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# 3 Second-Order Directional Derivative

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to appear in Journal of Optimization Theory and Applications, 2016

### circular cones

Jinchuan Zhou 1 Department of Mathematics

School of Science

Shandong University of Technology Zibo 255049, Shandong, P.R.China

E-mail: jinchuanzhou@163.com

Jingyong Tang2

College of Mathematics and Information Science Xinyang Normal University

Xinyang 464000, Henan, P.R.China E-mail: jingyongtang@163.com

Jein-Shan Chen 3 Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

June 28, 2015

(1st revised on March 9, 2016) (2nd revised on April 8, 2016)

Abstract. In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone. The

1The author’s work is supported by National Natural Science Foundation of China (11101248, 11271233), Shandong Province Natural Science Foundation (ZR2012AM016), and Young Teacher Sup- port Program of Shandong University of Technology.

2The author’s work is supported by Science Technology Research Projects of Education Department of Henan Province (13A110767).

3Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan.

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vector-valued function comes from applying a given real-valued function to the spec- tral decomposition associated with circular cone. In particular, we present the exact formula of second-order tangent set of circular cone by using the parabolic second-order directional derivative of projection operator. In addition, we also deal with the relation- ship of second-order differentiability between the vector-valued function and the given real-valued function. The results in this paper build fundamental bricks to the character- izations of second-order necessary and sufficient conditions for circular cone optimization problems.

Keywords Parabolic second-order derivative, circular cone, second-order tangent set

### 1 Introduction

The parabolic second-order derivatives were originally introduced by Ben-Tal and Zowe in [1, 2], please refer to [3] for more details about properties of parabolic second-order derivatives. Usually the parabolic second-order derivatives can be employed to char- acterize the optimality conditions for various optimization problems, see [1, 4, 5, 6, 7]

and references therein. The so-called generalized parabolic second-order derivatives are studied in [4, 5, 8], whereas the parabolic second-order derivatives for certain types of functions are investigated in [5, 8, 9, 10]. In this paper, we mainly focus on the parabolic second-order directional derivative in the Hadamard sense for the vector-valued func- tions associated with circular cones. This vector-valued function, called circular cone function, comes from applying a given real-valued function to the spectral decomposition associated with circular cone.

For the circular cone function, by using the basic tools of nonsmooth analysis, various properties such as directional derivative, differentiability, B-subdifferentiability, semis- moothness, and positive homogeneity, have been studied in [11, 12]. The aforementioned results can be regarded as the first-order type of differentiability analysis. Here, we further discuss the second-order type of differentiability analysis for the circular cone function. As mentioned above, the concept of parabolic second-order directional differ- entiability plays an important role in second-order necessary and sufficient conditions.

Recently, there was an investigation on the parabolic second-order directional derivative of singular values of matrices and symmetric matrix-valued functions in [10]. Inspired by this work, we study the parabolic second-order directional derivative for the vector-valued circular cone function. The relationship of parabolic second-order directional derivative between the vector-valued circular cone function and the given real-valued function is established, in which we do not require that the real-valued function is second-order dif- ferentiable. This allows us to apply our result to more general nonsmooth functions. For example, we obtain the exact formula of second-order tangent set by using the parabolic second-order directional differentiability of projection operator associated with circular

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cone, which is corresponding to the nonsmooth max-type function. In addition, we study the relationship of second-order differentiability between circular cone function and the given real-valued function. It is surprising that, not like the first-order differentiability, the relationship in the second-order differentiability case really depends on the angle.

This further shows the essential role played by the angle in the circular cone setting.

### 2 Preliminaries

The n-dimensional circular cone is defined as

Lθ := {x = (x1, x2)T ∈ R × Rn−1 : cos θkxk ≤ x1},

which is a non-symmetrical cone in the standard inner product. In our previous works [12, 13, 14, 15], we have explored some important features about circular cone, such as characterizing its tangent cone, normal cone, and second-order regularity, etc.. In particular, the spectral decomposition associated with Lθ was discovered, i.e., for any x = (x1, x2) ∈ R × Rn−1, one has

x = λ1(x)u(1)x + λ2(x)u(2)x , (1) where

λ1(x) := x1− kx2k cot θ, λ2(x) := x1+ kx2k tan θ and

u(1)x := 1 1 + cot2θ

1 0

0 cot θ · I

  1

−¯x2



, u(2)x := 1 1 + tan2θ

1 0

0 tan θ · I

  1

¯ x2



with ¯x2 := x2/kx2k if x2 6= 0, and ¯x2 being any vector w ∈ Rn−1 satisfying kwk = 1 if x2 = 0. With this spectral decomposition (1), we can define a vector-valued function associated with circular cone as below. More specifically, for a given real-valued function f : R → R, the circular cone function fLθ : Rn→ Rn is defined as

fLθ(x) := f (λ1(x)) u(1)x + f (λ2(x)) u(2)x .

