Parabolic Second-Order Directional Differentiability in the Hadamard Sense of the Vector-Valued Functions Associated with Circular Cones

DOI 10.1007/s10957-016-0935-9

Parabolic Second-Order Directional Differentiability in the Hadamard Sense of the Vector-Valued Functions Associated with Circular Cones

Jinchuan Zhou¹ · Jingyong Tang² · Jein-Shan Chen³

Received: 29 June 2015 / Accepted: 9 April 2016

© Springer Science+Business Media New York 2017

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 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 relationship 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 characterizations of second-order necessary and sufficient conditions for circular cone optimization problems.

Keywords Parabolic second-order derivative· Circular cone · Second-order tangent set

Mathematics Subject Classification 90C30· 49J52 · 46G05

Communicated by Byung-Soo Lee.

B

1 Department of Mathematics, School of Science, Shandong University of Technology, Zibo 255049, Shandong, People’s Republic of China

2 College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan, People’s Republic of China

3 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan

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 characterize the optimality conditions for various optimization problems; see [1,4–7]

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 [5,8–10]. In this paper, we mainly focus on the par- abolic second-order directional derivative in the Hadamard sense for the vector-valued functions associated with circular cones. This vector-valued function, called circular cone function, comes from applying a given real-valued function to the spectral decom- position associated with circular cone.

For the circular cone function, by using the basic tools of nonsmooth analysis, var- ious properties such as directional derivative, differentiability, B-subdifferentiability, semismoothness, and positive homogeneity have been studied in [11,12]. The afore- mentioned 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 differentiability plays an important role in second-order necessary and suf- ficient conditions. Recently, there was an investigation on the parabolic second-order directional derivative of singular values of matrices and symmetric matrix-valued func- tions 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 differentiable. 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 differentiabil- ity 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= (x1, x2)^T ∈ R × Rⁿ⁻¹ : cos θx ≤ x1

,

which is a nonsymmetric cone in the standard inner product. In our previous works [12–15], we have explored some important features about circular cone, such as characterizing its tangent cone, normal cone, and second-order regularity. In par-

ticular, the spectral decomposition associated withL_θ 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− x2 cot θ, λ2(x) := x1+ x2 tan θ and

u⁽¹⁾_x := 1 1+ cot²θ

1 0

0 cotθ · I

  1

− ¯x2



, u⁽²⁾_x := 1 1+ tan²θ

1 0

0 tanθ · I

 1

¯x2



with¯x2:= x2/x2 if x2= 0, and ¯x2being any vectorw ∈ Rⁿ⁻¹satisfyingw = 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^Lθ : Rⁿ→ Rⁿis defined as

f^Lθ(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:

ψ^(x; d, w) := lim

t↓0 w^→w

ψ

x+ td +¹₂t²w^

− ψ(x) − tψ^(x; d)

1

2t² . (2)

To the contrary, the functionψ is said to be parabolical second-order directionally differentiable at x, ifw^is fixed to bew 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, whenψ is locally Lipschitz at x, these two concepts coincide. It is known that ifψ 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)+tψ^(x; d)+1

2t²ψ^(x; d, w)+o t²

. (3)

At the first glance on (3), the concept of parabolical second-order directional differ- entiability in the Hadamard sense is likely to say thatψ 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 nonnega- tive orthant, a polyhedral 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 nonpolyhedral 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 nonpolyhedral properties of nonpolyhedral cones. This point can be seen in Sect.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 program- ming; see [16] for more information.

3 Second-Order Directional Derivative

For subsequent analysis, we will frequently use the second-order derivative of¯x := _x^x at x = 0. To this end, we present the second-order derivative of ¯x in below theorem.

For convenience of notation, we also denoteΦ(x) := ¯x for x = 0, which does not cause any confusion from the context.

Theorem 3.1 Let a functionΦ : Rⁿ→ Rⁿbe given asΦ(x) := _x^x for x = 0. Then, the functionΦ is second-order continuous differentiable at x = 0 with

J Φ(x) = I− ¯x ¯x^T

x

and

J²Φ(x)(w, w) = −2

¯x^Tw

x² 

w + w^T

3¯x ¯x^T − I

x³ 

wx, ∀ w ∈ Rⁿ.

Proof It is clear thatΦ is second-order continuous differentiable because of x = 0.

The Jacobian ofΦ at x = 0 is obtained from direct calculation. To obtain the second- order derivative, for any given a∈ Rⁿ, we defineψ : Rⁿ→ R as

ψ(x) := Φ(x)^Ta= x^Ta

x.

