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

*2**Mathematics 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∈R^{𝑛} 𝑓 (𝜁)
s.t. ℎ (𝜁) = 0,

𝑔 (𝜁) ∈ K,

(1)

where𝑓 : R^{𝑛} → R, ℎ : R^{𝑛} → R^{𝑚}, and𝑔 : R^{𝑛} → R^{𝑛}are
given twice continuously differentiable functions, andK is
the Cartesian product of some second-order cones, that is,

K := K^{𝑛}^{1}× K^{𝑛}^{2}× ⋅ ⋅ ⋅ × K^{𝑛}^{𝑟}, (2)

with𝑛_{1}+ ⋅ ⋅ ⋅ + 𝑛_{𝑟} = 𝑛 and K^{𝑛}^{𝑗} being the second-order cone
(SOC) inR^{𝑛}^{𝑗}defined by

K^{𝑛}^{𝑗} := {(𝑥_{𝑗1}, 𝑥_{𝑗2}) ∈ R × R^{𝑛}^{𝑗}^{−1} | 𝑥_{𝑗1}≥ 𝑥^{𝑗2}} . (3)

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

𝜁,𝑥∈Rmin^{𝑛} 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 : R^{𝑛}× R^{𝑛}× R^{𝑚}× R^{𝑛}× 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 byN_{K}(𝑥) the normal cone of K at 𝑥 in the sense
of convex analysis [1]:

N_{K}(𝑥) = { {𝑑 ∈ R^{𝑛}: ⟨𝑑, 𝑧 − 𝑥⟩ ≤ 0 ∀𝑧 ∈ K} , if 𝑥 ∈ K,

0, if𝑥 ∉ K.

(6)

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

J_{𝜁,𝑥}𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) = 0, ℎ (𝜁) = 0,

𝑔 (𝜁) − 𝑥 = 0, −𝑦 ∈ N_{K}(𝑥) , (7)
whereJ_{𝜁,𝑥}𝐿(𝜁, 𝑥, 𝜇, 𝑠, 𝑦) is the derivative of 𝐿 at (𝜁, 𝑥, 𝜇, 𝑠, 𝑦)
with respect to(𝜁, 𝑥). Recall that 𝜙^{soc}is an SOC complemen-
tarity function associated with the coneK if

𝜙^{soc}(𝑥, 𝑦) = 0 ⇐⇒ 𝑥 ∈ K, 𝑦 ∈ K,

⟨𝑥, 𝑦⟩ = 0 ⇐⇒ −𝑦 ∈ N_{K}(𝑥) . (8)
With an SOC complementarity function𝜙^{soc}associated with
K, we may reformulate the KKT optimality conditions in (7)
as the following nonsmooth system:

𝐸 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦) :=[[[ [

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

]] ] ]

= 0. (9)

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

𝜙_{NR}(𝑥, 𝑦) := 𝑥 − Π_{K}(𝑥 − 𝑦) , ∀𝑥, 𝑦 ∈ R^{𝑛},

𝜙_{FB}(𝑥, 𝑦) := (𝑥 + 𝑦) − √𝑥^{2}+ 𝑦^{2}, ∀𝑥, 𝑦 ∈ R^{𝑛}, (10)
whereΠ_{K}(⋅) is the projection operator onto the closed convex
coneK, 𝑥^{2}= 𝑥 ∘ 𝑥 means the Jordan product of 𝑥 and itself,
and√𝑥 denotes the unique square root of 𝑥 ∈ K. It turns out
that the FB SOC complementarity function𝜙_{FB}enjoys almost
all favorable properties of the NR SOC complementarity
function𝜙_{NR}(see [2]). Also, the squared norm of𝜙_{FB}induces
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:

𝐸_{FB}(𝜁, 𝑥, 𝜇, 𝑠, 𝑦) :=[[[
[

J_{𝜁,𝑥}𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦)
ℎ (𝜁)
𝑔 (𝜁) − 𝑥
𝜙_{FB}(𝑥, 𝑦)

]] ] ]

= 0. (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

𝐸_{NR}(𝜁, 𝑥, 𝜇, 𝑠, 𝑦) :=[[[
[

J_{𝜁,𝑥}𝐿 (𝜁, 𝑥, 𝜇, 𝑠, 𝑦)
ℎ (𝜁)
𝑔 (𝜁) − 𝑥
𝜙_{NR}(𝑥, 𝑦)

]] ] ]

= 0. (12)

They showed that the strong regularity of the KKT point,
the nonsingularity of Clarke’s Jacobian of𝐸_{NR} 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
𝐸_{FB}. 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 of𝐸_{FB}at the KKT point, and the strong
regularity of the KKT point are equivalent to each other. This,
on the one hand, gives a new characterization for the strong
regularity of the KKT point and, on the other hand, provides
a mild condition to guarantee the quadratic convergence rate
of the semismooth Newton method [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,R^{𝑛} (𝑛 > 1) denotes the space of 𝑛-
dimensional real column vectors, and R^{𝑛}^{1} × ⋅ ⋅ ⋅ × R^{𝑛}^{𝑟} is
identified withR^{𝑛}^{1}^{+⋅⋅⋅+𝑛}^{𝑟}. Thus,(𝑥_{1}, . . . , 𝑥_{𝑟}) ∈ R^{𝑛}^{1}×⋅ ⋅ ⋅×R^{𝑛}^{𝑟}is
viewed as a column vector inR^{𝑛}^{1}^{+⋅⋅⋅+𝑛}^{𝑟}**. The notations int K**^{𝑛},
**bd K**^{𝑛}**, and bd**^{+}K^{𝑛}denote the interior, the boundary, and the
boundary excluding the origin ofK^{𝑛}, respectively. For any
𝑥 ∈ R^{𝑛}, we write𝑥 ⪰_{K}^{𝑛} 0 (resp., 𝑥 ≻_{K}^{𝑛} 0) if 𝑥 ∈ K^{𝑛}(resp.,
**𝑥 ∈ int K**^{𝑛}). For any given real symmetric matrix𝐴, we write
𝐴 ⪰ 0 (resp., 𝐴 ≻ 0) if 𝐴 is positive semidefinite (resp.,
positive definite). In addition,J_{𝜔}𝑓(𝜔) and J^{2}_{𝜔𝜔}𝑓(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 of*𝑥 = (𝑥_{1}, 𝑥_{2}), 𝑦 = (𝑦_{1},
𝑦_{2}) ∈ R × R^{𝑛−1}is given by

