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to appear in Abstract and Applied Analysis, 2013

Nonsingularity conditions for FB system of reformulating nonlinear second-order cone programming

1

Shaohua Pan, Shujun Bi and Jein-Shan Chen§

June 2, 2012

Abstract. This paper is a counterpart of [2]. Specifically, for a locally optimal solution to the nonlinear second-order cone programming (SOCP), 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.

Key words: nonlinear second-order cone programming; FB nonsmooth system; non- singularity; Clarke’s generalized Jacobian; strong regularity.

1 Introduction

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

ζmin∈IRn f (ζ)

s.t. h(ζ) = 0, (1)

g(ζ)∈ K,

where f : IRn → IR, h : IRn → IRm and g : IRn → IRn are given twice continuously differentiable functions, andK is the Cartesian product of some second-order cones, i.e.,

K := Kn1 × Kn2 × · · · × Knr

1This work was supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001), the Fundamental Research Funds for the Central Universities, and Project of Liaoning Innovative Research Team in UniversityWT2010004

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

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

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

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with n1 +· · · + nr = n and Knj being the second-order cone (SOC) in IRnj defined by Knj :={

(xj1, xj2)∈ IR × IRnj−1 | xj1 ≥ ∥xj2} .

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

ζ,xmin∈IRnf (ζ)

s.t. h(ζ) = 0, (2)

g(ζ)− x = 0, x ∈ K.

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

Let L : IRn× IRn× IRm× IRn× K → IR be the Lagrangian function of problem (2) L(ζ, x, µ, s, y) := f (ζ) +⟨µ, h(ζ)⟩ + ⟨g(ζ) − x, s⟩ − ⟨x, y⟩,

and denote by NK(x) the normal cone of K at x in the sense of convex analysis [19]:

NK(x) =

{ {d ∈ IRn: ⟨d, z − x⟩ ≤ 0 ∀z ∈ K} if x ∈ K,

if x /∈ K.

Then the Karush-Kuhn-Tucker (KKT) conditions for (2) take the following form Jζ,xL(ζ, x, µ, s, y) = 0, h(ζ) = 0, g(ζ)− x = 0 and − y ∈ NK(x), (3) where Jζ,xL(ζ, x, µ, s, y) is the derivative of L at (ζ, x, µ, s, y) with respect to (ζ, x).

Recall that ϕsoc is an SOC complementarity function associated with the cone K if ϕsoc(x, y) = 0 ⇐⇒ x ∈ K, y ∈ K, ⟨x, y⟩ = 0 ⇐⇒ −y ∈ NK(x). (4) With an SOC complementarity function ϕsoc associated withK, we may reformulate the KKT optimality conditions in (3) 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. (5)

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

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

ϕFB(x, y) := (x + y)−

x2 + y2 ∀x, y ∈ IRn, (6)

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where ΠK(·) is the projection operator onto the closed convex cone K, 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 function ϕFB enjoys almost all favorable properties of the NR SOC complementarity function ϕNR (see [22]). Also, the squared norm of ϕFB induces a continuously differentiable merit function with globally Lipschitz continuous derivative [6, 7]. This greatly facilitates the globalization of the semismooth Newton method [16, 17] 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. (7)

Recently, with the help of [3, Theorem 30] and [5, Lemma 11], Wang and Zhang [23]

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. (8)

They showed that the strong regularity of the KKT point, the nonsingularity of Clarke’s Jacobian of ENR at the KKT point, and the strong second order sufficient condition and constraint nondegeneracy [3], are all equivalent. These nonsingularity conditions are better structured than those of [14] for the nonsingularity of the B-subdifferential of the NR system. Then, it is natural to ask: 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 to now one even does not know whether the B-subdifferential of the FB system is nonsingular or not without the strict complementarity assumption.

