to appear in Journal of Optimization Theory and Applications, 2016

### Parabolic second-order directional differentiability in the Hadamard sense of the vector-valued functions associated with

### 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 Tang^{2}

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.

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

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

L_{θ} := {x = (x_{1}, x_{2})^{T} ∈ R × R^{n−1} : cos θkxk ≤ x_{1}},

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 = (x_{1}, x_{2}) ∈ R × R^{n−1}, one has

x = λ_{1}(x)u^{(1)}_{x} + λ_{2}(x)u^{(2)}_{x} , (1)
where

λ_{1}(x) := x_{1}− kx_{2}k cot θ, λ_{2}(x) := x_{1}+ kx_{2}k tan θ
and

u^{(1)}_{x} := 1
1 + cot^{2}θ

1 0

0 cot θ · I

1

−¯x_{2}

, u^{(2)}_{x} := 1
1 + tan^{2}θ

1 0

0 tan θ · I

1

¯
x_{2}

with ¯x_{2} := x_{2}/kx_{2}k if x_{2} 6= 0, and ¯x_{2} being any vector w ∈ R^{n−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 f^{L}^{θ} : R^{n}→ R^{n} is defined as

f^{L}^{θ}(x) := f (λ_{1}(x)) u^{(1)}_{x} + f (λ_{2}(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:

ψ^{00}(x; d, w) := lim

t↓0 w0→w

ψ(x + td +^{1}_{2}t^{2}w^{0}) − ψ(x) − tψ^{0}(x; d)

1

2t^{2} . (2)

To the contrast, the function ψ is said to be parabolical second-order directionally differ-
entiable at x, if w^{0} is fixed to be w in (2). Generally speaking, the concept of parabolical

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

2t^{2}w + o(t^{2})

= ψ(x) + tψ^{0}(x; d) + 1

2t^{2}ψ^{00}(x; d, w) + o(t^{2}). (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 +^{1}_{2}t^{2}w 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 := _{kxk}^{x} 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 : R^{n} → R^{n} be given as Φ(x) := _{kxk}^{x} for x 6= 0. Then,
the function Φ is second-order continuous differentiable at x 6= 0 with

J Φ(x) = I − ¯x¯x^{T}
kxk
and

J^{2}Φ(x)(w, w) = −2 ¯x^{T}w
kxk^{2}

w + w^{T} 3¯x¯x^{T} − I
kxk^{3}

wx, ∀w ∈ R^{n}.

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

derivative, for any given a ∈ R^{n}, we define ψ : R^{n}→ R as
ψ(x) := Φ(x)^{T}a = x^{T}a

kxk.

We also denote h(x) := a^{T}x 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, J^{2}h(x) = O,
and

J g(x) = − x¯

kxk^{2}, J^{2}g(x) = −(I − ¯x¯x^{T}) − 2¯x¯x^{T}

kxk^{3} = 3¯x¯x^{T} − I
kxk^{3} .
Hence, from the chain rule, we have J ψ(x) = g(x)J h(x) + h(x)J g(x) and

J^{2}ψ(x) = J g(x)^{T}J h(x) + h(x)J^{2}g(x) + g(x)J^{2}h(x) + J h(x)^{T}J g(x),
which implies

J^{2}ψ(x)(w, w) = 2J g(x)(w)J h(x)(w) + h(x)J^{2}g(x)(w, w) + g(x)J^{2}h(x)(w, w)

= 2J g(x)(w)J h(x)(w) + h(x)J^{2}g(x)(w, w)

= a^{T}

−2x¯^{T}w

kxk^{2}w + w^{T} 3¯x¯x^{T} − I
kxk^{3}

wx

. (4)

On the other hand, we see that J^{2}ψ(x)(w, w) = a^{T}J^{2}Φ(x)(w, w). Since a ∈ R^{n} is
arbitrary, this together with (4) yields

J^{2}Φ(x)(w, w) = −2x¯^{T}w

kxk^{2}w + w^{T} 3¯x¯x^{T} − I
kxk^{3}

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 ∈ R^{n} with spectral decomposition x = λ_{1}(x)u^{(1)}x + λ_{2}(x)u^{(2)}x given
as in (1). Then, the parabolic second-order directional differentiability in the Hadamard
sense of λ_{i}(x) for i = 1, 2 reduces to the parabolic second-order directional differentiability.

