝑥1𝑔1+ 𝑦11) (𝑥21+ 𝑦12) √𝑥21+ 𝑦12

= (𝑔2+ ℎ2) −𝑥1𝑔2+ 𝑦12

√𝑥21+ 𝑦12 ,

(36) where the last equality is using𝑥1𝑦2= 𝑦1𝑥2byLemma 5. The above two limits imply

𝜙FB󸀠 ((𝑥, 𝑦); (𝑔, ℎ)) = (𝑔 + ℎ) − 𝑥1

√𝑥21+ 𝑦21𝑔 − 𝑦1

√𝑥21+ 𝑦12ℎ.

(37) (c.2):(𝑥 + 𝑡𝑔)2+ (𝑦 + 𝑡ℎ)2∈ int K𝑛for sufficiently small 𝑡 > 0. Let 𝑢 = (𝑢1, 𝑢2) := (𝑥+𝑡𝑔)2+(𝑦+𝑡ℎ)2with the spectral values𝜇1, 𝜇2. An elementary calculation gives

𝑢1= 󵄩󵄩󵄩󵄩𝑥 + 𝑡𝑔󵄩󵄩󵄩󵄩2+ 󵄩󵄩󵄩󵄩𝑦 + 𝑡ℎ󵄩󵄩󵄩󵄩2= 𝑤1+ 2𝑡 (𝑥𝑇𝑔 + 𝑦𝑇ℎ) + 𝑡2(󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2) , (38) 𝑢2= 2 (𝑥1+ 𝑡𝑔1) (𝑥2+ 𝑡𝑔2) + 2 (𝑦1+ 𝑡ℎ1) (𝑦2+ 𝑡ℎ2)

= 𝑤2+ 2𝑡 (𝑥1𝑔2+ 𝑔1𝑥2+ 𝑦12+ ℎ1𝑦2) + 2𝑡2(𝑔1𝑔2+ ℎ12) .

(39)

Also, since𝑤2 ̸= 0, applying the Taylor formula of ‖ ⋅ ‖ at 𝑤2 andLemma 6(a) yields

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 + 𝑤𝑇2(𝑢2− 𝑤2)

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 + 𝑜(𝑡)

= 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 + 2𝑡(𝑥𝑇𝑔 + 𝑦𝑇ℎ) + 𝑜 (𝑡) .

(40)

Now using the definition of𝜙FBand noting that𝜆1 = 0 and 𝑤2 ̸= 0, we have that

𝜙FB(𝑥 + 𝑡𝑔, 𝑦 + 𝑡ℎ) − 𝜙FB(𝑥, 𝑦)

= (𝑥 + 𝑡𝑔 + 𝑦 + 𝑡ℎ) − √𝑢 − (𝑥 + 𝑦) + √𝑤

= 𝑡 (𝑔 + ℎ) − (

√𝜇1+ √𝜇2− √𝜆2

√𝜇2− √𝜇1 2 2

𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 −√𝜆2 2

𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

) ,

(41)

(6)

which in turn implies that 𝜙󸀠FB((𝑥, 𝑦); (𝑔, ℎ))

= (𝑔 + ℎ) − ( lim

𝑡↓0

√𝜇1+ √𝜇2− √𝜆2 2𝑡 lim𝑡↓0(√𝜇2− √𝜇1

2𝑡 𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 −√𝜆2 2𝑡

𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩) ) .

(42) We first calculate lim𝑡↓0((√𝜇2− √𝜆2)/𝑡). Using (38) and (40), it is easy to see that

𝜇2− 𝜆2= (𝑢1− 𝑤1) + (󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩)

= 4𝑡 (𝑥𝑇𝑔 + 𝑦𝑇ℎ) + 𝑜 (𝑡) , (43) and consequently,

lim𝑡↓0

√𝜇2− √𝜆2

𝑡 = lim

𝑡↓0

𝜇2− 𝜆2

𝑡 ⋅ 1

√𝜇2+ √𝜆2

=𝑥𝑇𝑔 + 𝑦𝑇

2√𝜆2 = 𝑥𝑇𝑔 + 𝑦𝑇

√𝑥21+ 𝑦12 .