In this work, for a locally optimal solution to the nonlinear SOCP (2), under Robin- son’s constraint qualification, we show that the strong second-order sufficient condition and constraint nondegeneracy introduced in [3], the nonsingularity of Clarke’s Jacobian of EFB 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 [16, 17] for the FB sys- tem. Note that parallel results are obtained recently for the FB system of the nonlinear semidefinite programming (see [2]), however, we do not duplicate them. As will be seen in Section 3 and Section 4, the analysis techniques here are totally different from those in [2]. It seems hard to put them together in a unified framework under the Euclidean Jor- dan algebra. The main reason causing this is due to completely different analysis when

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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, IRn (n > 1) denotes the space of n-dimensional real column vectors, and IRn1× · · · × IRnr is identified with IRn1+···+nr. Thus, (x1, . . . , xr)∈ IRn1 × · · · × IRnr is viewed as a column vector in IRn1+···+nr. The notations intKn, bdKn and bd+Kn denote the interior, the boundary, and the boundary excluding the origin of Kn, respectively. For any x ∈ IRn, we write x Kn 0 (respectively, x≻Kn 0) if x ∈ Kn (respectively, x ∈ intKn). For any given real symmetric matrix A, we write A ≽ 0 (respectively, A ≻ 0) if A is positive semidefinite (respectively, positive definite). In addition, Jωf (ω) and Jωω2 f (ω) denote the derivative and the second order derivative, respectively, of a twice differentiable function f with respect to the variable ω.

2 Preliminary results

First we recall from [11] the definition of Jordan product and spectral factorization.

Definition 2.1 The Jordan product of x = (x1, x2), y = (y1, y2)∈ IR× IRn−1 is given by x◦ y := (⟨x, y⟩, x1y2+ y1x2). (9) Unlike scalar or matrix multiplication, the Jordan product is not associative in general.

The identity element under this product is e := (1, 0, . . . , 0)T ∈ IRn, i.e., e◦ x = x for all x∈ IRn. For each x = (x1, x2)∈ IR × IRn−1, we define the associated arrow matrix by

Lx:=

[ x1 xT2 x2 x1I

]

. (10)

Then it is easy to verify that Lxy = x◦ y for any x, y ∈ IRn. Recall that each x = (x1, x2)∈ IR × IRn−1 admits a spectral factorization, associated withKn, of the form

x = λ1(x)u(1)x + λ2(x)u(2)x , (11) where λ1(x), λ2(x) ∈ IR and u(1)x , u(2)x ∈ IRn are the spectral values and the associated spectral vectors of x, respectively, with respect to the Jordan product, defined by

λi(x) := x1+ (−1)i∥x2∥, u(i)x := 1 2

( 1

(−1)ix˜2

)

for i = 1, 2, (12) with ˜x2 = ∥xx2

2 if x2 ̸= 0 and otherwise being any vector in IRn−1 satisfying ∥˜x2∥ = 1.

Definition 2.2 The determinant of a vector x∈IRn is defined as det(x) := λ1(x)λ2(x), and a vector x is said to be invertible if its determinant det(x) is nonzero.

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By the formula of spectral factorization, it is easy to compute that the projection of x∈ IRn onto the closed convex cone Kn, denoted by ΠKn(x), has the expression

ΠKn(x) = max(0, λ1(x))u(1)x + max(0, λ2(x))u(2)x .

Define|x| := 2ΠKn(x)− x. Then, using the expression of ΠKn(x), it follows that

|x| = |λ1(x)|u(1)x +2(x)|u(2)x . The spectral factorization of the vectors x, x2,

x and the matrix Lx have various interesting properties (see [12]). We list several properties that we will use later.

Property 2.1 For any x = (x1, x2)∈IR×IRn−1 with spectral factorization (11), we have (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) If x∈ intKn, then 0 < λ1(x)≤ λ2(x) and Lx is invertible with

L−1x = 1 det(x)

x1 −xT2

−x2

det(x)

x1 I +x2xT2 x1

 . (13)

(d) Lx ≽ 0 (respectively, Lx ≻ 0) if and only if x ∈ Kn (respectively, x∈ intKn).

The following lemma states a result for the arrow matrices associated with x, y ∈ IRn and z Kn

x2+ y2, which will be used in the next section to characterize an important property for the elements of Clarke’s Jacobian of ϕFB at a general point.