𝑥 ∘ 𝑦 := (⟨𝑥, 𝑦⟩ , 𝑥_{1}𝑦_{2}+ 𝑦_{1}𝑥_{2}) . (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} ∈ R^{𝑛}, that is,𝑒 ∘ 𝑥 = 𝑥 for all

𝑥 ∈ R^{𝑛}. For each𝑥 = (𝑥_{1}, 𝑥_{2}) ∈ R × R^{𝑛−1}, we define the
associated arrow matrix by

𝐿_{𝑥}:= [𝑥^{1} 𝑥_{2}^{𝑇}

𝑥_{2} 𝑥_{1}𝐼] . (14)

Then it is easy to verify that𝐿_{𝑥}𝑦 = 𝑥 ∘ 𝑦 for any 𝑥, 𝑦 ∈ R^{𝑛}.
Recall that each𝑥 = (𝑥_{1}, 𝑥_{2}) ∈ R × R^{𝑛−1}admits a spectral
factorization, associated withK^{𝑛}, of the form

𝑥 = 𝜆_{1}(𝑥) 𝑢^{(1)}_{𝑥} + 𝜆_{2}(𝑥) 𝑢^{(2)}_{𝑥} , (15)
where𝜆_{1}(𝑥), 𝜆_{2}(𝑥) ∈ R and 𝑢^{(1)}_{𝑥} , 𝑢^{(2)}_{𝑥} ∈ R^{𝑛}are the spectral
values and the associated spectral vectors of𝑥, respectively,
with respect to the Jordan product, defined by

𝜆_{𝑖}(𝑥) := 𝑥_{1}+ (−1)^{𝑖}𝑥^{2},
𝑢^{(𝑖)}_{𝑥} := 1

2( 1

(−1)^{𝑖}̃𝑥_{2}) , for 𝑖 = 1, 2, (16)
with̃𝑥_{2}= 𝑥_{2}/‖𝑥_{2}‖ if 𝑥_{2} ̸= 0 and otherwise being any vector in
R^{𝑛−1}satisfying‖̃𝑥_{2}‖ = 1.

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

By the formula of spectral factorization, it is easy to
compute that the projection of𝑥 ∈ R^{𝑛}onto the closed convex
coneK^{𝑛}, denoted byΠ_{K}^{𝑛}(𝑥), has the expression

Π_{K}^{𝑛}(𝑥) = max (0, 𝜆_{1}(𝑥)) 𝑢^{(1)}_{𝑥} + max (0, 𝜆_{2}(𝑥)) 𝑢^{(2)}_{𝑥} . (17)
Define|𝑥| := 2Π_{K}^{𝑛}(𝑥) − 𝑥. Then, using the expression of
Π_{K}^{𝑛}(𝑥), it follows that

|𝑥| = 𝜆^{1}(𝑥) 𝑢^{(1)}𝑥 + 𝜆^{2}(𝑥) 𝑢^{(2)}𝑥 . (18)
The spectral factorization of the vectors𝑥, 𝑥^{2},√𝑥 and the
matrix𝐿_{𝑥}have various interesting properties (see [13]). We
list several properties that we will use later.

*Property 3. For any*𝑥 = (𝑥_{1}, 𝑥_{2}) ∈ R × R^{𝑛−1}with spectral
factorization (15), we have the following.

(a)𝑥^{2}= 𝜆^{2}_{1}(𝑥)𝑢^{(1)}_{𝑥} + 𝜆^{2}_{2}(𝑥)𝑢^{(2)}_{𝑥} ∈ K^{𝑛}.

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

√𝜆_{1}(𝑥)𝑢^{(1)}_{𝑥} + √𝜆_{2}(𝑥)𝑢^{(2)}_{𝑥} .

(c) If**𝑥 ∈ int K**^{𝑛}, then0 < 𝜆_{1}(𝑥) ≤ 𝜆_{2}(𝑥) and 𝐿_{𝑥} is
invertible with

𝐿^{−1}_{𝑥} = 1
det(𝑥)[[

[

𝑥_{1} −𝑥^{𝑇}_{2}

−𝑥_{2} det(𝑥)

𝑥_{1} 𝐼 + 𝑥_{2}𝑥^{𝑇}_{2}
𝑥_{1}

]] ]

. (19)

(d)𝐿_{𝑥} ⪰ 0 (resp., 𝐿_{𝑥} ≻ 0) if and only if 𝑥 ∈ K^{𝑛} (resp.,
**𝑥 ∈ int K**^{𝑛}).