Moreover, given d, w ∈ R^{n}, we have

λ^{00}_{1}(x; d, w) =

w_{1}−

¯

x^{T}_{2}w_{2}+^{kd}^{2}^{k}^{2}_{kx}^{−(¯}^{x}^{T}^{2}^{d}^{2}^{)}^{2}

2k

cot θ, if x_{2} 6= 0,

w_{1}− ¯d^{T}_{2}w_{2}cot θ, if x_{2} = 0, d_{2} 6= 0,
w_{1}− kw_{2}k cot θ, if x_{2} = 0, d_{2} = 0,
and

λ^{00}_{2}(x; d, w) =

w_{1}+

¯

x^{T}_{2}w_{2}+^{kd}^{2}^{k}^{2}_{kx}^{−(¯}^{x}^{T}^{2}^{d}^{2}^{)}^{2}

2k

tan θ, if x_{2} 6= 0,

w_{1}+ ¯d^{T}_{2}w_{2}tan θ, if x_{2} = 0, d_{2} 6= 0,
w1+ kw2k tan θ, if x2 = 0, d2 = 0.

Proof. Note that λ_{i}(x) for i = 1, 2 is Lipschitz continuous [12], hence the parabolic
second-order directional differentiability in the Hadamard sense of λ_{i}(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 x_{2} 6= 0, then x + td + ^{1}_{2}t^{2}w = (x_{1} + td_{1} + ^{1}_{2}t^{2}w_{1}, x_{2} + td_{2} + ^{1}_{2}t^{2}w_{2}). Note that
λ^{0}_{1}(x; d) = d_{1}− ¯x^{T}_{2}d_{2}cot θ and

kx_{2}+ td_{2}+ 1

2t^{2}w_{2}k = kx_{2}k + t¯x^{T}_{2}d_{2}+ 1
2t^{2}

¯

x^{T}_{2}w_{2}+ kd_{2}k^{2}− (¯x^{T}_{2}d_{2})^{2}
kx_{2}k

+ o(t^{2}).

Thus, we obtain

λ_{1}(x + td + ^{1}_{2}t^{2}w) − λ_{1}(x) − tλ^{0}_{1}(x; d)

1

2t^{2} → w_{1}−

¯

x^{T}_{2}w_{2}+ kd_{2}k^{2}− (¯x^{T}_{2}d_{2})^{2}
kx_{2}k

cot θ.

(ii) If x_{2} = 0 and d_{2} 6= 0, then x + td + ^{1}_{2}t^{2}w = (x_{1} + td_{1} + ^{1}_{2}t^{2}w_{1}, td_{2} + ^{1}_{2}t^{2}w_{2}) and
λ^{0}_{1}(x; d) = d_{1}− kd_{2}k cot θ. Hence

λ1(x + td +^{1}_{2}t^{2}w) − λ1(x) − tλ^{0}_{1}(x; d)

1

2t^{2} → w_{1}− ¯d^{T}_{2}w_{2}cot θ.

(iii) If x2 = 0 and d2 = 0, then x + td + ^{1}_{2}t^{2}w = (x1 + td1 + ^{1}_{2}t^{2}w1,^{1}_{2}t^{2}w2). Thus,
λ^{0}_{1}(x; d) = d_{1} and

λ_{1}(x + td + ^{1}_{2}t^{2}w) − λ_{1}(x) − tλ^{0}_{1}(x; d)

1

2t^{2} → w_{1}− kw_{2}k cot θ.

From all the above, the formula of λ^{00}_{1}(x; d, w) is proved. Similar arguments can be applied
to obtain the formula of λ^{00}_{2}(x; d, w). 2

The relationship of parabolic second-order directional differentiability in the Hadamard
sense between f^{L}^{θ} and f is given below.

Theorem 3.3. Suppose that f : R → R. Then, f^{L}^{θ} 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 λ_{i}(x) in the Hadamard sense for i = 1, 2. Moreover,
(a) if x_{2} = 0 and d_{2} = 0, then

(f^{L}^{θ})^{00}(x; d, w) = f^{00}(x_{1}; d_{1}, w_{1}− kw_{2}k cot θ) u^{(1)}_{w} + f^{00}(x_{1}; d_{1}, w_{1}+ kw_{2}k tan θ) u^{(2)}_{w} ;

(b) if x_{2} = 0 and d_{2} 6= 0, then

(f^{L}^{θ})^{00}(x; d, w) = f^{00} x_{1}; d_{1}− kd_{2}k cot θ, w_{1}− ¯d^{T}_{2}w_{2}cot θ u^{(1)}_{d}
+f^{00} x_{1}; d_{1}+ kd_{2}k tan θ, w_{1}+ ¯d^{T}_{2}w_{2}tan θ u^{(2)}_{d}

+ 1

tan θ + cot θ

f^{0}(x1; d1+ kd2k tan θ) − f^{0}(x1; d1− kd2k cot θ)