Lemma 2.1 For any given x, y∈ IRn and z Kn 0, if z2 Kn x2+ y2, then [L−1z Lx L−1z Ly]

2 ≤ 1,

where ∥A∥2 means the spectral norm of a real matrix A. Consequently, it holds that

∥L−1z Lx△u + L−1z Ly△v∥ ≤

∥△u∥2+∥△v∥2 ∀△u, △v ∈ IRn. Proof. Let A = [L−1z Lx L−1z Ly]. From [12, Proposition 3.4], it follows that

AAT = L−1z (L2x+ L2y)L−1z ≼ L−1z L2zL−1z = I.

This shows that ∥A∥2 ≤ 1, and the first part follows. Note that for any ξ ∈ IR2n,

∥Aξ∥2 = ξTATAξ≤ λmax(ATA)∥ξ∥2 ≤ ∥ξ∥2.

By letting ξ = (△u, △v) ∈ IRn× IRn, we immediately obtain the second part. 2 The following two lemmas state the properties of x, y with x2+ y2 ∈ bdKn which are often used in the subsequent sections. The proof of Lemma 2.2 is given in [6, Lemma 2].

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Lemma 2.2 For any x = (x1, x2), y = (y1, y2)∈ IR ×IRn−1 with x2+ y2∈bdKn, we have x21 =∥x22, y12 =∥y22, x1y1 = xT2y2, x1y2 = y1x2.

Lemma 2.3 For any x = (x1, x2), y = (y1, y2)∈ IR × IRn−1, let w = (w1, w2) := x2+ y2. (a) If w ∈ bdKn, then for any g = (g1, g2), h = (h1, h2)∈ IR × IRn−1, it holds that

(x1x2+ y1y2)T(x1g2+ g1x2+ y1h2+ h1y2) = (x21+ y12)(xTg + yTh).

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

∥w2 = x2, xT2w2

∥w2 = x1, y1w2

∥w2 = y2, yT2w2

∥w2 = y1; and consequently the expression of ϕFB(x, y) can be simplified as

ϕFB(x, y) =

x1+ y1

x21+ y12 x2+ y2 x1x2+ y1y2

x21+ y12

 . (14)

Proof. (a) The result is direct by the equalities of Lemma 2.2 since x2+ y2 ∈ bdKn. (b) Since w ∈ bd+Kn, we must have w2 = 2(x1x2 + y1y2) ̸= 0. Using Lemma 2.2, w2 = 2(x1x2+ y1y2) and ∥w2∥ = w1 = 2(x21+ y21), we easily obtain the first part. Note that ϕFB(x, y) = (x + y)−√

w. Using Property 2.1(b) and Lemma 2.2 yields (14). 2 When x, y ∈ bdKnsatisfy the complementary condition, we have the following result.

Lemma 2.4 For any given x = (x1, x2), y = (y1, y2) ∈ IR × IRn−1, if x, y ∈ bdKn and

⟨x, y⟩ = 0, then there exists a constant α > 0 such that x1 = αy1 and x2 =−αy2. Proof. Since x, y ∈ bdKn, we have that x1 =∥x2∥ and y1 =∥y2∥, and consequently,

0 = ⟨x, y⟩ = x1y1+ xT2y2 =∥x2∥∥y2∥ + xT2y2.

This means that there exists α > 0 such that x2 =−αy2, and then x1 = αy1. 2 Next we recall from [21] the strong regularity for a solution of generalized equation

0∈ ϕ(z) + ND(z), (15)

where ϕ is a continuously differentiable mapping from a finite dimensional real vector spaceZ to itself, D is a closed convex set in Z, and ND(z) is the normal cone of D at z.

As will be shown in Sec. 4, the KKT condition (3) can be written in the form of (15).

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Definition 2.3 We say that ¯z is a strongly regular solution of the generalized equation (15) if there exist neighborhood B of the origin 0 ∈ Z and V of ¯z such that for every δ∈ B, the linearized generalized equation δ ∈ ϕ(¯z) + Jzϕ(¯z)(z− ¯z) + ND(z) has a unique solution in V, denoted by zV(δ), and the mapping zV :B → V is Lipschitz continuous.