Let X, Y be normed vector spaces and consider x, d, w ∈ X. Assume that ψ : X → Y is directionally differentiable. The function ψ is said to be parabolical second-order directionally differentiable in the Hadamard sense at x, if ψ is directionally differentiable at x and for any d, w ∈ X the following limit exists:

ψ00(x; d, w) := lim

t↓0 w0→w

ψ(x + td +12t2w0) − ψ(x) − tψ0(x; d)

1

2t2 . (2)

To the contrast, the function ψ is said to be parabolical second-order directionally differ- entiable at x, if w0 is fixed to be w in (2). Generally speaking, the concept of parabolical

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second-order directional differentiability in the Hadamard sense is stronger than that of parabolical second-order directional differentiability. However, when ψ is locally Lipschitz at x, then these two concepts coincide. It is known that if ψ is parabolical second-order directional differentiability in the Hadamard sense at x along d, w, then

ψ



x + td + 1

2t2w + o(t2)



= ψ(x) + tψ0(x; d) + 1

2t2ψ00(x; d, w) + o(t2). (3) At the first glance on (3), the concept of parabolical second-order directional differentia- bility in the Hadamard sense is likely to say that ψ has a second-order Taylor expansion along some directions. In fact, for the expression (3), the main difference lies on the appearance of w. Why do we need such expansion (3)? We say a few words about it.

For standard nonlinear programming, corresponding to the nonnegative orthant, a poly- hedral is targeted. Hence, considering the way x + td, a radial line, is enough. However, for optimization problems involved the circular cones, second-order cones or semidefinite matrices cones, they are all non-polyhedral cones. Thus, we need to describe the curves thereon. To this end, the curved approach x + td +12t2w is needed, which, to some extent, reflects the non-polyhedral properties of non-polyhedral cones. This point can be seen in Section 3, where the parabolic second-order directional derivative is used to study the second-order tangent sets of circular cones. The exact expression of second-order tangent set is important for describing the second-order necessary and sufficient conditions for conic programming, since its support function is appeared in the second-order necessary and sufficient conditions for conic programming; see [16] for more information.

### 3 Second-Order Directional Derivative

For subsequent analysis, we will frequently use the second-order derivative of ¯x := kxkx at x 6= 0. To this end, we present the second-order derivative of ¯x in below theorem. For convenience of notation, we also denote Φ(x) := ¯x for x 6= 0, which does not cause any confusion from the context.

Theorem 3.1. Let a function Φ : Rn → Rn be given as Φ(x) := kxkx for x 6= 0. Then, the function Φ is second-order continuous differentiable at x 6= 0 with

J Φ(x) = I − ¯x¯xT kxk and

J2Φ(x)(w, w) = −2 ¯xTw kxk2



w + wT  3¯x¯xT − I kxk3



wx, ∀w ∈ Rn.

Proof. It is clear that Φ is second-order continuous differentiable because of x 6= 0. The Jacobian of Φ at x 6= 0 is obtained from direct calculation. To obtain the second-order

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derivative, for any given a ∈ Rn, we define ψ : Rn→ R as ψ(x) := Φ(x)Ta = xTa

kxk.

