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 Zhou**^{1}**· Jingyong Tang**^{2}**·**
**Jein-Shan Chen**^{3}

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

Jein-Shan Chen jschen@math.ntnu.edu.tw1 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= (x*1*, x*2*)** ^{T}* ∈ R × R

^{n}^{−1}

*: cos θx ≤ x*1

*,*

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 with*L** _{θ}* 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*− x*2* cot θ, λ*2*(x) := x*1*+ x*2* 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*/x*2* if x*2*= 0, and ¯x*2being any vector*w ∈ R*^{n}^{−1}satisfying*w = 1 if*
*x*2= 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

*→ R*

^{n}*is defined as*

^{n}*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*^{}*(x; d, w) := lim*

*t*↓0
*w*^{}*→w*

*ψ*

*x+ td +*^{1}_{2}*t*^{2}*w*^{}

*− ψ(x) − tψ*^{}*(x; d)*

1

2*t*^{2} *.* (2)

To the contrary, the function*ψ is said to be parabolical second-order directionally*
*differentiable at x, ifw*^{}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, 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

2*t*^{2}*w + o*
*t*^{2}

*= ψ(x)+tψ*^{}*(x; d)+*1

2*t*^{2}*ψ*^{}*(x; d, w)+o*
*t*^{2}

*. (3)*

At the first glance on (3), the concept of parabolical second-order directional differ-
entiability 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 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 +*^{1}_{2}*t*^{2}*w 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 denote*Φ(x) := ¯x for x = 0, which does not*
cause any confusion from the context.

**Theorem 3.1 Let a function**Φ : R* ^{n}*→ R

^{n}*be 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*

*J*^{2}*Φ(x)(w, w) = −2*

*¯x*^{T}*w*

*x*^{2}

*w + w*^{T}

3*¯x ¯x*^{T}*− I*

*x*^{3}

*wx, ∀ w ∈ R*^{n}*.*

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

The Jacobian of*Φ at x = 0 is obtained from direct calculation. To obtain the second-*
*order derivative, for any given a*∈ R* ^{n}*, we define

*ψ : R*

*→ R as*

^{n}*ψ(x) := Φ(x)*^{T}*a*= *x*^{T}*a*

*x.*

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

*J g(x) = −* *¯x*

*x*^{2}*, J*^{2}*g(x) = −*

*I− ¯x ¯x*^{T}

*− 2 ¯x ¯x*^{T}

*x*^{3} =3*¯x ¯x*^{T}*− I*

*x*^{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}

−2*¯x*^{T}*w*

*x*^{2}*w + w*^{T}

3*¯x ¯x*^{T}*− I*

*x*^{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) = −2¯x*^{T}*w*

*x*^{2}*w + w*^{T}

3*¯x ¯x*^{T}*− I*

*x*^{3}

*wx,*

which is the desired result.

Next, we characterize the parabolic second-order directional derivative of the spec-
tral values*λ**i**(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 ofλ**i**(x) for i = 1, 2 reduces to the parabolic second-order direc-*
*tional differentiability. Moreover, given d, w ∈ R*^{n}*, we have*

*λ*^{}1*(x; d, w) =*

⎧⎪

⎪⎨

⎪⎪

⎩
*w*1−

*¯x*_{2}^{T}*w*2+^{d}^{2}^{}^{2}_{x}^{−}^{}_{2}^{¯x}_{}^{2}^{T}^{d}^{2}^{}^{2}

cot*θ, if x*2*= 0,*

*w*1*− ¯d*_{2}^{T}*w*2cot*θ,* *if x*2*= 0, d*2*= 0,*
*w*1*− w*2* cot θ,* *if x*2*= 0, d*2*= 0,*
*and*

*λ*^{}2*(x; d, w) =*

⎧⎪

⎪⎨

⎪⎪

⎩
*w*1+

*¯x*_{2}^{T}*w*2+^{d}^{2}^{}^{2}_{x}^{−}^{}_{2}^{¯x}_{}^{T}^{2}^{d}^{2}^{}^{2}

tan*θ, if x*2*= 0,*

*w*1*+ ¯d*_{2}^{T}*w*2tan*θ,* *if x*2*= 0, d*2*= 0,*
*w*1*+ w*2* tan θ,* *if x*2*= 0, d*2*= 0.*

*Proof Note thatλ**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 fol- lowing three cases.