We also denote h(x) := a^Tx and g(x) := 1/x so that ψ(x) = h(x)g(x). Since x= 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

x², J²g(x) = −

I− ¯x ¯x^T

− 2 ¯x ¯x^T

x³ =3¯x ¯x^T − I

x³ . Hence, from the chain rule, we haveJ 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)

= a^T



−2¯x^Tw

x²w + w^T

3¯x ¯x^T − I

x³ 

wx



. (4)

On the other hand, we see thatJ²ψ(x)(w, w) = a^TJ²Φ(x)(w, w). Since a ∈ Rⁿis arbitrary, this together with (4) yields

J²Φ(x)(w, w) = −2¯x^Tw

x²w + w^T

3¯x ¯x^T − I

x³ 

wx,

which is the desired result. 

Next, we characterize the parabolic second-order directional derivative of the spec- tral valuesλi(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 ofλi(x) for i = 1, 2 reduces to the parabolic second-order direc- tional differentiability. Moreover, given d, w ∈ Rⁿ, we have

λ^1(x; d, w) =

⎧⎪

⎪⎨

⎪⎪

⎩ w1−



¯x₂^Tw2+^d22_x^−₂^¯x_^2Td22 

cotθ, if x2= 0,

w1− ¯d₂^Tw2cotθ, if x2= 0, d2= 0, w1− w2 cot θ, if x2= 0, d2= 0, and

λ^2(x; d, w) =

⎧⎪

⎪⎨

⎪⎪

⎩ w1+



¯x₂^Tw2+^d22_x^−₂^¯x_^T2d22 

tanθ, if x2= 0,

w1+ ¯d₂^Tw2tanθ, if x2= 0, d2= 0, w1+ w2 tan θ, if x2= 0, d2= 0.

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

To compute the parabolic second-order directional derivative, we consider the fol- lowing three cases.

(i) If x2= 0, then x + td +¹₂t²w = (x1+ td1+¹₂t²w1, x2+ td2+¹₂t²w2). Note thatλ^₁(x; d) = d1− ¯x₂^Td2cotθ and

x2+ td2+1

2t²w2 = x2 + t ¯x2^Td2+1 2t²



¯x2^Tw2+d2²−

¯x₂^Td2

2

x2



+ o t²

.

Thus, we obtain λ1

x+ td +¹₂t²w

− λ1(x) − tλ^₁(x; d)

1 2t²

→ w1−



¯x₂^Tw2+d2²−

¯x₂^Td2

2

x2

 cotθ.

(ii) If x2= 0 and d2= 0, then x + td +¹₂t²w = (x1+ td1+¹₂t²w1, td2+¹₂t²w2) andλ^₁(x; d) = d1− d2 cot θ. Hence,

λ1

x+ td +¹₂t²w

− λ1(x) − tλ^₁(x; d)

1

2t² → w1− ¯d₂^Tw2cotθ.

(iii) If x2= 0 and d2= 0, then x + td +¹₂t²w = (x1+ td1+¹₂t²w1,¹₂t²w2). Thus, λ^₁(x; d) = d1and

λ1

x+ td +¹₂t²w

− λ1(x) − tλ^₁(x; d)

1

2t² → w1− w2 cot θ.

From all the above, the formula ofλ^₁(x; d, w) is proved. Similar arguments can be

applied to obtain the formula ofλ^₂(x; d, w). 

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

Theorem 3.3 Suppose that f : R → R. Then, f^Lθ is parabolic second-order direc- tionally differentiable at x in the Hadamard sense if and only if f is parabolic second-order directionally differentiable atλi(x) in the Hadamard sense for i = 1, 2.

Moreover,

(a) if x2= 0 and d2= 0, then

f^{Lθ }

(x; d, w) = f^(x1; d1, w1− w2 cot θ) u⁽¹⁾_w + f^(x1; d1, w1+ w2 tan θ) u⁽²⁾_w ; (b) if x2= 0 and d2= 0, then

f^Lθ



(x; d, w)

= f^

x₁; d1− d2 cot θ, w1− ¯d₂^Tw2cotθ u⁽¹⁾_d

+ f^

x₁; d1+ d2 tan θ, w1+ ¯d₂^Tw2tanθ u⁽²⁾_d

+ 1

tanθ + cot θ



f^(x1; d1+ d2 tan θ) − f^(x1; d1− d2 cot θ) 