We also denote h(x) := aTx and g(x) := 1/kxk so that ψ(x) = h(x)g(x). Since x 6= 0, it is clear that g and h are twice continuously differentiable at x with J h(x) = a, J2h(x) = O, and

J g(x) = − x¯

kxk2, J2g(x) = −(I − ¯x¯xT) − 2¯x¯xT

kxk3 = 3¯x¯xT − I kxk3 . Hence, from the chain rule, we have J ψ(x) = g(x)J h(x) + h(x)J g(x) and

J2ψ(x) = J g(x)TJ h(x) + h(x)J2g(x) + g(x)J2h(x) + J h(x)TJ g(x), which implies

J2ψ(x)(w, w) = 2J g(x)(w)J h(x)(w) + h(x)J2g(x)(w, w) + g(x)J2h(x)(w, w)

= 2J g(x)(w)J h(x)(w) + h(x)J2g(x)(w, w)

= aT



−2x¯Tw

kxk2w + wT  3¯x¯xT − I kxk3

 wx



. (4)

On the other hand, we see that J2ψ(x)(w, w) = aTJ2Φ(x)(w, w). Since a ∈ Rn is arbitrary, this together with (4) yields

J2Φ(x)(w, w) = −2x¯Tw

kxk2w + wT  3¯x¯xT − I kxk3

 wx, which is the desired result. 2

Next, we characterize the parabolic second-order directional derivative of the spectral values λi(x) for i = 1, 2.

Theorem 3.2. Let x ∈ Rn with spectral decomposition x = λ1(x)u(1)x + λ2(x)u(2)x given as in (1). Then, the parabolic second-order directional differentiability in the Hadamard sense of λi(x) for i = 1, 2 reduces to the parabolic second-order directional differentiability.

Moreover, given d, w ∈ Rn, we have

λ001(x; d, w) =





w1−

¯

xT2w2+kd2k2kx−(¯xT2d2)2

2k



cot θ, if x2 6= 0,

w1− ¯dT2w2cot θ, if x2 = 0, d2 6= 0, w1− kw2k cot θ, if x2 = 0, d2 = 0, and

λ002(x; d, w) =





w1+

¯

xT2w2+kd2k2kx−(¯xT2d2)2

2k



tan θ, if x2 6= 0,

w1+ ¯dT2w2tan θ, if x2 = 0, d2 6= 0, w1+ kw2k tan θ, if x2 = 0, d2 = 0.

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Proof. Note that λi(x) for i = 1, 2 is Lipschitz continuous [12], hence the parabolic second-order directional differentiability in the Hadamard sense of λi(x) for i = 1, 2 reduces to the parabolic second-order directional differentiability.

To compute the parabolic second-order directional derivative, we consider the follow- ing three cases.

(i) If x2 6= 0, then x + td + 12t2w = (x1 + td1 + 12t2w1, x2 + td2 + 12t2w2). Note that λ01(x; d) = d1− ¯xT2d2cot θ and

kx2+ td2+ 1

2t2w2k = kx2k + t¯xT2d2+ 1 2t2



¯

xT2w2+ kd2k2− (¯xT2d2)2 kx2k



+ o(t2).

Thus, we obtain

λ1(x + td + 12t2w) − λ1(x) − tλ01(x; d)

1

2t2 → w1



¯

xT2w2+ kd2k2− (¯xT2d2)2 kx2k

 cot θ.

(ii) If x2 = 0 and d2 6= 0, then x + td + 12t2w = (x1 + td1 + 12t2w1, td2 + 12t2w2) and λ01(x; d) = d1− kd2k cot θ. Hence

λ1(x + td +12t2w) − λ1(x) − tλ01(x; d)

1

2t2 → w1− ¯dT2w2cot θ.

(iii) If x2 = 0 and d2 = 0, then x + td + 12t2w = (x1 + td1 + 12t2w1,12t2w2). Thus, λ01(x; d) = d1 and

λ1(x + td + 12t2w) − λ1(x) − tλ01(x; d)

1

2t2 → w1− kw2k cot θ.

From all the above, the formula of λ001(x; d, w) is proved. Similar arguments can be applied to obtain the formula of λ002(x; d, w). 2

The relationship of parabolic second-order directional differentiability in the Hadamard sense between fLθ and f is given below.

Theorem 3.3. Suppose that f : R → R. Then, fLθ is parabolic second-order direction- ally differentiable at x in the Hadamard sense if and only if f is parabolic second-order directionally differentiable at λi(x) in the Hadamard sense for i = 1, 2. Moreover, (a) if x2 = 0 and d2 = 0, then

(fLθ)00(x; d, w) = f00(x1; d1, w1− kw2k cot θ) u(1)w + f00(x1; d1, w1+ kw2k tan θ) u(2)w ;

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(b) if x2 = 0 and d2 6= 0, then

(fLθ)00(x; d, w) = f00 x1; d1− kd2k cot θ, w1− ¯dT2w2cot θ u(1)d +f00 x1; d1+ kd2k tan θ, w1+ ¯dT2w2tan θ u(2)d