The following lemma states a result for the arrow matrices
associated with𝑥, 𝑦 ∈ R^{𝑛}and𝑧 ⪰_{K}^{𝑛}√𝑥^{2}+ 𝑦^{2}, which will be
used in the next section to characterize an important property
for the elements of Clarke’s Jacobian of𝜙_{FB}at a general point.

**Lemma 4. For any given 𝑥, 𝑦 ∈ R**^{𝑛}*and*𝑧 ≻_{K}^{𝑛}*0, if 𝑧*^{2}⪰_{K}^{𝑛}𝑥^{2}+
𝑦^{2}*, then*

[𝐿^{−1}^{𝑧} 𝐿_{𝑥} 𝐿^{−1}_{𝑧} 𝐿_{𝑦}]2≤ 1, (20)
*where* ‖𝐴‖_{2} *means the spectral norm of a real matrix* *𝐴.*

*Consequently, it holds that*

𝐿^{−1}^{𝑧} 𝐿_{𝑥}Δ𝑢 + 𝐿^{−1}_{𝑧} 𝐿_{𝑦}Δ𝑣 ≤ √‖Δ𝑢‖^{2}+ ‖Δ𝑣‖^{2}, ∀Δ𝑢, Δ𝑣 ∈ R^{𝑛}.
(21)
*Proof. Let*𝐴 = [𝐿^{−1}_{𝑧} 𝐿_{𝑥} 𝐿^{−1}_{𝑧} 𝐿_{𝑦}]. From [13, Proposition 3.4], it
follows that

𝐴𝐴^{𝑇}= 𝐿^{−1}_{𝑧} (𝐿^{2}_{𝑥}+ 𝐿^{2}_{𝑦}) 𝐿^{−1}_{𝑧} ⪯ 𝐿^{−1}_{𝑧} 𝐿^{2}_{𝑧}𝐿^{−1}_{𝑧} = 𝐼. (22)
This shows that‖𝐴‖_{2}≤ 1, and the first part follows. Note that,
for any𝜉 ∈ R^{2𝑛},

𝐴𝜉^{2}= 𝜉^{𝑇}𝐴^{𝑇}𝐴𝜉 ≤ 𝜆_{max}(𝐴^{𝑇}𝐴) 𝜉^{2}≤ 𝜉^{2}. (23)
By letting𝜉 = (Δ𝑢, Δ𝑣) ∈ R^{𝑛}×R^{𝑛}, we immediately obtain the
second part.

The following two lemmas state the properties of𝑥, 𝑦 with
𝑥^{2} + 𝑦^{2} **∈ bd K**^{𝑛} which are often used in the subsequent
sections. The proof ofLemma 5is given in [3, Lemma 2].

**Lemma 5. For any 𝑥 = (𝑥**_{1}, 𝑥_{2}), 𝑦 = (𝑦_{1}, 𝑦_{2}) ∈ R × R^{𝑛−1}
*with*𝑥^{2}+ 𝑦^{2}**∈ bd K**^{𝑛}*, one has*

𝑥^{2}_{1}= 𝑥^{2}^{2}, 𝑦_{1}^{2}= 𝑦^{2}^{2},

𝑥_{1}𝑦_{1}= 𝑥^{𝑇}_{2}𝑦_{2}, 𝑥_{1}𝑦_{2}= 𝑦_{1}𝑥_{2}. (24)
**Lemma 6. For any 𝑥 = (𝑥**_{1}, 𝑥_{2}), 𝑦 = (𝑦_{1}, 𝑦_{2}) ∈ R × R^{𝑛−1}*, let*
𝑤 = (𝑤_{1}, 𝑤_{2}) := 𝑥^{2}+ 𝑦^{2}*.*

*(a) If***𝑤 ∈ bd K**^{𝑛}*, then for any*𝑔 = (𝑔_{1}, 𝑔_{2}), ℎ = (ℎ_{1}, ℎ_{2}) ∈
R × R^{𝑛−1}*, it holds that*

(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2})^{𝑇}(𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})

= (𝑥^{2}_{1}+ 𝑦^{2}_{1}) (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ) . (25)

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

𝑤^{2} = 𝑥^{2}, 𝑥^{𝑇}_{2}𝑤_{2}

𝑤^{2} = 𝑥^{1},
𝑦_{1}𝑤_{2}

𝑤^{2} = 𝑦^{2}, 𝑦_{2}^{𝑇}𝑤_{2}

𝑤^{2} = 𝑦^{1};

(26)

*and consequently the expression of*𝜙_{FB}*(𝑥, 𝑦) can be*
*simplified as*

𝜙_{FB}(𝑥, 𝑦) = (

𝑥_{1}+ 𝑦_{1}− √𝑥^{2}_{1}+ 𝑦_{1}^{2}
𝑥_{2}+ 𝑦_{2}−𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2}

√𝑥^{2}_{1}+ 𝑦_{1}^{2}

) . (27)

*Proof. (a) The result is direct by the equalities of*Lemma 5
since𝑥^{2}+ 𝑦^{2}**∈ bd K**^{𝑛}.

(b) Since **𝑤 ∈ bd**^{+}K^{𝑛}, we must have𝑤_{2} = 2(𝑥_{1}𝑥_{2} +
𝑦_{1}𝑦_{2}) ̸= 0. UsingLemma 5,𝑤_{2} = 2(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2}) and ‖𝑤_{2}‖ =
𝑤_{1} = 2(𝑥^{2}_{1} + 𝑦_{1}^{2}), we easily obtain the first part. Note that
𝜙_{FB}(𝑥, 𝑦) = (𝑥 + 𝑦) − √𝑤. UsingProperty 3(b) andLemma 5
yields (27).