J eΦ(d)w;

(c) if x_{2} 6= 0, then

(f^{L}^{θ})^{00}(x; d, w) = f^{00} x_{1}− kx_{2}k cot θ; d_{1}− ¯x^{T}_{2}d_{2}cot θ, w_{1}− ¯x^{T}_{2}w_{2} + d^{T}_{2}J Φ(x_{2})d_{2} cot θ u^{(1)}_{x}
+f^{00} x_{1}+ kx_{2}k tan θ; d_{1}+ ¯x^{T}_{2}d_{2}tan θ, w_{1}+ ¯x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2} tan θ u^{(2)}_{x}

+ 2

cot θ + tan θΓ_{1}J eΦ(x)d + 1

cot θ + tan θΓ_{2}

J eΦ(x)w + J^{2}Φ(x)(d, d)e

,

where

Γ_{1} := f^{0}(x_{1}+ kx_{2}k tan θ; d_{1}+ ¯x^{T}_{2}d_{2}tan θ) − f^{0}(x_{1}− kx_{2}k cot θ; d_{1}− ¯x^{T}_{2}d_{2}cot θ)
Γ2 := f (x1+ kx2k tan θ) − f (x1− kx2k cot θ)

and eΦ(x) := (1, Φ(x_{2}))^{T} for all x ∈ R^{n} with x_{2} 6= 0.

Proof. “⇐” Suppose that f is parabolic second-order directionally differentiable at λ_{i}(x)
for i = 1, 2 in the Hadamard sense. Given d, w ∈ R^{n} and w^{0} → w, we consider the
following four cases. First we denote z := x + td + ^{1}_{2}t^{2}w^{0}.

Case 1: For x_{2} = 0, d_{2} = 0, and w_{2} = 0, we have f^{L}^{θ}(x) = f (x_{1}), 0 = f (x_{1})u^{(1)}z +
f (x_{1})u^{(2)}z and

(f^{L}^{θ})^{0}(x; d) = (f^{0}(x_{1}; d_{1}), 0) = f^{0}(x_{1}; d_{1})u^{(1)}_{z} + f^{0}(x_{1}; d_{1})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
f^{L}^{θ}(x + td +^{1}_{2}t^{2}w^{0}) − f^{L}^{θ}(x) − t(f^{L}^{θ})^{0}(x; d)

1

2t^{2} → f^{00}(x_{1}; d_{1}, w_{1})u^{(1)}_{ξ} + f^{00}(x_{1}; d_{1}, w_{1})u^{(2)}_{ξ}

= (f^{00}(x_{1}; d_{1}, w_{1}), 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) − tf^{0}(x1; d1)

1

2t^{2} → f^{00}(x1; d1, w1− kw2k cot θ)
and f (λ_{2}(z)) − f (x_{1}) − tf^{0}(x_{1}; d_{1})

1

2t^{2} → f^{00}(x_{1}; d_{1}, w_{1}+ kw_{2}k tan θ).

Note that u^{(i)}z → u^{(i)}w for i = 1, 2. Therefore, we also conclude that
f^{L}^{θ}(x + td +^{1}_{2}t^{2}w^{0}) − f^{L}^{θ}(x) − t(f^{L}^{θ})^{0}(x; d)

1
2t^{2}

→ f^{00}(x_{1}; d_{1}, w_{1}− kw_{2}k cot θ)u^{(1)}_{w} + f^{00}(x_{1}; d_{1}, w_{1}+ kw_{2}k tan θ)u^{(2)}_{w} .
In summary, from Cases 1 and 2, we see that under x_{2} = 0 and d_{2} = 0

(f^{L}^{θ})^{00}(x; d, w) = f^{00}(x_{1}; d_{1}, w_{1}− kw_{2}k cot θ)u^{(1)}_{w} + f^{00}(x_{1}; d_{1}, w_{1}+ kw_{2}k tan θ)u^{(2)}_{w} .
Case 3: For x_{2} = 0, d_{2} 6= 0, we have

(f^{L}^{θ})^{0}(x; d) = f^{0}(x_{1}; d_{1}− kd_{2}k cot θ)u^{(1)}_{d} + f^{0}(x_{1}; d_{1}+ kd_{2}k tan θ)u^{(2)}_{d} .
Note that

f (x1+ td1+ 1

2t^{2}w_{1}^{0} − tkd2+1

2tw^{0}_{2}k cot θ) (5)