(44)

We next calculate lim𝑡↓0(√𝜇1/𝑡). Since 𝑤1− ‖𝑤2‖ = 0, using (38)-(39) andLemma 6(a),

𝜇1= (𝑢1− 𝑤1) − (󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩)

= (𝑢1− 𝑤1) −󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

= 2𝑡 (𝑥𝑇𝑔 + 𝑦𝑇ℎ) − 4𝑡𝑤𝑇2(𝑥1𝑔2+ 𝑔1𝑥2+ 𝑦12+ ℎ1𝑦2)

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

+ 𝑡2(󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2)

− 4𝑡2󵄩󵄩󵄩󵄩𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2󵄩󵄩󵄩󵄩2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

− 4𝑡2𝑤2𝑇(𝑔1𝑔2+ ℎ12)

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 + 𝑜(𝑡2)

= 2𝑡 (𝑥𝑇𝑔 + 𝑦𝑇ℎ) − 8𝑡(𝑥21+ 𝑦12) (𝑥𝑇𝑔 + 𝑦𝑇ℎ)

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

+ 𝑡2(󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2) + 𝑜 (𝑡2)

− 4𝑡2󵄩󵄩󵄩󵄩𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2󵄩󵄩󵄩󵄩2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

− 8𝑡2(𝑥1𝑥2+ 𝑦1𝑦2)𝑇(𝑔1𝑔2+ ℎ12)

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 .

(45)

Using‖𝑤2‖ = 2(𝑥21+ 𝑦21) and (40), we simplify the sum of the first two terms in (45) as

2𝑡 (𝑥𝑇𝑔 + 𝑦𝑇ℎ) −4𝑡 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩(𝑥𝑇𝑔 + 𝑦𝑇ℎ)

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

= 2𝑡 (𝑥𝑇𝑔 + 𝑦𝑇ℎ)󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

= 4𝑡2(𝑥𝑇𝑔 + 𝑦𝑇ℎ)2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩 + 𝑜(𝑡2) .

(46)

Then, from (45) and‖𝑤2‖ = 2(𝑥21+ 𝑦12), we obtain that

lim𝑡↓0

𝜇1 𝑡2

= (𝑥21+ 𝑦12) (󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2) − 󵄩󵄩󵄩󵄩𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2󵄩󵄩󵄩󵄩2 𝑥21+ 𝑦12

+(𝑥𝑇𝑔 + 𝑦𝑇ℎ)2− 2(𝑥1𝑥2+ 𝑦1𝑦2)𝑇(𝑔1𝑔2+ ℎ12)

𝑥21+ 𝑦21 .

(47)

We next make simplification for the numerator of the right hand side of (47). Note that

(𝑥21+ 𝑦12) (󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2) − 󵄩󵄩󵄩󵄩𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2󵄩󵄩󵄩󵄩2

= (𝑥21+ 𝑦12) (󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2) − 󵄩󵄩󵄩󵄩𝑔1𝑥2+ 𝑥1𝑔2󵄩󵄩󵄩󵄩2

− 󵄩󵄩󵄩󵄩𝑦12+ ℎ1𝑦2󵄩󵄩󵄩󵄩2− 2(𝑔1𝑥2+ 𝑥1𝑔2)𝑇(𝑦12+ ℎ1𝑦2)

= 𝑥12‖ℎ‖2+ 𝑦21󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2− 2𝑥1𝑔1𝑥𝑇2𝑔2− 2𝑦11𝑦2𝑇2

− 2(𝑔1𝑥2+ 𝑥1𝑔2)𝑇(𝑦12+ ℎ1𝑦2) , (𝑥𝑇𝑔 + 𝑦𝑇ℎ)2− 2(𝑥1𝑥2+ 𝑦1𝑦2)𝑇(𝑔1𝑔2+ ℎ12)

= (𝑥1𝑔1+ 𝑥𝑇2𝑔2)2+ (𝑦11+ 𝑦2𝑇2)2+ 2𝑥𝑇𝑔𝑦𝑇

− 2(𝑥1𝑥2+ 𝑦1𝑦2)𝑇(𝑔1𝑔2+ ℎ12)

= (𝑥1𝑔1)2+ (𝑥𝑇2𝑔2)2+ (𝑦11)2+ (𝑦2𝑇2)2 + 2𝑥𝑇𝑔𝑦𝑇ℎ − 2𝑥11𝑥𝑇22− 2𝑔1𝑦1𝑔𝑇2𝑦2.