+ 1

tan θ + cot θ



f0(x1; d1+ kd2k tan θ) − f0(x1; d1− kd2k cot θ)



J eΦ(d)w;

(c) if x2 6= 0, then

(fLθ)00(x; d, w) = f00 x1− kx2k cot θ; d1− ¯xT2d2cot θ, w1− ¯xT2w2 + dT2J Φ(x2)d2 cot θ u(1)x +f00 x1+ kx2k tan θ; d1+ ¯xT2d2tan θ, w1+ ¯xT2w2+ dT2J Φ(x2)d2 tan θ u(2)x

+ 2

cot θ + tan θΓ1J eΦ(x)d + 1

cot θ + tan θΓ2



J eΦ(x)w + J2Φ(x)(d, d)e

 ,

where

Γ1 := f0(x1+ kx2k tan θ; d1+ ¯xT2d2tan θ) − f0(x1− kx2k cot θ; d1− ¯xT2d2cot θ) Γ2 := f (x1+ kx2k tan θ) − f (x1− kx2k cot θ)

and eΦ(x) := (1, Φ(x2))T for all x ∈ Rn with x2 6= 0.

Proof. “⇐” Suppose that f is parabolic second-order directionally differentiable at λi(x) for i = 1, 2 in the Hadamard sense. Given d, w ∈ Rn and w0 → w, we consider the following four cases. First we denote z := x + td + 12t2w0.

Case 1: For x2 = 0, d2 = 0, and w2 = 0, we have fLθ(x) = f (x1), 0 = f (x1)u(1)z + f (x1)u(2)z and

(fLθ)0(x; d) = (f0(x1; d1), 0) = f0(x1; d1)u(1)z + f0(x1; d1)u(2)z .

Note that u(i)z → u(i)ξ as i = 1, 2 for some ξ ∈ {(1, w) : kwk = 1}. Thus, we conclude that fLθ(x + td +12t2w0) − fLθ(x) − t(fLθ)0(x; d)

1

2t2 → f00(x1; d1, w1)u(1)ξ + f00(x1; d1, w1)u(2)ξ

= (f00(x1; d1, w1), 0).

Case 2: For x2 = 0, d2 = 0, and w2 6= 0, since f is parabolic second-order directionally differentiable, we have

f (λ1(z)) − f (x1) − tf0(x1; d1)

1

2t2 → f00(x1; d1, w1− kw2k cot θ) and f (λ2(z)) − f (x1) − tf0(x1; d1)

1

2t2 → f00(x1; d1, w1+ kw2k tan θ).

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Note that u(i)z → u(i)w for i = 1, 2. Therefore, we also conclude that fLθ(x + td +12t2w0) − fLθ(x) − t(fLθ)0(x; d)

1 2t2

→ f00(x1; d1, w1− kw2k cot θ)u(1)w + f00(x1; d1, w1+ kw2k tan θ)u(2)w . In summary, from Cases 1 and 2, we see that under x2 = 0 and d2 = 0

(fLθ)00(x; d, w) = f00(x1; d1, w1− kw2k cot θ)u(1)w + f00(x1; d1, w1+ kw2k tan θ)u(2)w . Case 3: For x2 = 0, d2 6= 0, we have

(fLθ)0(x; d) = f0(x1; d1− kd2k cot θ)u(1)d + f0(x1; d1+ kd2k tan θ)u(2)d . Note that

f (x1+ td1+ 1

2t2w10 − tkd2+1

2tw02k cot θ) (5)

= f



x1+ td1 +1

2t2w01− th

kd2k cot θ +1

2t ¯dT2w02cot θ + o(t)i

= f



x1+ td1 +1

2t2w1− th

kd2k cot θ +1

2t ¯dT2w2cot θi

+ o(t2)



= f (x1) + tf0(x1; d1 − kd2k cot θ) + 1

2t2f00(x1; d1 − kd2k cot θ, w1− ¯dT2w2cot θ) + o(t2), where we use the facts that w0 → w and f is parabolic second-order directionally differ- entiable at λ1(x) in the Hadamard sense. Similarly, we obtain

f (x1+ td1+ 1

2t2w10 + tkd2+ 1

2tw20k tan θ) (6)

= f (x1) + tf0(x1; d1 + kd2k tan θ) + 1

2t2f00(x1; d1+ kd2k tan θ, w1 + ¯dT2w2tan θ) + o(t2).