*(i) If x*2*= 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*^{}_{1}*(x; d) = d*1*− ¯x*_{2}^{T}*d*2cot*θ and*

*x*2*+ td*2+1

2*t*^{2}*w*2* = x*2* + t ¯x*2^{T}*d*2+1
2*t*^{2}

*¯x*2^{T}*w*2+*d*2^{2}−

*¯x*_{2}^{T}*d*2

2

*x*2

*+ o*
*t*^{2}

*.*

Thus, we obtain
*λ*1

*x+ td +*^{1}_{2}*t*^{2}*w*

*− λ*1*(x) − tλ*^{}_{1}*(x; d)*

1
2*t*^{2}

*→ w*1−

*¯x*_{2}^{T}*w*2+*d*2^{2}−

*¯x*_{2}^{T}*d*2

2

*x*2

cot*θ.*

*(ii) If x*2*= 0 and d*2*= 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*^{}_{1}*(x; d) = d*1*− d*2* cot θ. Hence,*

*λ*1

*x+ td +*^{1}_{2}*t*^{2}*w*

*− λ*1*(x) − tλ*^{}_{1}*(x; d)*

1

2*t*^{2} *→ w*1*− ¯d*_{2}^{T}*w*2cot*θ.*

*(iii) If x*2*= 0 and d*2*= 0, then x + td +*^{1}_{2}*t*^{2}*w = (x*1*+ td*1+^{1}_{2}*t*^{2}*w*1*,*^{1}_{2}*t*^{2}*w*2*). Thus,*
*λ*^{}_{1}*(x; d) = d*1and

*λ*1

*x+ td +*^{1}_{2}*t*^{2}*w*

*− λ*1*(x) − tλ*^{}_{1}*(x; d)*

1

2*t*^{2} *→ w*1*− w*2* cot θ.*

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

applied to obtain the formula of*λ*^{}_{2}*(x; d, w).*

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 direc-*
*tionally differentiable at x in the Hadamard sense if and only if f is parabolic*
*second-order directionally differentiable atλ**i**(x) in the Hadamard sense for i = 1, 2.*

*Moreover,*

*(a) if x*2*= 0 and d*2*= 0, then*

*f*^{L}^{θ}^{}

*(x; d, w) = f*^{}*(x*1*; d*1*, w*1*− w*2* cot θ) u*^{(1)}_{w}*+ f*^{}*(x*1*; d*1*, w*1*+ w*2* tan θ) u*^{(2)}* _{w}* ;

*(b) if x*2

*= 0 and d*2

*= 0, then*

*f*^{L}^{θ}

*(x; d, w)*

*= f*^{}

*x*_{1}*; d*1*− d*2* cot θ, w*1*− ¯d*_{2}^{T}*w*2cot*θ*
*u*^{(1)}_{d}

*+ f*^{}

*x*_{1}*; d*1*+ d*2* tan θ, w*1*+ ¯d*_{2}^{T}*w*2tan*θ*
*u*^{(2)}_{d}

+ 1

tan*θ + cot θ*

*f*^{}*(x*1*; d*1*+ d*2* tan θ) − f*^{}*(x*1*; d*1*− d*2* cot θ)*

*J Φ(d)w;*

*(c) if x*2*= 0, then*

*f*^{L}^{θ}

*(x; d, w)*

*= f*^{}

*x*_{1}*− x*2* cot θ; d*1*− ¯x*_{2}^{T}*d*_{2}cot*θ, w*1−

*¯x*_{2}^{T}*w*2*+ d*_{2}^{T}*J Φ(x*2*)d*2

cot*θ*
*u*^{(1)}_{x}*+ f*^{}

*x*_{1}*+ x*2* tan θ; d*1*+ ¯x*_{2}^{T}*d*_{2}tan*θ, w*1+

*¯x*_{2}^{T}*w*2*+ d*_{2}^{T}*J Φ(x*2*)d*2

tan*θ*
*u*^{(2)}_{x}

+ 2

cot*θ + tan θΓ*1*J Φ(x)d +* 1
cot*θ + tan θΓ*2

*J Φ(x)w + J*^{2}*Φ(x)(d, d)*

*,*

*where*

*Γ*1*:= f*^{}

*x*1*+ x*2* tan θ; d*1*+ ¯x*2^{T}*d*2tan*θ*

*− f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*2^{T}*d*2cot*θ*
*Γ*2*:= f (x*^{1}*+ x*2* tan θ) − f (x*^{1}*− x*2* cot θ)*
*and Φ(x) := (1, Φ(x*2*))*^{T}*for all x*∈ R^{n}*with x*2*= 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*

^{}

*→ w, we consider*

*the following four cases. First we denote z:= x + td +*

^{1}

_{2}

*t*

^{2}

*w*

^{}.