(48)

(7)

Therefore, adding the last two equalities and usingLemma 5 yield that

(𝑥12+ 𝑦21) (󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2) − 󵄩󵄩󵄩󵄩𝑔1𝑥2+ 𝑥1𝑔2+ 𝑦12+ ℎ1𝑦2󵄩󵄩󵄩󵄩2 + (𝑥𝑇𝑔 + 𝑦𝑇ℎ)2− 2(𝑥1𝑥2+ 𝑦1𝑦2)𝑇(𝑔1𝑔2+ ℎ12)

= (𝑥21‖ℎ‖2− 2𝑥11𝑥𝑇22) + (𝑦12󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2− 2𝑔1𝑦1𝑔𝑇2𝑦2) + ((𝑥1𝑔1)2+ (𝑥𝑇2𝑔2)2− 2𝑥1𝑔1𝑥𝑇2𝑔2)

+ ((𝑦11)2+ (𝑦𝑇22)2− 2𝑦11𝑦2𝑇2) + 2𝑥𝑇𝑔𝑦𝑇

− 2(𝑔1𝑥2+ 𝑥1𝑔2)𝑇(𝑦12+ ℎ1𝑦2)

= 󵄩󵄩󵄩󵄩𝑥12− ℎ1𝑥2󵄩󵄩󵄩󵄩2+ 󵄩󵄩󵄩󵄩𝑔1𝑦2− 𝑦1𝑔2󵄩󵄩󵄩󵄩2+ (𝑥1𝑔1− 𝑥𝑇2𝑔2)2 + (𝑦11− 𝑦𝑇22)2+ 2 (𝑔1𝑥1+ 𝑔𝑇2𝑥2) (𝑦11+ 𝑦𝑇22)

− 2(𝑔1𝑥2+ 𝑥1𝑔2)𝑇(𝑦12+ ℎ1𝑦2)

= 󵄩󵄩󵄩󵄩𝑥12− ℎ1𝑥2󵄩󵄩󵄩󵄩2+ 󵄩󵄩󵄩󵄩𝑔1𝑦2− 𝑦1𝑔2󵄩󵄩󵄩󵄩2+ (𝑥1𝑔1− 𝑥𝑇2𝑔2)2 + (𝑦11− 𝑦𝑇22)2+ 2(𝑥12− ℎ1𝑥2)𝑇(𝑔1𝑦2− 𝑔2𝑦1) + 2 (𝑥1𝑔1− 𝑥𝑇2𝑔2) (𝑦11− 𝑦2𝑇2)

= 󵄩󵄩󵄩󵄩𝑥12− ℎ1𝑥2+ 𝑔1𝑦2− 𝑦1𝑔2󵄩󵄩󵄩󵄩2 + (𝑥1𝑔1− 𝑥𝑇2𝑔2+ 𝑦11− 𝑦2𝑇2)2.

(49)

Combining this equality with (47) and using the definition of 𝜑 in (33), we readily get

lim𝑡↓0

√𝜇1

𝑡 = 𝜑 (𝑔, ℎ) . (50)

We next calculate lim𝑡↓0[((√𝜇2 − √𝜇1)/2𝑡)(𝑢2/‖𝑢2‖) − ((√𝜆2/2𝑡)(𝑤2/‖𝑤2‖))]. To this end, we also need to take a look at‖𝑤2‖𝑢2− ‖𝑢2‖𝑤2. From (38)-(39) and (40), it follows that

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩𝑢2− 󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩𝑤2

= 2𝑡 󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩[(𝑥1𝑔2+ 𝑔1𝑥2+ 𝑦12+ ℎ1𝑦2)

− (𝑥𝑇𝑔 + 𝑦𝑇ℎ) ̃𝑤2] + 𝑜 (𝑡) .