When**𝑥, 𝑦 ∈ bd K**^{𝑛}satisfies the complementary condi-
tion, we have the following result.

**Lemma 7. For any given 𝑥 = (𝑥**_{1}, 𝑥_{2}), 𝑦 = (𝑦_{1}, 𝑦_{2}) ∈ R ×
R^{𝑛−1}*, if***𝑥, 𝑦 ∈ bd K**^{𝑛} *and⟨𝑥, 𝑦⟩ = 0, then there exists a*
*constant𝛼 > 0 such that 𝑥*_{1}= 𝛼𝑦_{1}*and*𝑥_{2}= −𝛼𝑦_{2}*.*

*Proof. Since***𝑥, 𝑦 ∈ bd K**^{𝑛}, we have that𝑥_{1} = ‖𝑥_{2}‖ and 𝑦_{1} =

‖𝑦_{2}‖, and consequently,

0 = ⟨𝑥, 𝑦⟩ = 𝑥_{1}𝑦_{1}+ 𝑥^{𝑇}_{2}𝑦_{2}= 𝑥^{2}𝑦^{2} + 𝑥^{𝑇}2𝑦_{2}. (28)
This means that there exists𝛼 > 0 such that 𝑥_{2} = −𝛼𝑦_{2}, and
then𝑥_{1}= 𝛼𝑦_{1}.

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

0 ∈ 𝜙 (𝑧) + N_{𝐷}(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 N_{𝐷}(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 that*𝑧 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) + J_{𝑧}𝜙(𝑧)(𝑧 − 𝑧) +
N_{𝐷}(𝑧) has a unique solution in V, denoted by 𝑧_{V}(𝛿), and the
mapping𝑧_{V}: 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 ⊂
R^{𝑛} be an open set and Ξ : 𝑆 → R^{𝑛} a locally Lipschitz
continuous function on 𝑆. By Rademacher’s theorem, Ξ is
almost everywhere𝐹(r´echet)-differentiable in 𝑆. We denote
by 𝑆_{Ξ} the set of points in 𝑆 where Ξ is 𝐹-differentiable.

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

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

𝜕_{𝐵}Ξ (𝑦) := {𝑉 : 𝑉 = lim

𝑘 → ∞J_{𝑦}Ξ (𝑦^{𝑘}) , 𝑦^{𝑘}→ 𝑦, 𝑦^{𝑘}∈ 𝑆_{Ξ}} .
(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 ∈ R^{𝑛} (𝑛 > 1), we write 𝑥 = (𝑥_{1}, 𝑥_{2}), where 𝑥_{1} is the
first component of𝑥 and 𝑥_{2}is a column vector consisting of
the remaining𝑛 − 1 entries of 𝑥. For any 𝑥 = (𝑥_{1}, 𝑥_{2}), 𝑦 =
(𝑦_{1}, 𝑦_{2}) ∈ R × R^{𝑛−1}, let

𝑤 = 𝑤 (𝑥, 𝑦) := 𝑥^{2}+ 𝑦^{2}, 𝑤̃_{2}:= 𝑤_{2}

𝑤^{2},
if 𝑤_{2} ̸= 0, 𝑧 = 𝑧 (𝑥, 𝑦) = √𝑤 (𝑥, 𝑦).

(31)

**3. Directional Derivative and** **𝐵-Subdifferential**

The function𝜙_{FB} 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 𝜙_{FB} at a general point
coincides with that of its directional derivative function at the
origin. Throughout this section, we assume thatK = K^{𝑛}.
**Proposition 9. For any given 𝑥 = (𝑥**_{1}, 𝑥_{2}), 𝑦 = (𝑦_{1}, 𝑦_{2}) ∈
R×R^{𝑛−1}*, the directional derivative*𝜙_{FB}^{} *((𝑥, 𝑦); (𝑔, ℎ)) of 𝜙*_{FB}*at*
*(𝑥, 𝑦) with the direction (𝑔, ℎ) has the following form.*

*(a) If(𝑥, 𝑦) = (0, 0), then 𝜙*_{FB}^{} ((𝑥, 𝑦); (𝑔, ℎ)) = 𝜙_{FB}*(𝑔, ℎ).*

*(b) If*𝑥^{2}+ 𝑦^{2} **∈ int K**^{𝑛}*, then*𝜙^{}_{FB}((𝑥, 𝑦); (𝑔, ℎ)) = (𝐼 −
𝐿^{−1}_{𝑧} 𝐿_{𝑥})𝑔 + (𝐼 − 𝐿^{−1}_{𝑧} 𝐿_{𝑦}*)ℎ.*

*(c) If*𝑥^{2}+ 𝑦^{2}**∈ bd**^{+}K^{𝑛}*, then*
𝜙^{}_{FB}((𝑥, 𝑦); (𝑔, ℎ))

= (𝑔 + ℎ) −𝜑 (𝑔, ℎ)

2 ( 1−̃𝑤_{2}) + 𝑥^{𝑇}_{2}𝑔_{2}+ 𝑦_{2}^{𝑇}ℎ_{2}
2√𝑥^{2}_{1}+ 𝑦_{1}^{2} ( 0𝑤̃_{2})

− 1

2√𝑥^{2}_{1}+ 𝑦_{1}^{2}( 𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ

2𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 2𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}) ,
(32)
*where*𝑔 = (𝑔_{1}, 𝑔_{2}), ℎ = (ℎ_{1}, ℎ_{2}) ∈ R × R^{𝑛−1}*, and*
𝜑 : R^{𝑛}× R^{𝑛} *→ R is defined by*

𝜑 (𝑔, ℎ)

:=√(𝑥_{1}𝑔_{1}−𝑥^{𝑇}_{2}𝑔_{2}+𝑦_{1}ℎ_{1}−𝑦_{2}^{𝑇}ℎ_{2})^{2}+𝑥^{1}ℎ_{2}−ℎ_{1}𝑥_{2}+𝑔_{1}𝑦_{2}−𝑦_{1}𝑔_{2}^{2}

√𝑥^{2}_{1}+ 𝑦_{1}^{2} .