Thus, the first component of f(x+td+

1

2t2w0)−f(x)−t(f)0(x;d)

1

2t2 converges to

1

1 + cot2θf00(x1; d1− kd2k cot θ, w1− ¯dT2w2cot θ) + 1

1 + tan2θf00(x1; d1+ kd2k tan θ, w1 + ¯dT2w2tan θ).

In addition, according to Theorem 3.1, we know d2+12tw02

kd2+12tw02k = Φ(d2+1 2tw02)

= Φ(d2) + 1

2tJ Φ(d2)w20 + 1

8t2J2Φ(d2)(w20, w02) + o(t2)

= Φ(d2) + 1

2tJ Φ(d2)w20 + 1

8t2J2Φ(d2)(w2, w2) + o(t2). (7)

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Hence, it follows from (5), (6), and (7) that

−f (λ1(z))Φ(d2 +1

2tw20) + f (x1)Φ(d2) + tf0(x1; d1 − kd2k cot θ)Φ(d2)

= −1

2tf (x1)J Φ(d2)w20 − 1 2t2



f00(x1; d1− kd2k cot θ, w1− ¯dT2w2cot θ)Φ(d2) +f0(x1; d1− kd2k cot θ)J Φ(d2)w02+1

4f (x1)J2Φ(d2)(w2, w2)



+ o(t2) and

f (λ2(z))Φ(d2+1

2tw02) − f (x1)Φ(d2) − tf0(x1; d1+ kd2k tan θ)Φ(d2)

= 1

2tf (x1)J Φ(d2)w02+1 2t2



f00(x1; d1+ kd2k tan θ, w1+ ¯dT2w2tan θ)Φ(d2) +f0(x1; d1 + kd2k tan θ)J Φ(d2)w02+1

4f (x1)J2Φ(d2)(w2, w2)



+ o(t2).

Thus, the second component of f(x+td+

1

2t2w0)−f(x)−t(f)0(x;d)

1

2t2 converges to

1 tan θ + ctanθ



κ1J Φ(d2)w2+ κ2Φ(d2)

 , where

κ1 := f0(x1; d1+ kd2k tan θ) − f0(x1; d1− kd2k cot θ)

κ2 := f00(x1; d1+ kd2k tan θ, w1 + ¯dT2w2tan θ) − f00(x1; d1− kd2k cot θ, w1− ¯dT2w2cot θ).

To sum up, we can conclude that (fLθ)00(x; d, w)

= f00(x1; d1− kd2k, w1− ¯dT2w2cot θ)u(1)d + f00(x1; d1+ kd2k tan θ, w1+ ¯dT2w2tan θ)u(2)d

+ 1

tan θ + cot θ



f0(x1; d1+ kd2k tan θ) − f0(x1; d1− kd2k cot θ)



J eΦ(d)w.

Case 4: For x2 6= 0, under this case, we know

(fLθ)0(x; d) = f0 λ1(x); d1− ¯xT2d2cot θ u(1)x + f0 λ2(x); d1+ ¯xT2d2tan θ u(2)x +f (λ2(x)) − f (λ1(x))

λ2(x) − λ1(x)

0 0

0 I − ¯x2T2

 d.

Note that

kx2+ td2+ 1

2t2w20k = kx2k + t¯xT2d2+ 1 2t2h

¯

xT2w20 + dT2J Φ(x2)d2i

+ o(t2)

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= kx2k + t¯xT2d2+ 1 2t2h

¯

xT2w2 + dT2J Φ(x2)d2i

+ o(t2).

Since f is parabolic second-order directionally differentiable at λ1(x) in the Hadamard sense, we have

f (x1+ td1+ 1

2t2w10 − kx2+ td2+ 1

2t2w20k cot θ)

= f (x1− kx2k cot θ) + tf0(x1− kx2k cot θ; d1 − ¯xT2d2cot θ) +1

2t2f00

x1− kx2k cot θ; d1− ¯xT2d2cot θ, w1−h

¯

xT2w2+ dT2J Φ(x2)d2i cot θ

+ o(t2).