*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}

_{}

*(x; d) =*

*f*^{}*(x*1*; d*1*) , 0*

*= f*^{}*(x*1*; d*1*) u*^{(1)}*z* *+ f*^{}*(x*1*; d*1*) 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*^{L}^{θ}

*x+ td +*^{1}_{2}*t*^{2}*w*^{}

*− f*^{L}^{θ}*(x) − t*
*f*^{L}^{θ}_{}

*(x; d)*

1
2*t*^{2}

*→ f*^{}*(x*1*; d*1*, w*1*) u*^{(1)}_{ξ}*+ f*^{}*(x*1*; d*1*, w*1*) u*^{(2)}_{ξ}

=

*f*^{}*(x*1*; d*1*, w*1*) , 0*
*.*

*Case 2: For x*2*= 0, d*2*= 0, and w*2*= 0, since f is parabolic second-order direction-*
ally differentiable, we have

*f(λ*1*(z)) − f (x*1*) − t f*^{}*(x*1*; d*1*)*

1

2*t*^{2} *→ f*^{}*(x*1*; d*1*, w*1*− w*2* cot θ)*

and

*f* *(λ*2*(z)) − f (x*^{1}*) − t f*^{}*(x*1*; d*1*)*

1

2*t*^{2} *→ f*^{}*(x*1*; d*1*, w*1*+ w*2* 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*^{}

*− f*^{L}^{θ}*(x) − t*
*f*^{L}^{θ}_{}

*(x; d)*

1
2*t*^{2}

*→ f*^{}*(x*1*; d*1*, w*1*− w*2* cot θ) u*^{(1)}_{w}*+ f*^{}*(x*1*; d*1*, w*1*+ w*2* 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}

^{}

*(x; d, w) = f*^{}*(x*1*; d*1*, w*1*− w*2* cot θ) u*^{(1)}_{w}*+ f*^{}*(x*1*; d*1*, w*1*+ w*2* tan θ) u*^{(2)}_{w}*.*

*Case 3: For x*2*= 0, d*2= 0, we have

*( f*^{L}^{θ}*)*^{}*(x; d) = f*^{}*(x*1*; d*1*− d*2* cot θ)u*^{(1)}_{d}*+ f*^{}*(x*1*; d*1*+ d*2* tan θ)u*^{(2)}_{d}*.*
Note that

*f*

*x*1*+ td*1+1

2*t*^{2}*w*_{1}^{} *− td*2+1

2*tw*^{}_{2}* cot θ*

*= f*

*x*1*+ td*1+1

2*t*^{2}*w*^{}_{1}*− t*

*d*2* cot θ +*1

2*t ¯d*_{2}^{T}*w*_{2}^{}cot*θ + o(t)*

*= f*

*x*1*+ td*1+1

2*t*^{2}*w*1*− t*

*d*2* cot θ +*1

2*t ¯d*_{2}^{T}*w*2cot*θ*
*+ o*

*t*^{2}

*= f (x*1*) + t f*^{}*(x*1*; d*1*− d*2* cot θ)*
+1

2*t*^{2}*f*^{}

*x*1*; d*1*− d*2* cot θ, w*1*− ¯d*2^{T}*w*2cot*θ*
*+ o*

*t*^{2}

*,* (5)

where we use the facts that*w*^{} *→ w and f is parabolic second-order directionally*
differentiable at*λ*1*(x) in the Hadamard sense. Similarly, we obtain*

*f*

*x*1*+ td*1+1

2*t*^{2}*w*^{}1*+ td*2+1

2*tw*^{}2* tan θ*

*= f (x*1*) + t f*^{}*(x*1*; d*1*+ d*2* tan θ)*
+1

2*t*^{2}*f*^{}

*x*1*; d*1*+ d*2* tan θ, w*1*+ ¯d*2^{T}*w*2tan*θ*
*+ o*

*t*^{2}

*.* (6)