To close this section, we recall from [8] Clarke’s (generalized) Jacobian of a locally Lipschitz mapping. Let S ⊂ IRn be an open set and Ξ : S → IRn be a locally Lipschitz continuous function on S. By Rademacher’s theorem, Ξ is almost everywhere F(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 [18], has the form

BΞ(y) :=

{

V : V = lim

k→∞JyΞ(yk), yk → y, yk ∈ SΞ

} .

For the concept of (strong) semismoothness, please refer to the literature [16, 17].

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

w = w(x, y) := x2+ y2, w˜2 := w2

∥w2 if w2 ̸= 0 and z = z(x, y) =

w(x, y). (16)

3 Directional derivative and B-subdifferential

The function ϕFB is directionally differentiable everywhere by [22, Corollary 3.3]. 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 that K = Kn.

Proposition 3.1 For any given x = (x1, x2), y = (y1, y2) ∈ IR × IRn−1, the directional derivative ϕ

FB((x,y); (g,h)) of ϕFB 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)) = ϕFB(g, h).

(b) If x2+ y2 ∈ intKn, then ϕ

FB((x, y); (g, h)) = (I− L−1z Lx) g + (I− L−1z Ly) h.

(c) If x2+ y2 ∈ bd+Kn, then ϕ

FB((x, y); (g, h)) = (g + h)− φ(g, h) 2

( 1

− ˜w2 )

+xT2g2+ yT2h2 2√

x21+ y21 ( 0

˜ w2

)

1

2√

x21+ y12

( xTg + yTh

2x1g2 + g1x2+ 2y1h2+ h1y2 )

(17)

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where g = (g1, g2), h = (h1, h2)∈ IR × IRn−1, and φ : IRn× IRn → IR is defined by φ(g, h) :=

(x1g1− xT2g2 + y1h1−yT2h2)2+∥x1h2− h1x2+g1y2− y1g22

x21 + y21 . (18)

Proof. Part (a) is immediate by noting that ϕFB is a positively homogeneous function.

Part (b) is due to [12, Proposition 5.2]. We next prove part (c) by two subcases as shown below. In the rest of proof, we let λ1, λ2 with λ1 ≤ λ2 denote the spectral values of w.

Since w = x2 + y2 ∈ bd+Kn, we have w2 ̸= 0, and from Lemma 2.3(b) it follows that w1 =∥w2∥ = 2∥x1x2+ y1y2∥ = 2∥x21we2+ y12we2∥ = 2(x21+ y12),

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

(c.1): (x + tg)2 + (y + th)2 ∈ bd+Kn for sufficiently small t > 0. In this case, from Lemma 2.3(b), we know that ϕFB(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

 .

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

limt↓0

FB(x + tg, y + th)]1− [ϕFB(x, y)]1

t

= (g1+ h1)− lim

t↓0

(x1+ tg1)2+ (y1+ th1)2

x21+ y12 t

= (g1+ h1) x1g1+ y1h1

x21+ y12 and

limt↓0

FB(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+ y1h2+ h1y2

x21+ y21

− lim

t↓0

[

x1x2+ y1y2 t

(x1+ tg1)2+ (y1+ th1)2 −x1x2+ y1y2 t

x21+ y21 ]

= (g2+ h2) g1x2+ x1g2+ y1h2+ h1y2

x21+ y21 +(x1x2+ y1y2)(x1g1+ y1h1) (x21+ y12)√

x21+ y12

= (g2+ h2) x1g2+ y1h2

x21+ y12

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where the last equality is using x1y2 = y1x2 by Lemma 2.2. The above two limits imply ϕFB((x, y); (g, h)) = (g + h)− x1

x21+ y12g− y1

x21+ y21h. (19)

(c.2): (x + tg)2 + (y + th)2 ∈ intKn for sufficiently small t > 0. Let u = (u1, u2) :=