(33)

*Proof. Part (a) is immediate by noting that*𝜙_{FB}is 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 let𝜆_{1}, 𝜆_{2} with𝜆_{1} ≤ 𝜆_{2}

denote the spectral values of𝑤. Since 𝑤 = 𝑥^{2}+ 𝑦^{2}**∈ bd**^{+}K^{𝑛},
we have𝑤_{2} ̸= 0, and fromLemma 6(b) it follows that

𝑤_{1}= 𝑤^{2} = 2𝑥^{1}𝑥_{2}+ 𝑦_{1}𝑦_{2}

= 2 𝑥^{2}^{1}𝑤̃_{2}+ 𝑦_{1}^{2}𝑤̃_{2} = 2 (𝑥^{2}^{1}+ 𝑦_{1}^{2}) ,

𝜆_{1}= 𝑤_{1}− 𝑤^{2} = 0, 𝜆_{2}= 𝑤_{1}+ 𝑤^{2} = 4(𝑥^{2}^{1}+ 𝑦_{1}^{2}) .
(34)

(c.1):(𝑥 + 𝑡𝑔)^{2}+ (𝑦 + 𝑡ℎ)^{2}**∈ bd**^{+}K^{𝑛}for sufficiently small
𝑡 > 0. In this case, fromLemma 6(b), we know that𝜙_{FB}(𝑥 +
𝑡𝑔, 𝑦 + 𝑡ℎ) has the following expression:

(

(𝑥_{1}+ 𝑦_{1}) + 𝑡 (𝑔_{1}+ ℎ_{1}) − √(𝑥_{1}+ 𝑡𝑔_{1})^{2}+ (𝑦_{1}+ 𝑡ℎ_{1})^{2}
(𝑥_{2}+ 𝑦_{2}) + 𝑡 (𝑔_{2}+ ℎ_{2})

−(𝑥_{1}+ 𝑡𝑔_{1}) (𝑥_{2}+ 𝑡𝑔_{2}) + (𝑦_{1}+ 𝑡ℎ_{1}) (𝑦_{2}+ 𝑡ℎ_{2})

√(𝑥_{1}+ 𝑡𝑔_{1})^{2}+ (𝑦_{1}+ 𝑡ℎ_{1})^{2}

) .

(35)

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

lim𝑡↓0

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

= (𝑔_{1}+ ℎ_{1}) − lim

𝑡↓0

√(𝑥_{1}+𝑡𝑔_{1})^{2}+ (𝑦_{1}+𝑡ℎ_{1})^{2}− √𝑥^{2}_{1}+𝑦^{2}_{1}
𝑡

= (𝑔_{1}+ ℎ_{1}) −𝑥_{1}𝑔_{1}+ 𝑦_{1}ℎ_{1}

√𝑥^{2}_{1}+ 𝑦_{1}^{2} ,

lim𝑡↓0

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

= (𝑔_{2}+ ℎ_{2})

− lim

𝑡↓0

[[ [

(𝑥_{1}+𝑡𝑔_{1}) (𝑥_{2}+𝑡𝑔_{2}) + (𝑦_{1}+𝑡ℎ_{1}) (𝑦_{2}+𝑡ℎ_{2})
𝑡√(𝑥_{1}+ 𝑡𝑔_{1})^{2}+ (𝑦_{1}+ 𝑡ℎ_{1})^{2}

−𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2}
𝑡√𝑥^{2}_{1}+ 𝑦_{1}^{2}

]] ]

= (𝑔_{2}+ ℎ_{2}) −𝑔_{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}

√𝑥^{2}_{1}+ 𝑦_{1}^{2}

− lim

𝑡↓0

[[ [

𝑥_{1}𝑥_{2}+𝑦_{1}𝑦_{2}

𝑡√(𝑥_{1}+𝑡𝑔_{1})^{2}+ (𝑦_{1}+𝑡ℎ_{1})^{2} −𝑥_{1}𝑥_{2}+𝑦_{1}𝑦_{2}
𝑡√𝑥^{2}_{1}+𝑦_{1}^{2}

]] ]

= (𝑔_{2}+ ℎ_{2}) −𝑔_{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}

√𝑥^{2}_{1}+ 𝑦_{1}^{2}

+(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2}) (𝑥_{1}𝑔_{1}+ 𝑦_{1}ℎ_{1})
(𝑥^{2}_{1}+ 𝑦_{1}^{2}) √𝑥^{2}_{1}+ 𝑦_{1}^{2}

= (𝑔_{2}+ ℎ_{2}) −𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}

√𝑥^{2}_{1}+ 𝑦_{1}^{2} ,

(36)
where the last equality is using𝑥_{1}𝑦_{2}= 𝑦_{1}𝑥_{2}byLemma 5. The
above two limits imply

𝜙_{FB}^{} ((𝑥, 𝑦); (𝑔, ℎ)) = (𝑔 + ℎ) − 𝑥_{1}

√𝑥^{2}_{1}+ 𝑦^{2}_{1}𝑔 − 𝑦_{1}

√𝑥^{2}_{1}+ 𝑦_{1}^{2}ℎ.