J Φ(d)w;

(c) if x2= 0, then

f^Lθ 

(x; d, w)

= f^

x₁− x2 cot θ; d1− ¯x₂^Td₂cotθ, w1−

¯x₂^Tw2+ d₂^TJ Φ(x2)d2

cotθ u⁽¹⁾_x + f^

x₁+ x2 tan θ; d1+ ¯x₂^Td₂tanθ, w1+

¯x₂^Tw2+ d₂^TJ Φ(x2)d2

tanθ u⁽²⁾_x

+ 2

cotθ + tan θΓ1J Φ(x)d + 1 cotθ + tan θΓ2



J Φ(x)w + J²Φ(x)(d, d) 

,

where

Γ1:= f^

x1+ x2 tan θ; d1+ ¯x2^Td2tanθ

− f^

x1− x2 cot θ; d1− ¯x2^Td2cotθ Γ2:= f (x¹+ x2 tan θ) − f (x¹− x2 cot θ) and Φ(x) := (1, Φ(x2))^T for all x∈ Rⁿwith x2= 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 ∈ Rⁿandw^→ w, we consider the following four cases. First we denote z:= x + td +¹₂t²w^.

Case 1: For x2= 0, d2= 0, and w2= 0, we have f^Lθ(x) =

f(x1), 0

= f (x1)u⁽¹⁾z + f(x1)u⁽²⁾z and

f^Lθ

_

(x; d) =

f^(x1; d1) , 0

= f^(x1; d1) u⁽¹⁾z + f^(x1; d1) u⁽²⁾z .

Note that u⁽ⁱ⁾z → u⁽ⁱ⁾_ξ as i = 1, 2 for some ξ ∈ {(1, w) : w = 1}. Thus, we conclude that

f^Lθ

x+ td +¹₂t²w^

− f^Lθ(x) − t f^Lθ_

(x; d)

1 2t²

→ f^(x1; d1, w1) u⁽¹⁾_ξ + f^(x1; d1, w1) u⁽²⁾_ξ

=

f^(x1; d1, w1) , 0 .

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

f(λ1(z)) − f (x1) − t f^(x1; d1)

1

2t² → f^(x1; d1, w1− w2 cot θ)

and

f (λ2(z)) − f (x¹) − t f^(x1; d1)

1

2t² → f^(x1; d1, w1+ w2 tan θ) . Note that u⁽ⁱ⁾z → u⁽ⁱ⁾w for i = 1, 2. Therefore, we also conclude that

f^Lθ

x+ td +¹₂t²w^

− f^Lθ(x) − t f^Lθ_

(x; d)

1 2t²

→ f^(x1; d1, w1− w2 cot θ) u⁽¹⁾_w + f^(x1; d1, w1+ w2 tan θ) u⁽²⁾_w . In summary, from Cases 1 and 2, we see that under x2= 0 and d2= 0

f^Lθ

^

(x; d, w) = f^(x1; d1, w1− w2 cot θ) u⁽¹⁾_w + f^(x1; d1, w1+ w2 tan θ) u⁽²⁾_w .

Case 3: For x2= 0, d2= 0, we have

( f^Lθ)^(x; d) = f^(x1; d1− d2 cot θ)u⁽¹⁾_d + f^(x1; d1+ d2 tan θ)u⁽²⁾_d . Note that

f



x1+ td1+1

2t²w₁^ − td2+1

2tw^₂ cot θ 

= f



x1+ td1+1

2t²w^₁− t

d2 cot θ +1

2t ¯d₂^Tw₂^cotθ + o(t)

= f



x1+ td1+1

2t²w1− t

d2 cot θ +1

2t ¯d₂^Tw2cotθ + o

t² 

= f (x1) + t f^(x1; d1− d2 cot θ) +1

2t²f^

x1; d1− d2 cot θ, w1− ¯d2^Tw2cotθ + o

t²

, (5)

where we use the facts thatw^ → w and f is parabolic second-order directionally differentiable atλ1(x) in the Hadamard sense. Similarly, we obtain

f



x1+ td1+1

2t²w^1+ td2+1

2tw^2 tan θ 

= f (x1) + t f^(x1; d1+ d2 tan θ) +1

2t²f^

x1; d1+ d2 tan θ, w1+ ¯d2^Tw2tanθ + o

t²

. (6)

Thus, the first component of ^fLθ(x+td+

1

2t²w^)− f^Lθ(x)−t( f^Lθ)^(x;d)

1

2t² converges to

1 1+ cot²θ f^

x1; d1− d2 cot θ, w1− ¯d₂^Tw2cotθ

+ 1

1+ tan²θ f^

x1; d1+ d2 tan θ, w1+ ¯d2^Tw2tanθ .