Besides, we know that Φ(x2 + td2+1

2t2w02) = Φ(x2) + tJ Φ(x2)d2+1 2t2



J Φ(x2)w02+ J2Φ(x2)(d2, d2)



+ o(t2)

= Φ(x2) + tJ Φ(x2)d2+1 2t2



J Φ(x2)w2+ J2Φ(x2)(d2, d2)



+ o(t2).

Thus, the first component of f(x+td+

1

2t2w0)−f(x)−t(f)0(x;d)

1

2t2 converges to

1

1 + cot2θf00

x1− kx2k cot θ; d1− ¯xT2d2cot θ, w1 −h

¯

xT2w2+ dT2J Φ(x2)d2i cot θ

+ 1

1 + tan2θf00

x1+ kx2k tan θ; d1 + ¯xT2d2tan θ, w1+h

¯

xT2w2+ dT2J Φ(x2)d2i

tan θ .

Moreover, the second component of f(x+td+

1

2t2w0)−f(x)−t(f)0(x;d)

1

2t2 converges to

− cot θ 1 + cot2θ



f (x1− kx2k cot θ)J Φ(x2)w2+ J2Φ(x2)(d2, d2) + 2f0(x1− kx2k cot θ; d1 − ¯xT2d2cot θ)J Φ(x2)d2

+ f00 x1− kx2k cot θ; d1− ¯xT2d2cot θ, w1− ¯xT2w2 + dT2J Φ(x2)d2 cot θΦ(x2)



+ tan θ

1 + tan2θ



f (x1+ kx2k tan θ)J Φ(x2)w2+ J2Φ(x2)(d2, d2) + 2f0(x1+ kx2k tan θ; d1+ ¯xT2d2tan θ)J Φ(x2)d2

+ f00(x1 + kx2k tan θ; d1+ ¯xT2d2tan θ, w1 + ¯xT2w2+ dT2J Φ(x2)d2 tan θ)Φ(x2)

 . To sum up, we can conclude that

(fLθ)00(x; d, w)

= f00



x1− kx2k cot θ; d1− ¯xT2d2cot θ, w1



¯

xT2w2+ dT2J Φ(x2)d2

 cot θ

 u1x

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+f00



x1+ kx2k tan θ; d1+ ¯xT2d2tan θ, w1+



¯

xT2w2+ dT2J Φ(x2)d2

 tan θ

 u2x

+ 2

cot θ + tan θΓ1J eΦ(x)d + 1

cot θ + tan θΓ2



J eΦ(x)w + J2Φ(x)(d, d)e

 ,

where we use the facts that J eΦ(x)w = (0, J Φ(x2)w2) and J2Φ(x)(d, d) = (0, Je 2Φ(x2)(d2, d2)).

“⇒” Suppose that fLθ is parabolic second-order directionally differentiable at x in the Hadamard sense. Given ˜d, ˜w ∈ R and ˜w0 → ˜w. To proceed, we also discuss the following two cases.

Case 1: For x2 = 0, let d = ˜de, w0 = ˜w0e, and w = ˜we. Denote z := x + td +12t2w0. Then f (x1+ t ˜d +12t20) − f (x1) − tf0(x1, ˜d)

1

2t2 = fLθ(z) − fLθ(x) − t(fLθ)0(x; d)

1

2t2 , e

. Thus, we obtain f00(x1; ˜d, ˜w) = h(fLθ)00(x; d, w), ei.

Case 2: For x2 6= 0, let d = ˜du(1)x , w0 = ˜w0u(1)x , and w = ˜wu(1)x . Then, we have x + td + 1

2t2w0 = (λ1(x) + t ˜d + 1

2t20)u(1)x + λ2(x)u(2)x with t > 0 satisfying t ˜d + 12t20 < λ2(x) − λ1(x). This implies

fLθ



x + td +1 2t2w0



= f (λ1(x) + t ˜d +1

2t20)u(1)x + f (λ2(x))u(2)x and (fLθ)0(x; d) = f01(x); ˜d)u(1)x . Thus,

f (λ1(x) + t ˜d +12t20) − f (λ1(x)) − tf01(x); ˜d)

1 2t2

= (1 + cot2θ) fLθ(x + td + 12t2w0) − fLθ(x) − t(fLθ)0(x; d)

1

2t2 , u1x

, which says

f001(x); ˜d, ˜w) = (1 + cot2θ)D

(fLθ)00(x; d, w), u(1)x E .