(37)
(c.2):(𝑥 + 𝑡𝑔)^{2}+ (𝑦 + 𝑡ℎ)^{2}**∈ int K**^{𝑛}for sufficiently small
𝑡 > 0. Let 𝑢 = (𝑢_{1}, 𝑢_{2}) := (𝑥+𝑡𝑔)^{2}+(𝑦+𝑡ℎ)^{2}with the spectral
values𝜇_{1}, 𝜇_{2}. An elementary calculation gives

𝑢_{1}= 𝑥 + 𝑡𝑔^{2}+ 𝑦 + 𝑡ℎ^{2}= 𝑤_{1}+ 2𝑡 (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ)
+ 𝑡^{2}(𝑔^{2}+ ‖ℎ‖^{2}) , (38)
𝑢_{2}= 2 (𝑥_{1}+ 𝑡𝑔_{1}) (𝑥_{2}+ 𝑡𝑔_{2}) + 2 (𝑦_{1}+ 𝑡ℎ_{1}) (𝑦_{2}+ 𝑡ℎ_{2})

= 𝑤_{2}+ 2𝑡 (𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})
+ 2𝑡^{2}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2}) .

(39)

Also, since𝑤_{2} ̸= 0, applying the Taylor formula of ‖ ⋅ ‖ at 𝑤_{2}
andLemma 6(a) yields

𝑢^{2} = 𝑤^{2} + 𝑤^{𝑇}^{2}(𝑢_{2}− 𝑤_{2})

𝑤^{2} + 𝑜(𝑡)

= 𝑤^{2} + 2𝑡(𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ) + 𝑜 (𝑡) .

(40)

Now using the definition of𝜙_{FB}and noting that𝜆_{1} = 0 and
𝑤_{2} ̸= 0, we have that

𝜙_{FB}(𝑥 + 𝑡𝑔, 𝑦 + 𝑡ℎ) − 𝜙_{FB}(𝑥, 𝑦)

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

= 𝑡 (𝑔 + ℎ) − (

√𝜇1+ √𝜇2− √𝜆_{2}

√𝜇2− √𝜇1 2 2

𝑢_{2}

𝑢^{2} −√𝜆_{2}
2

𝑤_{2}

𝑤^{2}

) ,

(41)

which in turn implies that
𝜙^{}_{FB}((𝑥, 𝑦); (𝑔, ℎ))

= (𝑔 + ℎ) − ( lim

𝑡↓0

√𝜇1+ √𝜇2− √𝜆_{2}
2𝑡
lim𝑡↓0(√𝜇_{2}− √𝜇1

2𝑡
𝑢_{2}

𝑢^{2} −√𝜆_{2}
2𝑡

𝑤_{2}

𝑤^{2})
) .

(42)
We first calculate lim_{𝑡↓0}((√𝜇2− √𝜆_{2})/𝑡). Using (38) and (40),
it is easy to see that

𝜇_{2}− 𝜆_{2}= (𝑢_{1}− 𝑤_{1}) + (𝑢^{2} − 𝑤^{2})

= 4𝑡 (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ) + 𝑜 (𝑡) , (43)
and consequently,

lim𝑡↓0

√𝜇_{2}− √𝜆_{2}

𝑡 = lim

𝑡↓0

𝜇_{2}− 𝜆_{2}

𝑡 ⋅ 1

√𝜇2+ √𝜆_{2}

=𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ

2√𝜆_{2} = 𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ

√𝑥^{2}_{1}+ 𝑦_{1}^{2} .

(44)

We next calculate lim_{𝑡↓0}(√𝜇1/𝑡). Since 𝑤_{1}− ‖𝑤_{2}‖ = 0, using
(38)-(39) andLemma 6(a),

𝜇_{1}= (𝑢_{1}− 𝑤_{1}) − (𝑢^{2} − 𝑤^{2})

= (𝑢_{1}− 𝑤_{1}) −𝑢^{2}^{2}− 𝑤^{2}^{2}

𝑢^{2} + 𝑤^{2}

= 2𝑡 (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ) − 4𝑡𝑤^{𝑇}_{2}(𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})

𝑢^{2} + 𝑤^{2}

+ 𝑡^{2}(𝑔^{2}+ ‖ℎ‖^{2})

− 4𝑡^{2}𝑔^{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}^{2}

𝑢^{2} + 𝑤^{2}

− 4𝑡^{2}𝑤_{2}^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})

𝑢^{2} + 𝑤^{2} + 𝑜(𝑡^{2})

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

𝑢^{2} + 𝑤^{2}

+ 𝑡^{2}(𝑔^{2}+ ‖ℎ‖^{2}) + 𝑜 (𝑡^{2})

− 4𝑡^{2}𝑔^{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}^{2}

𝑢^{2} + 𝑤^{2}

− 8𝑡^{2}(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2})^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})

𝑢^{2} + 𝑤^{2} .

(45)

Using‖𝑤_{2}‖ = 2(𝑥^{2}_{1}+ 𝑦^{2}_{1}) and (40), we simplify the sum of the
first two terms in (45) as

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

𝑢^{2} + 𝑤^{2}

= 2𝑡 (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ)𝑢^{2} − 𝑤^{2}

𝑢^{2} + 𝑤^{2}

= 4𝑡^{2}(𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ)^{2}

𝑢^{2} + 𝑤^{2} + 𝑜(𝑡^{2}) .