In addition, according to Theorem3.1, we know d2+¹₂tw^₂

d2+¹₂tw^₂ = Φ

 d2+1

2tw2^

= Φ(d2) +1

2tJ Φ(d2)w2^ +1

8t²J²Φ(d2) w2^, w2^

+ o t²

= Φ(d2) +1

2tJ Φ(d2)w2^ +1

8t²J²Φ(d2) (w², w2) + o t²

. (7)

Hence, it follows from (5) to (7) that

− f (λ1(z)) Φ

 d2+1

2tw^2

+ f (x1)Φ(d2) + t f^(x1; d1− d2 cot θ) Φ(d2)

= −1

2t f(x1)J Φ(d2)w2^ −1 2t²

 f^



x1; d1− d2 cot θ, w1− ¯d2^Tw2cotθ 

Φ(d2) + f^(x1; d1− d2 cot θ) J Φ(d²)w^2+1

4 f(x1)J²Φ(d2) (w², w2)

 + o

t²

and

f(λ2(z))Φ

 d2+1

2tw2^

− f (x1)Φ(d2) − t f^(x1; d1+ d2 tan θ)Φ(d2)

=1

2t f(x1)J Φ(d2)w₂^ +1 2t²

 f^



x1; d1+ d2 tan θ, w1+ ¯d₂^Tw2tanθ 

Φ(d2) + f^(x1; d1+ d2 tan θ) J Φ(d2)w₂^ +1

4f(x1)J²Φ(d2) (w2, w2)

 + o

t² .

Thus, the second component of ^fLθ(x+td+

1

2t²w^)− f^Lθ(x)−t( f^Lθ)^(x;d)

1

2t² converges to

1 tanθ + ctanθ



κ1J Φ(d2)w2+ κ2Φ(d2) 

,

where

κ1:= f^(x1; d1+ d2 tan θ) − f^(x1; d1− d2 cot θ) κ2:= f^

x1; d1+ d2 tan θ, w1+ ¯d2^Tw2tanθ

− f^

x1; d1− d2 cot θ, w1− ¯d2^Tw2cotθ .

To sum up, we can conclude that

f^{Lθ }

(x; d, w)

= f^

x1; d1− d2, w1− ¯d₂^Tw2cotθ u⁽¹⁾_d + f^

x1; d1+ d2 tan θ, w1+ ¯d2^Tw2tanθ u⁽²⁾_d

+ 1

tanθ + cot θ



f^(x1; d1+ d2 tan θ) − f^(x1; d1− d2 cot θ) 

J Φ(d)w.

Case 4: For x2= 0, under this case, we know ( f^Lθ)^(x; d) = f^

λ1(x); d1− ¯x2^Td2cotθ

u⁽¹⁾_x + f^

λ2(x); d1+ ¯x2^Td2tanθ u⁽²⁾_x + f(λ2(x)) − f (λ1(x))

λ2(x) − λ1(x)

0 0

0 I− ¯x2¯x₂^T

 d.

Note that

x2+ td2+1

2t²w^₂ = x2 + t ¯x₂^Td2+1 2t²

¯x₂^Tw^₂+ d₂^TJ Φ(x2)d2

+ o t²

= x2 + t ¯x₂^Td2+1 2t²

¯x₂^Tw2+ d₂^TJ Φ(x2)d2

+ o t²

.

Since f is parabolic second-order directionally differentiable atλ1(x) in the Hadamard sense, we have

f



x1+ td1+1

2t²w^1− x2+ td2+1

2t²w^2 cot θ 

= f (x1− x2 cot θ) + t f^

x1− x2 cot θ; d1− ¯x₂^Td2cotθ +1

2t²f^

x1− x2 cot θ; d1− ¯x₂^Td2cotθ, w1

−

¯x2^Tw2+ d2^TJ Φ(x2)d2

 cotθ

+ o t²

.