= f

x_{1}+ td_{1} +1

2t^{2}w^{0}_{1}− th

kd_{2}k cot θ +1

2t ¯d^{T}_{2}w^{0}_{2}cot θ + o(t)i

= f

x_{1}+ td_{1} +1

2t^{2}w_{1}− th

kd_{2}k cot θ +1

2t ¯d^{T}_{2}w_{2}cot θi

+ o(t^{2})

= f (x1) + tf^{0}(x1; d1 − kd2k cot θ) + 1

2t^{2}f^{00}(x1; d1 − kd2k cot θ, w1− ¯d^{T}_{2}w2cot θ) + o(t^{2}),
where we use the facts that w^{0} → w and f is parabolic second-order directionally differ-
entiable at λ_{1}(x) in the Hadamard sense. Similarly, we obtain

f (x_{1}+ td_{1}+ 1

2t^{2}w_{1}^{0} + tkd_{2}+ 1

2tw_{2}^{0}k tan θ) (6)

= f (x_{1}) + tf^{0}(x_{1}; d_{1} + kd_{2}k tan θ) + 1

2t^{2}f^{00}(x_{1}; d_{1}+ kd_{2}k tan θ, w_{1} + ¯d^{T}_{2}w_{2}tan θ) + o(t^{2}).

Thus, the first component of ^{f}^{Lθ}^{(x+td+}

1

2t^{2}w^{0})−f^{Lθ}(x)−t(f^{Lθ})^{0}(x;d)

1

2t^{2} converges to

1

1 + cot^{2}θf^{00}(x1; d1− kd2k cot θ, w1− ¯d^{T}_{2}w2cot θ) + 1

1 + tan^{2}θf^{00}(x1; d1+ kd2k tan θ, w1 + ¯d^{T}_{2}w2tan θ).

In addition, according to Theorem 3.1, we know
d_{2}+^{1}_{2}tw^{0}_{2}

kd_{2}+^{1}_{2}tw^{0}_{2}k = Φ(d2+1
2tw^{0}_{2})

= Φ(d_{2}) + 1

2tJ Φ(d_{2})w_{2}^{0} + 1

8t^{2}J^{2}Φ(d_{2})(w_{2}^{0}, w^{0}_{2}) + o(t^{2})

= Φ(d_{2}) + 1

2tJ Φ(d_{2})w_{2}^{0} + 1

8t^{2}J^{2}Φ(d_{2})(w_{2}, w_{2}) + o(t^{2}). (7)

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

−f (λ_{1}(z))Φ(d_{2} +1

2tw_{2}^{0}) + f (x_{1})Φ(d_{2}) + tf^{0}(x_{1}; d_{1} − kd_{2}k cot θ)Φ(d_{2})

= −1

2tf (x_{1})J Φ(d_{2})w_{2}^{0} − 1
2t^{2}

f^{00}(x_{1}; d_{1}− kd_{2}k cot θ, w_{1}− ¯d^{T}_{2}w_{2}cot θ)Φ(d_{2})
+f^{0}(x_{1}; d_{1}− kd_{2}k cot θ)J Φ(d_{2})w^{0}_{2}+1

4f (x_{1})J^{2}Φ(d_{2})(w_{2}, w_{2})

+ o(t^{2})
and

f (λ_{2}(z))Φ(d_{2}+1

2tw^{0}_{2}) − f (x_{1})Φ(d_{2}) − tf^{0}(x_{1}; d_{1}+ kd_{2}k tan θ)Φ(d_{2})

= 1

2tf (x_{1})J Φ(d_{2})w^{0}_{2}+1
2t^{2}

f^{00}(x_{1}; d_{1}+ kd_{2}k tan θ, w_{1}+ ¯d^{T}_{2}w_{2}tan θ)Φ(d_{2})
+f^{0}(x_{1}; d_{1} + kd_{2}k tan θ)J Φ(d_{2})w^{0}_{2}+1

4f (x_{1})J^{2}Φ(d_{2})(w_{2}, w_{2})

+ o(t^{2}).

Thus, the second component of ^{f}^{Lθ}^{(x+td+}

1

2t^{2}w^{0})−f^{Lθ}(x)−t(f^{Lθ})^{0}(x;d)

1

2t^{2} converges to

1 tan θ + ctanθ

κ_{1}J Φ(d_{2})w_{2}+ κ_{2}Φ(d_{2})

, where

κ_{1} := f^{0}(x_{1}; d_{1}+ kd_{2}k tan θ) − f^{0}(x_{1}; d_{1}− kd_{2}k cot θ)

κ_{2} := f^{00}(x_{1}; d_{1}+ kd_{2}k tan θ, w_{1} + ¯d^{T}_{2}w_{2}tan θ) − f^{00}(x_{1}; d_{1}− kd_{2}k cot θ, w_{1}− ¯d^{T}_{2}w_{2}cot θ).