(46)

Then, from (45) and‖𝑤_{2}‖ = 2(𝑥^{2}_{1}+ 𝑦_{1}^{2}), we obtain that

lim𝑡↓0

𝜇_{1}
𝑡^{2}

= (𝑥^{2}_{1}+ 𝑦_{1}^{2}) (𝑔^{2}+ ‖ℎ‖^{2}) − 𝑔^{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}^{2}
𝑥^{2}_{1}+ 𝑦_{1}^{2}

+(𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ)^{2}− 2(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2})^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})

𝑥^{2}_{1}+ 𝑦^{2}_{1} .

(47)

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

(𝑥^{2}_{1}+ 𝑦_{1}^{2}) (𝑔^{2}+ ‖ℎ‖^{2}) − 𝑔^{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}^{2}

= (𝑥^{2}_{1}+ 𝑦_{1}^{2}) (𝑔^{2}+ ‖ℎ‖^{2}) − 𝑔^{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}^{2}

− 𝑦^{1}ℎ_{2}+ ℎ_{1}𝑦_{2}^{2}− 2(𝑔_{1}𝑥_{2}+ 𝑥_{1}𝑔_{2})^{𝑇}(𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})

= 𝑥_{1}^{2}‖ℎ‖^{2}+ 𝑦^{2}_{1}𝑔^{2}− 2𝑥_{1}𝑔_{1}𝑥^{𝑇}_{2}𝑔_{2}− 2𝑦_{1}ℎ_{1}𝑦_{2}^{𝑇}ℎ_{2}

− 2(𝑔_{1}𝑥_{2}+ 𝑥_{1}𝑔_{2})^{𝑇}(𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}) ,
(𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ)^{2}− 2(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2})^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})

= (𝑥_{1}𝑔_{1}+ 𝑥^{𝑇}_{2}𝑔_{2})^{2}+ (𝑦_{1}ℎ_{1}+ 𝑦_{2}^{𝑇}ℎ_{2})^{2}+ 2𝑥^{𝑇}𝑔𝑦^{𝑇}ℎ

− 2(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2})^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})

= (𝑥_{1}𝑔_{1})^{2}+ (𝑥^{𝑇}_{2}𝑔_{2})^{2}+ (𝑦_{1}ℎ_{1})^{2}+ (𝑦_{2}^{𝑇}ℎ_{2})^{2}
+ 2𝑥^{𝑇}𝑔𝑦^{𝑇}ℎ − 2𝑥_{1}ℎ_{1}𝑥^{𝑇}_{2}ℎ_{2}− 2𝑔_{1}𝑦_{1}𝑔^{𝑇}_{2}𝑦_{2}.

(48)

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

(𝑥_{1}^{2}+ 𝑦^{2}_{1}) (𝑔^{2}+ ‖ℎ‖^{2}) − 𝑔^{1}𝑥_{2}+ 𝑥_{1}𝑔_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}^{2}
+ (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ)^{2}− 2(𝑥_{1}𝑥_{2}+ 𝑦_{1}𝑦_{2})^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})

= (𝑥^{2}_{1}‖ℎ‖^{2}− 2𝑥_{1}ℎ_{1}𝑥^{𝑇}_{2}ℎ_{2}) + (𝑦_{1}^{2}𝑔^{2}− 2𝑔_{1}𝑦_{1}𝑔^{𝑇}_{2}𝑦_{2})
+ ((𝑥_{1}𝑔_{1})^{2}+ (𝑥^{𝑇}_{2}𝑔_{2})^{2}− 2𝑥_{1}𝑔_{1}𝑥^{𝑇}_{2}𝑔_{2})

+ ((𝑦_{1}ℎ_{1})^{2}+ (𝑦^{𝑇}_{2}ℎ_{2})^{2}− 2𝑦_{1}ℎ_{1}𝑦_{2}^{𝑇}ℎ_{2}) + 2𝑥^{𝑇}𝑔𝑦^{𝑇}ℎ

− 2(𝑔_{1}𝑥_{2}+ 𝑥_{1}𝑔_{2})^{𝑇}(𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})

= 𝑥^{1}ℎ_{2}− ℎ_{1}𝑥_{2}^{2}+ 𝑔^{1}𝑦_{2}− 𝑦_{1}𝑔_{2}^{2}+ (𝑥_{1}𝑔_{1}− 𝑥^{𝑇}_{2}𝑔_{2})^{2}
+ (𝑦_{1}ℎ_{1}− 𝑦^{𝑇}_{2}ℎ_{2})^{2}+ 2 (𝑔_{1}𝑥_{1}+ 𝑔^{𝑇}_{2}𝑥_{2}) (𝑦_{1}ℎ_{1}+ 𝑦^{𝑇}_{2}ℎ_{2})

− 2(𝑔_{1}𝑥_{2}+ 𝑥_{1}𝑔_{2})^{𝑇}(𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})

= 𝑥^{1}ℎ_{2}− ℎ_{1}𝑥_{2}^{2}+ 𝑔^{1}𝑦_{2}− 𝑦_{1}𝑔_{2}^{2}+ (𝑥_{1}𝑔_{1}− 𝑥^{𝑇}_{2}𝑔_{2})^{2}
+ (𝑦_{1}ℎ_{1}− 𝑦^{𝑇}_{2}ℎ_{2})^{2}+ 2(𝑥_{1}ℎ_{2}− ℎ_{1}𝑥_{2})^{𝑇}(𝑔_{1}𝑦_{2}− 𝑔_{2}𝑦_{1})
+ 2 (𝑥_{1}𝑔_{1}− 𝑥^{𝑇}_{2}𝑔_{2}) (𝑦_{1}ℎ_{1}− 𝑦_{2}^{𝑇}ℎ_{2})

= 𝑥^{1}ℎ_{2}− ℎ_{1}𝑥_{2}+ 𝑔_{1}𝑦_{2}− 𝑦_{1}𝑔_{2}^{2}
+ (𝑥_{1}𝑔_{1}− 𝑥^{𝑇}_{2}𝑔_{2}+ 𝑦_{1}ℎ_{1}− 𝑦_{2}^{𝑇}ℎ_{2})^{2}.