Besides, we know that

Φ



x2+ td2+1 2t²w₂^

= Φ(x2) + tJ Φ(x2)d2+1 2t²



J Φ(x2)w₂^ + 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 ^fLθ(x+td+

1

2t²w^)− f^Lθ(x)−t( f^Lθ)^(x;d)

1

2t² converges to

1 1+ cot²θ f^

x1− x2 cot θ; d1− ¯x₂^Td2cotθ, w1

−

¯x2^Tw2+ d2^TJ Φ(x2)d2

 cotθ

+ 1

1+ tan²θ f^

x1+ x2 tan θ; d1+ ¯x2^Td2tanθ, w1

+

¯x2^Tw2+ d2^TJ Φ(x2)d2

 tanθ

.

Moreover, the second component of ^fLθ(x+td+

1

2t²w^)− f^Lθ(x)−t( f^Lθ)^(x;d)

1

2t² converges

to

− cotθ 1+ cot²θ



f (x1− x2 cot θ)

J Φ(x2)w2+ J²Φ(x2) (d², d2) + 2 f^

x1− x2 cot θ; d1− ¯x₂^Td2cotθ

J Φ(x2)d2

+ f^

x1− x2 cot θ; d1− ¯x2^Td2cotθ, w1

−

¯x2^Tw2+ d2^TJ Φ(x2)d2

cotθ Φ(x2)

+ tanθ 1+ tan²θ



f (x1+ x2 tan θ)

J Φ(x2)w2+ J²Φ(x2) (d², d2) + 2 f^

x1+ x2 tan θ; d1+ ¯x2^Td2tanθ

J Φ(x2)d2

+ f^

x1+ x2 tan θ; d1+ ¯x2^Td2tanθ, w1

+

¯x₂^Tw2+ d₂^TJ Φ(x2)d2

tanθ Φ(x2)

 .

To sum up, we can conclude that

f^Lθ^

(x; d, w)

= f^



x1− x2 cot θ; d1− ¯x₂^Td2cotθ, w1−



¯x₂^Tw2+ d₂^TJΦ(x2)d2

 cotθ

 u¹_x + f^



x1+ x2 tan θ; d1+ ¯x₂^Td2tanθ, w1+



¯x₂^Tw2+ d₂^TJΦ(x2)d2

 tanθ

 u²_x

+ 2

cotθ + tan θΓ1JΦ(x)d + 1 cotθ + tan θΓ2



JΦ(x)w + J²Φ(x)(d, d) 

,

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

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

Case 1: For x2= 0, let d = ˜de, w^= ˜w^e, andw = ˜we. Denote z := x + td +¹₂t²w^. Then

f

x1+ t ˜d +¹₂t²˜w^

− f (x1) − t f^(x1, ˜d)

1 2t²

=

f^Lθ(z) − f^Lθ(x) − t f^Lθ_

(x; d)

1

2t² , e

 .

Thus, we obtain f^(x1; ˜d, ˜w) = ( f^Lθ)^(x; d, w), e.

Case 2: For x2= 0, let d = ˜du⁽¹⁾x ,w^= ˜w^u⁽¹⁾x , andw = ˜wu⁽¹⁾x . Then, we have

x+ td +1 2t²w^=



λ1(x) + t ˜d +1 2t²˜w^

u⁽¹⁾_x + λ2(x)u⁽²⁾x

with t > 0 satisfying t ˜d +¹₂t²˜w^< λ2(x) − λ1(x). This implies

f^Lθ



x+ td + 1 2t²w^

= f



λ1(x) + t ˜d +1 2t²˜w^

u⁽¹⁾_x + f (λ2(x)) u⁽²⁾x

and( f^Lθ)^(x; d) = f^(λ1(x); ˜d)u⁽¹⁾x . Thus, f

λ1(x) + t ˜d +¹₂t²˜w^

− f (λ¹(x)) − t f^

λ1(x); ˜d

1 2t²

=

1+ cot²θ  f^Lθ

x+ td + ¹₂t²w^

− f^Lθ(x) − t f^Lθ_

(x; d)

1

2t² , u¹x

 ,

which says

f^

λ1(x); ˜d, ˜w

=

1+ cot²θ  f^Lθ



(x; d, w), u⁽¹⁾x

 .

The similar arguments can be used for f atλ2(x). From all the above, the proof is

complete. 

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^Lθ is just the projection operator associated with circular cone. For x ∈ L_θ, from [16], we know the tangent cone is given by

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

= 

d : Π_Lθ(x + td) − (x + td) = o(t), t ≥ 0

= 

d : Π_L^_θ(x; d) = d

, (8)

which, together with the formula ofΠ_L^_θ, yields

T_Lθ(x) =

⎧⎨

⎩

Rⁿ, if x∈ intL_θ,

L_θ, if x= 0,

d : d₂^Tx2− d1x1tan²θ ≤ 0

, if x ∈ bdLθ/{0}.