To sum up, we can conclude that
(f^{L}^{θ})^{00}(x; d, w)

= f^{00}(x_{1}; d_{1}− kd_{2}k, w_{1}− ¯d^{T}_{2}w_{2}cot θ)u^{(1)}_{d} + f^{00}(x_{1}; d_{1}+ kd_{2}k tan θ, w_{1}+ ¯d^{T}_{2}w_{2}tan θ)u^{(2)}_{d}

+ 1

tan θ + cot θ

f^{0}(x_{1}; d_{1}+ kd_{2}k tan θ) − f^{0}(x_{1}; d_{1}− kd_{2}k cot θ)

J eΦ(d)w.

Case 4: For x_{2} 6= 0, under this case, we know

(f^{L}^{θ})^{0}(x; d) = f^{0} λ_{1}(x); d_{1}− ¯x^{T}_{2}d_{2}cot θ u^{(1)}_{x} + f^{0} λ_{2}(x); d_{1}+ ¯x^{T}_{2}d_{2}tan θ u^{(2)}_{x}
+f (λ_{2}(x)) − f (λ_{1}(x))

λ_{2}(x) − λ_{1}(x)

0 0

0 I − ¯x_{2}x¯^{T}_{2}

d.

Note that

kx_{2}+ td_{2}+ 1

2t^{2}w_{2}^{0}k = kx_{2}k + t¯x^{T}_{2}d_{2}+ 1
2t^{2}h

¯

x^{T}_{2}w_{2}^{0} + d^{T}_{2}J Φ(x_{2})d_{2}i

+ o(t^{2})

= kx_{2}k + t¯x^{T}_{2}d_{2}+ 1
2t^{2}h

¯

x^{T}_{2}w_{2} + d^{T}_{2}J Φ(x_{2})d_{2}i

+ o(t^{2}).

Since f is parabolic second-order directionally differentiable at λ_{1}(x) in the Hadamard
sense, we have

f (x_{1}+ td_{1}+ 1

2t^{2}w_{1}^{0} − kx_{2}+ td_{2}+ 1

2t^{2}w_{2}^{0}k cot θ)

= f (x_{1}− kx_{2}k cot θ) + tf^{0}(x_{1}− kx_{2}k cot θ; d_{1} − ¯x^{T}_{2}d_{2}cot θ)
+1

2t^{2}f^{00}

x_{1}− kx_{2}k cot θ; d_{1}− ¯x^{T}_{2}d_{2}cot θ, w_{1}−h

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}i
cot θ

+ o(t^{2}).

Besides, we know that
Φ(x_{2} + td_{2}+1

2t^{2}w^{0}_{2}) = Φ(x_{2}) + tJ Φ(x_{2})d_{2}+1
2t^{2}

J Φ(x_{2})w^{0}_{2}+ J^{2}Φ(x_{2})(d_{2}, d_{2})

+ o(t^{2})

= Φ(x_{2}) + tJ Φ(x_{2})d_{2}+1
2t^{2}

J Φ(x_{2})w_{2}+ J^{2}Φ(x_{2})(d_{2}, d_{2})

+ o(t^{2}).

Thus, the first component of ^{f}^{Lθ}^{(x+td+}

1

2t^{2}w^{0})−f^{Lθ}(x)−t(f^{Lθ})^{0}(x;d)

1

2t^{2} converges to

1

1 + cot^{2}θf^{00}

x_{1}− kx_{2}k cot θ; d_{1}− ¯x^{T}_{2}d_{2}cot θ, w_{1} −h

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}i
cot θ

+ 1

1 + tan^{2}θf^{00}

x_{1}+ kx_{2}k tan θ; d_{1} + ¯x^{T}_{2}d_{2}tan θ, w_{1}+h

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}i

tan θ .

Moreover, the second component of ^{f}^{Lθ}^{(x+td+}

1

2t^{2}w^{0})−f^{Lθ}(x)−t(f^{Lθ})^{0}(x;d)

1

2t^{2} converges to

− cot θ
1 + cot^{2}θ

f (x_{1}− kx_{2}k cot θ)J Φ(x_{2})w_{2}+ J^{2}Φ(x_{2})(d_{2}, d_{2})
+ 2f^{0}(x1− kx2k cot θ; d1 − ¯x^{T}_{2}d2cot θ)J Φ(x2)d2

+ f^{00} x_{1}− kx_{2}k cot θ; d_{1}− ¯x^{T}_{2}d_{2}cot θ, w_{1}− ¯x^{T}_{2}w_{2} + d^{T}_{2}J Φ(x_{2})d_{2} cot θΦ(x_{2})