(51)

Together with (44) and (50), we have that

lim𝑡↓0 [√𝜇2− √𝜇1

2𝑡 𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 −√𝜆2 2𝑡

𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩]

= −lim

𝑡↓0

√𝜇1

2𝑡 𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + lim𝑡↓0 [√𝜇2 2𝑡

𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 −√𝜆2 2𝑡

𝑤2

󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩]

= −lim

𝑡↓0

√𝜇1 2𝑡

𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩 + lim𝑡↓0√𝜇2− √𝜆2 2𝑡

𝑢2

󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩

+ lim

𝑡↓0

√𝜆2(󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩𝑢2− 󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩𝑤2) 2𝑡 󵄩󵄩󵄩󵄩𝑢2󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑤2󵄩󵄩󵄩󵄩

= −𝜑 (𝑔, ℎ)

2 𝑤̃2+𝑥1𝑔2+ 𝑔1𝑥2+ 𝑦12+ ℎ1𝑦2

√𝑥21+ 𝑦12

−𝑥𝑇𝑔 + 𝑦𝑇ℎ 2√𝑥21+ 𝑦12𝑤̃2

= −𝜑 (𝑔, ℎ)

2 𝑤̃2+2𝑥1𝑔2+ 𝑔1𝑥2+ 2𝑦12+ ℎ1𝑦2 2√𝑥21+ 𝑦21

−𝑥𝑇2𝑔2+ 𝑦𝑇22 2√𝑥21+ 𝑦12 𝑤̃2,

(52) where the last equality is using𝑥1𝑤̃2 = 𝑥2and𝑦1𝑤̃2 = 𝑦2. Combining with (42), (44), and (50), a suitable rearrange- ment shows that𝜙󸀠FB((𝑥, 𝑦); (𝑔, ℎ)) has the expression (32).

Finally, we show that, when(𝑥+𝑡𝑔)2+(𝑦+𝑡ℎ)2∈ bd+K𝑛 for sufficiently small𝑡 > 0, the formula in (32) reduces to the one in (37). Indeed, an elementary calculation yields

𝜆1((𝑥 + 𝑡𝑔)2+ (𝑦 + 𝑡ℎ)2)

= [󵄩󵄩󵄩󵄩𝑥 + 𝑡𝑔󵄩󵄩󵄩󵄩2+ 󵄩󵄩󵄩󵄩𝑦 + 𝑡ℎ󵄩󵄩󵄩󵄩2]2

− 4󵄩󵄩󵄩󵄩(𝑥1+ 𝑡𝑔1) (𝑥2+ 𝑡𝑔2) + (𝑦1+ 𝑡ℎ1) (𝑦2+ 𝑡ℎ2)󵄩󵄩󵄩󵄩2

= 4𝑡2𝜑 (𝑔, ℎ) √𝑥21+ 𝑦12

+ 4𝑡3(𝑥𝑇𝑔 + 𝑦𝑇ℎ) (󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2)

− 8𝑡2(𝑥1𝑔2+ 𝑔1𝑥2+ 𝑦12+ ℎ1𝑦2)𝑇(𝑔1𝑔2+ ℎ12) + 𝑡4[(󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩2+ ‖ℎ‖2)2− 2󵄩󵄩󵄩󵄩𝑔1𝑔2+ ℎ12󵄩󵄩󵄩󵄩2]

= 4𝑡2𝜑 (𝑔, ℎ) √𝑥21+ 𝑦12+ 𝑜 (𝑡2) .

(53) This implies that if(𝑥+𝑡𝑔)2+(𝑦+𝑡ℎ)2∈ bd+K𝑛for sufficiently small𝑡 > 0, that is, 𝜆1((𝑥+𝑡𝑔)2+(𝑦+𝑡ℎ)2) = 0 for sufficiently small𝑡 > 0, then 𝜑(𝑔, ℎ) = 0, and hence 𝑥1𝑔1+𝑦11−(𝑥𝑇2𝑔2+

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