The similar arguments can be used for f at λ2(x). From all the above, the proof is complete. 2

### 4 Second-order Tangent Sets

In this section, we turn our attention to f being the special function f (t) = max{t, 0}. In this case, the corresponding fLθ is just the projection operator associated with circular cone. For x ∈ Lθ, from [16], we know the tangent cone is given by

TLθ(x) := {d : dist(x + td, Lθ) = o(t), t ≥ 0}

(12)

= {d : ΠLθ(x + td) − (x + td) = o(t), t ≥ 0}

= {d : Π0Lθ(x; d) = d}, (8)

which, together with the formula of Π0L

θ, yields

TLθ(x) =

Rn, if x ∈ intLθ,

Lθ, if x = 0,

{d : dT2x2− d1x1tan2θ ≤ 0}, if x ∈ bdLθ/{0}.

Definition 4.1. [16, Definition 3.28] The set limits TSi,2(x, d) :=



w ∈ Rn : dist



x + td +1 2t2w, S



= o(t2), t ≥ 0



and

TS2(x, d) :=



w ∈ Rn : ∃ tn↓ 0 such that dist



x + tnd +1 2t2nw, S



= o(t2n)



are called the inner and outer second-order tangent sets, respectively, to the set S at x in the direction d.

In [13], we have shown that the circular cone is second-order regular, which means TLi,2

θ(x; d) is equal to TL2

θ(x; d) for all d ∈ TLθ(x). Since the inner and outer second-order tangent sets are equal, we simply say that TL2

θ(x; d) is the second-order tangent set.

Next, we provide two different approaches to establish the exact formula of second-order tangent set of circular cone. One is following from the parabolic second-order directional derivative of the spectral value λ1(x), and the other is using the parabolic second-order directional derivative of projection operator ΠLθ.

Theorem 4.1. Given x ∈ Lθ and d ∈ TLθ(x), then

TL2θ(x, d) =

Rn, if d ∈ intTLθ(x), TLθ(d), if x = 0,

{w : wT2x2cot θ − w1x1tan θ ≤ d21tan θ − kd2k2cot θ}, otherwise.

Proof. First, we note that Lθ = {x : −λ1(x) ≤ 0}. With this, we have w ∈ TL2

θ(x; d) ⇐⇒ −λ1(x + td + 1

2t2w + o(t2)) ≤ 0

⇐⇒ −λ1(x) − tλ01(x; d) −1

2t2λ001(x; d, w) + o(t2) ≤ 0. (9) The case of x ∈ intLθ (corresponding to −λ1(x) < 0) or x ∈ bdLθ and d ∈ intTLθ(x) (corresponding to λ1(x) = 0 and −λ01(x; d) < 0) ensures that (9) holds for all w ∈ Rn.

(13)

For the case x = 0 and d = 0, it follows from Theorem 3.2 and (9) that w ∈ TL2

θ(x; d) =⇒ −w1+ kw2k cot θ ≤ 0 ⇐⇒ w ∈ Lθ.

Conversely, if w ∈ Lθ, then dist(12t2w, Lθ) = 0 due to Lθ is a cone, which implies w ∈ TL2

θ(x; d). Hence, TL2

θ(x; d) = TLθ(x).

For the case x = 0 and d ∈ bdTLθ(x)\{0} = bdLθ\{0}, it follows from Theorem 3.2 and (9) that

w ∈ TL2θ(x; d) =⇒ −w1d1tan2θ + dT2w2 ≤ 0 ⇐⇒ w ∈ TLθ(d).

Conversely, if w ∈ TLθ(d), then dist(d + tw, Lθ) = o(t) and hence dist(d + 12tw, Lθ) = o(12t) = o(t). Thus, we obtain dist(x + td +12t2w, Lθ) = dist(td +12t2w, Lθ) = o(t2), which means w ∈ TL2

θ(x; d).

The case remained is x ∈ bdLθ/{0} and d ∈ bdTLθ(x), i.e., x1 = kx2k cot θ and dT2x2 = d1x1tan2θ. Since x2 6= 0, −λ1 is second-order differentiable at x. Hence it follows from Theorem 3.2 that

TL2

θ(x; d) = {w : −λ001(x; d, w) ≤ 0} =w : −x1w1tan θ + xT2w2cot θ + kd2k2cot θ − d21tan θ ≤ 0 , where the last step is due to ¯xT2d2 = d1tan θ. 2

As below, we provide the second approach to establish the formula of second-order tangent set by using the parabolic second-order directional derivative of projection oper- ator associated with circular cone. To this end, we need a technical lemma.