+ tan θ

1 + tan^{2}θ

f (x_{1}+ kx_{2}k tan θ)J Φ(x_{2})w_{2}+ J^{2}Φ(x_{2})(d_{2}, d_{2})
+ 2f^{0}(x1+ kx2k tan θ; d1+ ¯x^{T}_{2}d2tan θ)J Φ(x2)d2

+ f^{00}(x_{1} + kx_{2}k tan θ; d_{1}+ ¯x^{T}_{2}d_{2}tan θ, w_{1} + ¯x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2} tan θ)Φ(x_{2})

. To sum up, we can conclude that

(f^{L}^{θ})^{00}(x; d, w)

= f^{00}

x_{1}− kx_{2}k cot θ; d_{1}− ¯x^{T}_{2}d_{2}cot θ, w_{1}−

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}

cot θ

u^{1}_{x}

+f^{00}

x_{1}+ kx_{2}k tan θ; d_{1}+ ¯x^{T}_{2}d_{2}tan θ, w_{1}+

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}

tan θ

u^{2}_{x}

+ 2

cot θ + tan θΓ_{1}J eΦ(x)d + 1

cot θ + tan θΓ_{2}

J eΦ(x)w + J^{2}Φ(x)(d, d)e

,

where we use the facts that J eΦ(x)w = (0, J Φ(x_{2})w_{2}) and J^{2}Φ(x)(d, d) = (0, Je ^{2}Φ(x_{2})(d_{2}, d_{2})).

“⇒” Suppose that f^{L}^{θ} is parabolic second-order directionally differentiable at x in the
Hadamard sense. Given ˜d, ˜w ∈ R and ˜w^{0} → ˜w. To proceed, we also discuss the following
two cases.

Case 1: For x_{2} = 0, let d = ˜de, w^{0} = ˜w^{0}e, and w = ˜we. Denote z := x + td +^{1}_{2}t^{2}w^{0}. Then
f (x_{1}+ t ˜d +^{1}_{2}t^{2}w˜^{0}) − f (x_{1}) − tf^{0}(x_{1}, ˜d)

1

2t^{2} = f^{L}^{θ}(z) − f^{L}^{θ}(x) − t(f^{L}^{θ})^{0}(x; d)

1

2t^{2} , e

.
Thus, we obtain f^{00}(x1; ˜d, ˜w) = h(f^{L}^{θ})^{00}(x; d, w), ei.

Case 2: For x_{2} 6= 0, let d = ˜du^{(1)}x , w^{0} = ˜w^{0}u^{(1)}x , and w = ˜wu^{(1)}x . Then, we have
x + td + 1

2t^{2}w^{0} = (λ_{1}(x) + t ˜d + 1

2t^{2}w˜^{0})u^{(1)}_{x} + λ_{2}(x)u^{(2)}_{x}
with t > 0 satisfying t ˜d + ^{1}_{2}t^{2}w˜^{0} < λ_{2}(x) − λ_{1}(x). This implies

f^{L}^{θ}

x + td +1
2t^{2}w^{0}

= f (λ1(x) + t ˜d +1

2t^{2}w˜^{0})u^{(1)}_{x} + f (λ2(x))u^{(2)}_{x}
and (f^{L}^{θ})^{0}(x; d) = f^{0}(λ1(x); ˜d)u^{(1)}x . Thus,

f (λ_{1}(x) + t ˜d +^{1}_{2}t^{2}w˜^{0}) − f (λ_{1}(x)) − tf^{0}(λ_{1}(x); ˜d)

1
2t^{2}

= (1 + cot^{2}θ) f^{L}^{θ}(x + td + ^{1}_{2}t^{2}w^{0}) − f^{L}^{θ}(x) − t(f^{L}^{θ})^{0}(x; d)

1

2t^{2} , u^{1}_{x}

, which says

f^{00}(λ_{1}(x); ˜d, ˜w) = (1 + cot^{2}θ)D

(f^{L}^{θ})^{00}(x; d, w), u^{(1)}_{x} E
.

The similar arguments can be used for f at λ_{2}(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 f^{L}^{θ} is just the projection operator associated with circular
cone. For x ∈ L_{θ}, from [16], we know the tangent cone is given by

T_{L}_{θ}(x) := {d : dist(x + td, L_{θ}) = o(t), t ≥ 0}

= {d : Π_{L}_{θ}(x + td) − (x + td) = o(t), t ≥ 0}

= {d : Π^{0}_{L}_{θ}(x; d) = d}, (8)

which, together with the formula of Π^{0}_{L}

θ, yields

TL_{θ}(x) =

R^{n}, if x ∈ intL_{θ},

L_{θ}, if x = 0,

{d : d^{T}_{2}x2− d1x1tan^{2}θ ≤ 0}, if x ∈ bdLθ/{0}.