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

1

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

1

2*t*^{2} converges to

1
1+ cot^{2}*θ* *f*^{}

*x*1*; d*1*− d*2* cot θ, w*1*− ¯d*_{2}^{T}*w*2cot*θ*

+ 1

1+ tan^{2}*θ* *f*^{}

*x*1*; d*1*+ d*2* tan θ, w*1*+ ¯d*2^{T}*w*2tan*θ*
*.*

In addition, according to Theorem3.1, we know
*d*2+^{1}_{2}*tw*^{}_{2}

*d*2+^{1}_{2}*tw*^{}_{2} *= Φ*

*d*2+1

2*tw*2^{}

*= Φ(d*2*) +*1

2*tJ Φ(d*2*)w*2^{} +1

8*t*^{2}*J*^{2}*Φ(d*2*)*
*w*2^{}*, w*2^{}

*+ o*
*t*^{2}

*= Φ(d*2*) +*1

2*tJ Φ(d*2*)w*2^{} +1

8*t*^{2}*J*^{2}*Φ(d*2*) (w*^{2}*, w*2*) + o*
*t*^{2}

*. (7)*

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

*− f (λ*1*(z)) Φ*

*d*2+1

2*tw*^{}2

*+ f (x*1*)Φ(d*2*) + t f*^{}*(x*1*; d*1*− d*2* cot θ) Φ(d*2*)*

= −1

2*t f(x*1*)J Φ(d*2*)w*2^{} −1
2*t*^{2}

*f*^{}

*x*1*; d*1*− d*2* cot θ, w*1*− ¯d*2^{T}*w*2cot*θ*

*Φ(d*2*)*
*+ f*^{}*(x*1*; d*1*− d*2* cot θ) J Φ(d*^{2}*)w*^{}2+1

4 *f(x*1*)J*^{2}*Φ(d*2*) (w*^{2}*, w*2*)*

*+ o*

*t*^{2}

and

*f(λ*2*(z))Φ*

*d*2+1

2*tw*2^{}

*− f (x*1*)Φ(d*2*) − t f*^{}*(x*1*; d*1*+ d*2* tan θ)Φ(d*2*)*

=1

2*t f(x*1*)J Φ(d*2*)w*_{2}^{} +1
2*t*^{2}

*f*^{}

*x*1*; d*1*+ d*2* tan θ, w*1*+ ¯d*_{2}^{T}*w*2tan*θ*

*Φ(d*2*)*
*+ f*^{}*(x*1*; d*1*+ d*2* tan θ) J Φ(d*2*)w*_{2}^{} +1

4*f(x*1*)J*^{2}*Φ(d*2*) (w*2*, w*2*)*

*+ o*

*t*^{2}
*.*

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

1

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

1

2*t*^{2} converges to

1
tan*θ + ctanθ*

*κ*1*J Φ(d*2*)w*2*+ κ*2*Φ(d*2*)*

*,*

where

*κ*1*:= f*^{}*(x*1*; d*1*+ d*2* tan θ) − f*^{}*(x*1*; d*1*− d*2* cot θ)*
*κ*2*:= f*^{}

*x*1*; d*1*+ d*2* tan θ, w*1*+ ¯d*2^{T}*w*2tan*θ*

*− f*^{}

*x*1*; d*1*− d*2* cot θ, w*1*− ¯d*2^{T}*w*2cot*θ*
*.*

To sum up, we can conclude that

*f*^{L}^{θ}^{}

*(x; d, w)*

*= f*^{}

*x*1*; d*1*− d*2*, w*1*− ¯d*_{2}^{T}*w*2cot*θ*
*u*^{(1)}_{d}*+ f*^{}

*x*1*; d*1*+ d*2* tan θ, w*1*+ ¯d*2^{T}*w*2tan*θ*
*u*^{(2)}_{d}

+ 1

tan*θ + cot θ*

*f*^{}*(x*1*; d*1*+ d*2* tan θ) − f*^{}*(x*1*; d*1*− d*2* cot θ)*

*J Φ(d)w.*

*Case 4: For x*2= 0, under this case, we know
*( f*^{L}^{θ}*)*^{}*(x; d) = f*^{}

*λ*1*(x); d*1*− ¯x*2^{T}*d*2cot*θ*

*u*^{(1)}_{x}*+ f*^{}

*λ*2*(x); d*1*+ ¯x*2^{T}*d*2tan*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*_{2}^{T}

*d.*

Note that

*x*2*+ td*2+1

2*t*^{2}*w*^{}_{2}* = x*2* + t ¯x*_{2}^{T}*d*2+1
2*t*^{2}

*¯x*_{2}^{T}*w*^{}_{2}*+ d*_{2}^{T}*J Φ(x*2*)d*2

*+ o*
*t*^{2}

*= x*2* + t ¯x*_{2}^{T}*d*2+1
2*t*^{2}

*¯x*_{2}^{T}*w*2*+ d*_{2}^{T}*J Φ(x*2*)d*2

*+ o*
*t*^{2}

*.*

*Since f is parabolic second-order directionally differentiable atλ*1*(x) in the Hadamard*
sense, we have

*f*

*x*1*+ td*1+1

2*t*^{2}*w*^{}1*− x*2*+ td*2+1

2*t*^{2}*w*^{}2* cot θ*

*= f (x*1*− x*2* cot θ) + t f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*_{2}^{T}*d*2cot*θ*
+1