Lemma 4.1. For x ∈ Lθ and d ∈ TLθ(x), we have

TL2θ(x, d) = {w : Π00Lθ(x; d, w) = w}.

Proof. The desired result follows from TL2

θ(x, d) = {w : dist(x + td + 1

2t2w, Lθ) = o(t2), t ≥ 0}

= {w : ΠLθ(x + td +1

2t2w) − (x + td + 1

2t2w) = o(t2), t ≥ 0}

= {w : ΠLθ(x + td +1

2t2w) − ΠLθ(x) − tΠ0L

θ(x; d) −1

2t2w = o(t2), t ≥ 0}

= {w : Π00Lθ(x; d, w) = w},

where the third step uses the fact that d = Π0L

θ(x; d) since d ∈ TLθ(x) by (8). 2 Recall first from [15] that ΠLθ, the projection operator, is the vector-valued function corresponding to f (t) = max{t, 0}. To present the second approach, we will also use the parabolic second-order directional derivative of the f (t) = max{t, 0}, which can be found in [10]. Now the second approach to prove Theorem 4.1 is given below.

(14)

Proof. Notice first that as x1 > kx2k cot θ or x1 = kx2k cot θ 6= 0 and d1 ≥ ¯xT2d2cot θ, then

2

tan θ + cot θΓ1J eΦ(x)d + 1

tan θ + cot θΓ2 J eΦ(x)w + J2Φ(x)(d, d)e 

=



0, w2



¯

xT2w2− (¯xT2d2)2

kx2k +kd2k2 kx2k

 x2 kx2k

T

. (10)

As x1 ≥ 0 and d1 ≥ kd2k cot θ, we know that 1

tan θ + cot θ



f0(x1; d1+kd2k tan θ)−f0(x1; d1−kd2k cot θ)



J eΦ(d)w =



0, w2− ¯dT2w22

T

. (11) We point it out that, in the above formulas (10) and (11), we have applied the parabolic second-order directional derivative of the max-type function f (t) = max{t, 0}. To pro- ceed, we discuss the following three cases.

Case 1: For d ∈ intTLθ(x), we keep going to discuss three subcases.

Subcase (1): x = 0. Under this subcase, we see d ∈ intLθ, i.e., d1 > kd2k cot θ. If d2 = 0, then d1 > 0 which yields

f00(x1; d1, w1− kw2k cot θ)u1w+ f00(x1; d1, w1+ kw2k tan θ)u2w = w, ∀w ∈ Rn. If d2 6= 0, it then follows from (11) that

(fLθ)00(x; d, w) = (w1− ¯dT2w2cot θ)u(1)d + (w1+ ¯dT2w2tan θ)u(2)d +



0, w2 − ¯dT2w22

T

= w.

Subcase (2): x ∈ intLθ. Under this subcase, it is clear that TLθ(x) = Rn. If x2 = 0, it follows from Theorem 3.3 that (fLθ)00(x; d, w) = w whenever d2 = 0 or d2 6= 0 due to x1 > 0 in this case. If x2 6= 0, from (10), we know that

(fLθ)00(x; d, w) =

w1



¯

xT2w2+ dT2J Φ(x2)d2



x2

kx2k

+

0 w2



¯

xT2w2+ dT2J Φ(x2)d2



x2

kx2k

= w.

Subcase (3): x ∈ bdLθ/{0}. Then d ∈ intTLθ(x) means dT2x2 < d1x1tan2θ = d1kx2k tan θ, i.e., ¯xT2d2cot θ < d1. Thus, (fLθ)00(x; d, w) = w for all w ∈ Rn by the similar argument as above.

In summary, we have TL2

θ(x, d) = Rn in this case.

Case 2: For x = 0, since d ∈ TLθ(x) = Lθ, we see that d1 ≥ kd2k cot θ. It only remains to show the case of d1 = kd2k cot θ. If d2 = 0, then d1 = 0, and hence

(fLθ)00(x; d, w) = f00(x1; d1, w1− kw2k cot θ)u1w+ f00(x1; d1, w1+ kw2k tan θ)u2w

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