Definition 4.1. [16, Definition 3.28] The set limits
T_{S}^{i,2}(x, d) :=

w ∈ R^{n} : dist

x + td +1
2t^{2}w, S

= o(t^{2}), t ≥ 0

and

T_{S}^{2}(x, d) :=

w ∈ R^{n} : ∃ t_{n}↓ 0 such that dist

x + t_{n}d +1
2t^{2}_{n}w, S

= o(t^{2}_{n})

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
T_{L}^{i,2}

θ(x; d) is equal to T_{L}^{2}

θ(x; d) for all d ∈ TL_{θ}(x). Since the inner and outer second-order
tangent sets are equal, we simply say that T_{L}^{2}

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 λ_{1}(x), and the other is using the parabolic second-order
directional derivative of projection operator Π_{L}_{θ}.

Theorem 4.1. Given x ∈ L_{θ} and d ∈ T_{L}_{θ}(x), then

T_{L}^{2}_{θ}(x, d) =

R^{n}, if d ∈ intTL_{θ}(x),
TL_{θ}(d), if x = 0,

{w : w^{T}_{2}x_{2}cot θ − w_{1}x_{1}tan θ ≤ d^{2}_{1}tan θ − kd_{2}k^{2}cot θ}, otherwise.

Proof. First, we note that L_{θ} = {x : −λ_{1}(x) ≤ 0}. With this, we have
w ∈ T_{L}^{2}

θ(x; d) ⇐⇒ −λ_{1}(x + td + 1

2t^{2}w + o(t^{2})) ≤ 0

⇐⇒ −λ_{1}(x) − tλ^{0}_{1}(x; d) −1

2t^{2}λ^{00}_{1}(x; d, w) + o(t^{2}) ≤ 0. (9)
The case of x ∈ intL_{θ} (corresponding to −λ_{1}(x) < 0) or x ∈ bdL_{θ} and d ∈ intTL_{θ}(x)
(corresponding to λ_{1}(x) = 0 and −λ^{0}_{1}(x; d) < 0) ensures that (9) holds for all w ∈ R^{n}.

For the case x = 0 and d = 0, it follows from Theorem 3.2 and (9) that
w ∈ T_{L}^{2}

θ(x; d) =⇒ −w_{1}+ kw_{2}k cot θ ≤ 0 ⇐⇒ w ∈ L_{θ}.

Conversely, if w ∈ Lθ, then dist(^{1}_{2}t^{2}w, Lθ) = 0 due to Lθ is a cone, which implies
w ∈ T_{L}^{2}

θ(x; d). Hence, T_{L}^{2}

θ(x; d) = T_{L}_{θ}(x).

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

w ∈ T_{L}^{2}_{θ}(x; d) =⇒ −w1d1tan^{2}θ + d^{T}_{2}w2 ≤ 0 ⇐⇒ w ∈ TL_{θ}(d).

Conversely, if w ∈ T_{L}_{θ}(d), then dist(d + tw, L_{θ}) = o(t) and hence dist(d + ^{1}_{2}tw, L_{θ}) =
o(^{1}_{2}t) = o(t). Thus, we obtain dist(x + td +^{1}_{2}t^{2}w, L_{θ}) = dist(td +^{1}_{2}t^{2}w, L_{θ}) = o(t^{2}), which
means w ∈ T_{L}^{2}

θ(x; d).

The case remained is x ∈ bdL_{θ}/{0} and d ∈ bdTL_{θ}(x), i.e., x_{1} = kx_{2}k cot θ and d^{T}_{2}x_{2} =
d_{1}x_{1}tan^{2}θ. Since x_{2} 6= 0, −λ_{1} is second-order differentiable at x. Hence it follows from
Theorem 3.2 that

T_{L}^{2}

θ(x; d) = {w : −λ^{00}_{1}(x; d, w) ≤ 0} =w : −x_{1}w_{1}tan θ + x^{T}_{2}w_{2}cot θ + kd_{2}k^{2}cot θ − d^{2}_{1}tan θ ≤ 0 ,
where the last step is due to ¯x^{T}_{2}d_{2} = d_{1}tan 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 ∈ TL_{θ}(x), we have

T_{L}^{2}_{θ}(x, d) = {w : Π^{00}_{L}_{θ}(x; d, w) = w}.