Definition 4.1 [16, Definition 3.28] The set limits T_S^i,2(x, d) :=

w ∈ Rⁿ : dist



x+ td +1 2t²w, S

= o t²

, t ≥ 0

and

T_S²(x, d) :=

w ∈ Rⁿ : ∃ tn ↓ 0 such that dist



x+ tnd+1 2t_n²w, S

= o t_n²

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_L²_θ(x; d) for all d ∈ T_Lθ(x). Since the inner and outer second- order tangent sets are equal, we simply say that T_L²

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 valueλ1(x), and the other is using the parabolic second-order directional derivative of projection operatorΠ_Lθ.

Theorem 4.1 Given x ∈ L_θand d∈ T_Lθ(x), then

T_L²_θ(x, d) =

⎧⎨

⎩

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

w : w₂^Tx2cotθ − w1x1tanθ ≤ d₁²tanθ − d2²cotθ

, otherwise.

Proof First, we note thatL_θ = {x : −λ1(x) ≤ 0}. With this, we have w ∈ T_L²_θ(x; d) ⇐⇒ −λ1



x+ td +1

2t²w + o t²

≤ 0

⇐⇒ −λ1(x) − tλ^1(x; d) −1

2t²λ^1(x; d, w) + o t²

≤ 0. (9)

The case of x∈ intL_θ(corresponding to−λ1(x) < 0) or x ∈ bdL_θand d∈ intT_Lθ(x) (corresponding toλ1(x) = 0 and −λ^₁(x; d) < 0) ensures that (9) holds for allw ∈ Rⁿ.

For the case x= 0 and d = 0, it follows from Theorem3.2and (9) that w ∈ T_L²_θ(x; d) ⇒ −w1+ w2 cot θ ≤ 0 ⇐⇒ w ∈ Lθ.

Conversely, ifw ∈ L_θ, then dist(¹₂t²w, L_θ) = 0 due to L_θ is a cone, which implies w ∈ T_L²_θ(x; d). Hence, T_L²_θ(x; d) = T_Lθ(x).

For the case x= 0 and d ∈ bdT_Lθ(x)\{0} = bdL_θ\{0}, it follows from Theorem3.2 and (9) that

w ∈ T_L²_θ(x; d) ⇒ −w1d1tan²θ + d2^Tw2≤ 0 ⇐⇒ w ∈ T_Lθ(d).

Conversely, if w ∈ T_Lθ(d), then dist(d + tw, L_θ) = o(t), and hence, dist(d +

1

2tw, Lθ) = o(¹₂t) = o(t). Thus, we obtain dist(x + td + ¹₂t²w, Lθ) = dist(td +

1

2t²w, L_θ) = o(t²), which means w ∈ T_L²_θ(x; d).

The case remained is x ∈ bdL_θ/{0} and d ∈ bdT_Lθ(x), i.e., x1 = x2 cot θ and d₂^Tx2= d1x1tan²θ. Since x2= 0, −λ1is second-order differentiable at x. Hence, it follows from Theorem3.2that

T_L²

θ(x; d) =

w : −λ^1(x; d, w) ≤ 0

=

w : −x1w1tanθ + x2^Tw2cotθ + d2²cotθ − d1²tanθ ≤ 0 ,

where the last step is due to ¯x₂^Td2= d1tanθ. 

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 operator associated with circular cone. To this end, we need a technical lemma.

Lemma 4.1 For x ∈ L_θ and d∈ T_Lθ(x), we have T_L²

θ(x, d) =

w : Π_L^_θ(x; d, w) = w .

Proof The desired result follows from

T_L²_θ(x, d) =

w : dist



x+ td +1 2t²w, L_θ

= o t²

, t ≥ 0

=

w : Π_Lθ



x+ td +1 2t²w

−



x+ td +1 2t²w

= o t²

, t ≥ 0

=

w : Π_Lθ



x+ td +1 2t²w

− Π_Lθ(x) − tΠ_L^_θ(x; d) −1 2t²w

= o t²

, t ≥ 0

=

w : Π_L^_θ(x; d, w) = w ,

where the third step uses the fact that d= Π_L^_θ(x; d) since d ∈ T_Lθ(x) by (8).