(x + tg)2+ (y + th)2 with the spectral values µ1, µ2. An elementary calculation gives u1 = ∥x + tg∥2+∥y + th∥2 = w1+ 2t(xTg + yTh) + t2(∥g∥2+∥h∥2), (20) u2 = 2(x1+ tg1)(x2 + tg2) + 2(y1+ th1)(y2+ th2)

= w2+ 2t(x1g2+ g1x2+ y1h2+ h1y2) + 2t2(g1g2+ h1h2). (21) Also, since w2 ̸= 0, applying the Taylor formula of ∥ · ∥ at w2 and Lemma 2.3(a) yields

∥u2∥ = ∥w2∥ +wT2(u2− w2)

∥w2 + o(t) =∥w2∥ + 2t(xTg + yTh) + o(t). (22) Now using the definition of ϕFB and noting that λ1 = 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)−



õ1+

µ2−√ λ2

2

µ2− √µ1

2

u2

∥u2

√λ2 2

w2

∥w2



 ,

which in turn implies that

ϕ

FB((x, y); (g, h)) = (g + h)−



limt↓0

õ1+

µ2−√ λ2 2t

limt↓0

(√µ2 − √µ1

2t

u2

∥u2

√λ2 2t

w2

∥w2 )



 . (23)

We first calculate limt↓0µ2

λ2

t . Using equations (20) and (22), it is easy to see that µ2− λ2 = (u1− w1) + (∥u2∥ − ∥w2∥) = 4t(xTg + yTh) + o(t),

and consequently, lim

t↓0

√µ2−√ λ2

t = lim

t↓0

µ2− λ2

t · 1

√µ2+ λ2

= xTg + yTh 2

λ2

= xTg + yTh

x21+ y12 . (24)

We next calculate limt↓0tµ1. Since w1 − ∥w2∥ = 0, using (20)-(21) and Lemma 2.3(a), µ1 = (u1− w1)− (∥u2∥ − ∥w2∥) = (u1− w1) ∥u22− ∥w22

∥u2∥ + ∥w2

(10)

= 2t(xTg + yTh)− 4tw2T(x1g2+ g1x2+ y1h2 + h1y2)

∥u2∥ + ∥w2 + t2(∥g∥2+∥h∥2)

−4t2∥g1x2+ x1g2+ y1h2+ h1y22

∥u2∥ + ∥w2 − 4t2w2T(g1g2+ h1h2)

∥u2∥ + ∥w2 + o(t2)

= 2t(xTg + yTh)− 8t(x21+ y21)(xTg + yTh)

∥u2∥ + ∥w2 + t2(∥g∥2+∥h∥2) + o(t2)

−4t2∥g1x2+ x1g2+ y1h2+ h1y22

∥u2∥ + ∥w2 − 8t2(x1x2+ y1y2)T(g1g2+ h1h2)

∥u2∥ + ∥w2 . (25) Using∥w2∥ = 2(x21+ y12) and (22), we simplify the sum of the first two terms in (25) 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).

Then, from equation (25) and ∥w2∥ = 2(x21+ y12), we obtain that lim

t↓0

µ1

t2 = (x21+ y12)(∥g∥2 +∥h∥2)− ∥g1x2+ x1g2+ y1h2+ h1y22 x21 + y21

+(xTg + yTh)2− 2(x1x2+ y1y2)T(g1g2+ h1h2)

x21+ y21 . (26)

We next make simplification for the numerator of the right hand side of (26). Note that (x21+ y12)(∥g∥2+∥h∥2)− ∥g1x2+ x1g2+ y1h2+ h1y22

= (x21+ y12)(∥g∥2+∥h∥2)− ∥g1x2+ x1g22− ∥y1h2+ h1y22

−2(g1x2+ x1g2)T(y1h2+ h1y2)

= x21∥h∥2+ y21∥g∥2− 2x1g1xT2g2− 2y1h1y2Th2− 2(g1x2+ x1g2)T(y1h2+ h1y2) and

(xTg + yTh)2 − 2(x1x2+ y1y2)T(g1g2+ h1h2)