Proof. The desired result follows from
T_{L}^{2}

θ(x, d) = {w : dist(x + td + 1

2t^{2}w, L_{θ}) = o(t^{2}), t ≥ 0}

= {w : ΠL_{θ}(x + td +1

2t^{2}w) − (x + td + 1

2t^{2}w) = o(t^{2}), t ≥ 0}

= {w : ΠL_{θ}(x + td +1

2t^{2}w) − ΠL_{θ}(x) − tΠ^{0}_{L}

θ(x; d) −1

2t^{2}w = o(t^{2}), t ≥ 0}

= {w : Π^{00}_{L}_{θ}(x; d, w) = w},

where the third step uses the fact that d = Π^{0}_{L}

θ(x; d) since d ∈ TL_{θ}(x) by (8). 2
Recall first from [15] that ΠL_{θ}, 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.

Proof. Notice first that as x_{1} > kx_{2}k cot θ or x_{1} = kx_{2}k cot θ 6= 0 and d_{1} ≥ ¯x^{T}_{2}d_{2}cot θ,
then

2

tan θ + cot θΓ_{1}J eΦ(x)d + 1

tan θ + cot θΓ_{2} J eΦ(x)w + J^{2}Φ(x)(d, d)e

=

0, w_{2}−

¯

x^{T}_{2}w_{2}− (¯x^{T}_{2}d_{2})^{2}

kx_{2}k +kd_{2}k^{2}
kx_{2}k

x_{2}
kx_{2}k

T

. (10)

As x_{1} ≥ 0 and d_{1} ≥ kd_{2}k cot θ, we know that
1

tan θ + cot θ

f^{0}(x_{1}; d_{1}+kd_{2}k tan θ)−f^{0}(x_{1}; d_{1}−kd_{2}k cot θ)

J eΦ(d)w =

0, w_{2}− ¯d^{T}_{2}w_{2}d¯_{2}

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 ∈ intT_{L}_{θ}(x), we keep going to discuss three subcases.

Subcase (1): x = 0. Under this subcase, we see d ∈ intL_{θ}, i.e., d_{1} > kd_{2}k cot θ. If d_{2} = 0,
then d_{1} > 0 which yields

f^{00}(x_{1}; d_{1}, w_{1}− kw_{2}k cot θ)u^{1}_{w}+ f^{00}(x_{1}; d_{1}, w_{1}+ kw_{2}k tan θ)u^{2}_{w} = w, ∀w ∈ R^{n}.
If d_{2} 6= 0, it then follows from (11) that

(f^{L}^{θ})^{00}(x; d, w) = (w_{1}− ¯d^{T}_{2}w_{2}cot θ)u^{(1)}_{d} + (w_{1}+ ¯d^{T}_{2}w_{2}tan θ)u^{(2)}_{d} +

0, w_{2} − ¯d^{T}_{2}w_{2}d¯_{2}

T

= w.

Subcase (2): x ∈ intL_{θ}. Under this subcase, it is clear that T_{L}_{θ}(x) = R^{n}. If x_{2} = 0, it
follows from Theorem 3.3 that (f^{L}^{θ})^{00}(x; d, w) = w whenever d_{2} = 0 or d_{2} 6= 0 due to
x_{1} > 0 in this case. If x_{2} 6= 0, from (10), we know that

(f^{L}^{θ})^{00}(x; d, w) =

w1

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}

x2

kx2k

+

0
w_{2}−

¯

x^{T}_{2}w_{2}+ d^{T}_{2}J Φ(x_{2})d_{2}

x2

kx2k

= w.

Subcase (3): x ∈ bdL_{θ}/{0}. Then d ∈ intTL_{θ}(x) means d^{T}_{2}x_{2} < d_{1}x_{1}tan^{2}θ = d_{1}kx_{2}k tan θ,
i.e., ¯x^{T}_{2}d_{2}cot θ < d_{1}. Thus, (f^{L}^{θ})^{00}(x; d, w) = w for all w ∈ R^{n} by the similar argument
as above.

In summary, we have T_{L}^{2}

θ(x, d) = R^{n} in this case.

Case 2: For x = 0, since d ∈ TL_{θ}(x) = L_{θ}, we see that d_{1} ≥ kd_{2}k cot θ. It only remains
to show the case of d_{1} = kd_{2}k cot θ. If d_{2} = 0, then d_{1} = 0, and hence

(f^{L}^{θ})^{00}(x; d, w) = f^{00}(x_{1}; d_{1}, w_{1}− kw_{2}k cot θ)u^{1}_{w}+ f^{00}(x_{1}; d_{1}, w_{1}+ kw_{2}k tan θ)u^{2}_{w}