(49)

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

lim𝑡↓0

√𝜇_{1}

𝑡 = 𝜑 (𝑔, ℎ) . (50)

We next calculate lim_{𝑡↓0}[((√𝜇2 − √𝜇1)/2𝑡)(𝑢_{2}/‖𝑢_{2}‖) −
((√𝜆_{2}/2𝑡)(𝑤_{2}/‖𝑤_{2}‖))]. To this end, we also need to take a look
at‖𝑤_{2}‖𝑢_{2}− ‖𝑢_{2}‖𝑤_{2}. From (38)-(39) and (40), it follows that

𝑤^{2}𝑢^{2}− 𝑢^{2}𝑤^{2}

= 2𝑡 𝑤^{2}[(𝑥^{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})

− (𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ) ̃𝑤_{2}] + 𝑜 (𝑡) .

(51)

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

lim𝑡↓0 [√𝜇_{2}− √𝜇1

2𝑡
𝑢_{2}

𝑢^{2} −√𝜆_{2}
2𝑡

𝑤_{2}

𝑤^{2}]

= −lim

𝑡↓0

√𝜇1

2𝑡
𝑢_{2}

𝑢^{2} + lim^{𝑡↓0} [√𝜇_{2}
2𝑡

𝑢_{2}

𝑢^{2} −√𝜆_{2}
2𝑡

𝑤_{2}

𝑤^{2}]

= −lim

𝑡↓0

√𝜇_{1}
2𝑡

𝑢_{2}

𝑢^{2} + lim^{𝑡↓0}√𝜇_{2}− √𝜆_{2}
2𝑡

𝑢_{2}

𝑢^{2}

+ lim

𝑡↓0

√𝜆_{2}(𝑤^{2}𝑢^{2}− 𝑢^{2}𝑤^{2})
2𝑡 𝑢^{2}𝑤^{2}

= −𝜑 (𝑔, ℎ)

2 𝑤̃_{2}+𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}

√𝑥^{2}_{1}+ 𝑦_{1}^{2}

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

= −𝜑 (𝑔, ℎ)

2 𝑤̃_{2}+2𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 2𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2}
2√𝑥^{2}_{1}+ 𝑦^{2}_{1}

−𝑥^{𝑇}_{2}𝑔_{2}+ 𝑦^{𝑇}_{2}ℎ_{2}
2√𝑥^{2}_{1}+ 𝑦_{1}^{2} 𝑤̃_{2},

(52)
where the last equality is using𝑥_{1}𝑤̃_{2} = 𝑥_{2}and𝑦_{1}𝑤̃_{2} = 𝑦_{2}.
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**^{+}K^{𝑛}
for sufficiently small𝑡 > 0, the formula in (32) reduces to the
one in (37). Indeed, an elementary calculation yields

𝜆_{1}((𝑥 + 𝑡𝑔)^{2}+ (𝑦 + 𝑡ℎ)^{2})

= [𝑥 + 𝑡𝑔^{2}+ 𝑦 + 𝑡ℎ^{2}]^{2}

− 4(𝑥^{1}+ 𝑡𝑔_{1}) (𝑥_{2}+ 𝑡𝑔_{2}) + (𝑦_{1}+ 𝑡ℎ_{1}) (𝑦_{2}+ 𝑡ℎ_{2})^{2}

= 4𝑡^{2}𝜑 (𝑔, ℎ) √𝑥^{2}_{1}+ 𝑦_{1}^{2}

+ 4𝑡^{3}(𝑥^{𝑇}𝑔 + 𝑦^{𝑇}ℎ) (𝑔^{2}+ ‖ℎ‖^{2})

− 8𝑡^{2}(𝑥_{1}𝑔_{2}+ 𝑔_{1}𝑥_{2}+ 𝑦_{1}ℎ_{2}+ ℎ_{1}𝑦_{2})^{𝑇}(𝑔_{1}𝑔_{2}+ ℎ_{1}ℎ_{2})
+ 𝑡^{4}[(𝑔^{2}+ ‖ℎ‖^{2})^{2}− 2𝑔^{1}𝑔_{2}+ ℎ_{1}ℎ_{2}^{2}]

= 4𝑡^{2}𝜑 (𝑔, ℎ) √𝑥^{2}_{1}+ 𝑦_{1}^{2}+ 𝑜 (𝑡^{2}) .

(53)
This implies that if(𝑥+𝑡𝑔)^{2}+(𝑦+𝑡ℎ)^{2}**∈ bd**^{+}K^{𝑛}for sufficiently
small𝑡 > 0, that is, 𝜆_{1}((𝑥+𝑡𝑔)^{2}+(𝑦+𝑡ℎ)^{2}) = 0 for sufficiently
small𝑡 > 0, then 𝜑(𝑔, ℎ) = 0, and hence 𝑥_{1}𝑔_{1}+𝑦_{1}ℎ_{1}−(𝑥^{𝑇}_{2}𝑔_{2}+