= (x1g1+ xT2g2)2+ (y1h1+ y2Th2)2+ 2xTgyTh− 2(x1x2+ y1y2)T(g1g2+ h1h2)

= (x1g1)2+ (xT2g2)2+ (y1h1)2+ (y2Th2)2+ 2xTgyTh− 2x1h1xT2h2− 2g1y1g2Ty2. Therefore, adding the last two equalities and using Lemma 2.2 yields that

(x21+ y12)(∥g∥2+∥h∥2)− ∥g1x2+ x1g2+ y1h2+ h1y22 +(xTg + yTh)2− 2(x1x2+ y1y2)T(g1g2+ h1h2)

= (x21∥h∥2− 2x1h1xT2h2) + (y12∥g∥2− 2g1y1g2Ty2) +(

(x1g1)2+ (xT2g2)2− 2x1g1xT2g2) +(

(y1h1)2 + (yT2h2)2− 2y1h1y2Th2)

+ 2xTgyTh− 2(g1x2+ x1g2)T(y1h2+ h1y2)

(11)

= ∥x1h2− h1x22+∥g1y2− y1g22 + (x1g1 − xT2g2)2+ (y1h1− yT2h2)2 +2(g1x1+ g2Tx2)(y1h1+ y2Th2)− 2(g1x2+ x1g2)T(y1h2+ h1y2)

= ∥x1h2− h1x22+∥g1y2− y1g22 + (x1g1 − xT2g2)2+ (y1h1− yT2h2)2 +2(x1h2− h1x2)T(g1y2 − g2y1) + 2(x1g1− xT2g2)(y1h1− yT2h2)

= ∥x1h2− h1x2 + g1y2− y1g22+ (x1g1 − xT2g2+ y1h1− yT2h2)2.

Combining this equality with (26) and using the definition of φ in (18), we readily get limt↓0

õ1

t = φ(g, h). (27)

We next calculate limt↓0

[µ2−√µ1

2t u2

∥u2 2tλ2∥ww22]

. To this end, we also need to take a look at ∥w2∥u2− ∥u2∥w2. From equations (20)-(21) and (22), it follows that

∥w2∥u2− ∥u2∥w2 = 2t∥w2[

(x1g2+ g1x2+ y1h2 + h1y2)− (xTg + yTh) ˜w2]

+ o(t).

Together with equations (24) and (27), we have that lim

t↓0

[√µ2− √µ1

2t

u2

∥u2∥−

√λ2 2t

w2

∥w2 ]

= − lim

t↓0

õ1 2t

u2

∥u2 + lim

t↓0

[õ2 2t

u2

∥u2∥−

√λ2 2t

w2

∥w2 ]

= − lim

t↓0

õ1 2t

u2

∥u2 + lim

t↓0

√µ2−√ λ2 2t

u2

∥u2 + lim

t↓0

√λ2(∥w2∥u2− ∥u2∥w2) 2t∥u2∥∥w2

= −φ(g, h)

2 w˜2+ x1g2+ g1x2+ y1h2+ h1y2

x21+ y21 xTg + yTh 2√

x21+ y12w˜2

= −φ(g, h)

2 w˜2+ 2x1g2+ g1x2+ 2y1h2+ h1y2 2√

x21+ y21 xT2g2+ y2Th2 2√

x21+ y21 w˜2,

where the last equality is using x1w˜2 = x2 and y1w˜2 = y2. Combining with (23), (24) and (27), a suitable rearrangement shows that ϕ

FB((x, y); (g, h)) has the expression (17).

Finally, we show that when (x + tg)2+ (y + th)2 ∈ bd+Knfor sufficiently small t > 0, the formula in (17) reduces to the one in (19). 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+ y21 + 4t3(xTg + yTh)(∥g∥2+∥h∥2)

−8t2(x1g2+ g1x2+ y1h2+ h1y2)T(g1g2 + h1h2) +t4[

(∥g∥2+∥h∥2)2− 2∥g1g2+ h1h22]

= 4t2φ(g, h)

x21+ y21 + o(t2).

參考文獻

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