2*t*^{2}*f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*_{2}^{T}*d*2cot*θ, w*1

−

*¯x*2^{T}*w*2*+ d*2^{T}*J Φ(x*2*)d*2

cot*θ*

*+ o*
*t*^{2}

*.*

Besides, we know that

*Φ*

*x*2*+ td*2+1
2*t*^{2}*w*_{2}^{}

*= Φ(x*2*) + tJ Φ(x*2*)d*2+1
2*t*^{2}

*J Φ(x*2*)w*_{2}^{} *+ J*^{2}*Φ(x*2*) (d*2*, d*2*)*

*+ o*
*t*^{2}

*= Φ(x*2*) + tJ Φ(x*2*)d*2+1
2*t*^{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

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

1

2*t*^{2} converges to

1
1+ cot^{2}*θ* *f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*_{2}^{T}*d*2cot*θ, w*1

−

*¯x*2^{T}*w*2*+ d*2^{T}*J Φ(x*2*)d*2

cot*θ*

+ 1

1+ tan^{2}*θ* *f*^{}

*x*1*+ x*2* tan θ; d*1*+ ¯x*2^{T}*d*2tan*θ, w*1

+

*¯x*2^{T}*w*2*+ d*2^{T}*J Φ(x*2*)d*2

tan*θ*

*.*

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

1

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

1

2*t*^{2} converges

to

− cot*θ*
1+ cot^{2}*θ*

*f* *(x*1*− x*2* cot θ)*

*J Φ(x*2*)w*2*+ J*^{2}*Φ(x*2*) (d*^{2}*, d*2*)*
*+ 2 f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*_{2}^{T}*d*2cot*θ*

*J Φ(x*2*)d*2

*+ f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*2^{T}*d*2cot*θ, w*1

−

*¯x*2^{T}*w*2*+ d*2^{T}*J Φ(x*2*)d*2

cot*θ*
*Φ(x*2*)*

+ tan*θ*
1+ tan^{2}*θ*

*f* *(x*1*+ x*2* tan θ)*

*J Φ(x*2*)w*2*+ J*^{2}*Φ(x*2*) (d*^{2}*, d*2*)*
*+ 2 f*^{}

*x*1*+ x*2* tan θ; d*1*+ ¯x*2^{T}*d*2tan*θ*

*J Φ(x*2*)d*2

*+ f*^{}

*x*1*+ x*2* tan θ; d*1*+ ¯x*2^{T}*d*2tan*θ, w*1

+

*¯x*_{2}^{T}*w*2*+ d*_{2}^{T}*J Φ(x*2*)d*2

tan*θ*
*Φ(x*2*)*

*.*

To sum up, we can conclude that

*f*^{L}^{θ}^{}

*(x; d, w)*

*= f*^{}

*x*1*− x*2* cot θ; d*1*− ¯x*_{2}^{T}*d*2cot*θ, w*1−

*¯x*_{2}^{T}*w*2*+ d*_{2}^{T}*JΦ(x*2*)d*2

cot*θ*

*u*^{1}_{x}*+ f*^{}

*x*1*+ x*2* tan θ; d*1*+ ¯x*_{2}^{T}*d*2tan*θ, w*1+

*¯x*_{2}^{T}*w*2*+ d*_{2}^{T}*JΦ(x*2*)d*2

tan*θ*

*u*^{2}_{x}

+ 2

cot*θ + tan θΓ*1*JΦ(x)d +* 1
cot*θ + tan θΓ*2

*JΦ(x)w +* *J*^{2}*Φ(x)(d, d)*

*,*

where we use the facts that *J Φ(x)w = (0, J Φ(x*2*)w*2*) and J*^{2}*Φ(x)(d, d) =*
*(0, J*^{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*^{} *→ ˜w. To proceed, we also discuss the*
following two cases.

*Case 1: For x*2*= 0, let d = ˜de, w*^{}*= ˜w*^{}*e, andw = ˜we. Denote z := x + td +*^{1}_{2}*t*^{2}*w*^{}.
Then

*f*

*x*1*+ t ˜d +*^{1}_{2}*t*^{2}*˜w*^{}

*− f (x*1*) − t f*^{}*(x*1*, ˜d)*

1
2*t*^{2}

=

*f*^{L}^{θ}*(z) − f*^{L}^{θ}*(x) − t*
*f*^{L}^{θ}_{}

*(x; d)*

1

2*t*^{2} *, e*

*.*

*Thus, we obtain f*^{}*(x*1*; ˜d, ˜w) = ( f*^{L}^{θ}*)*^{}*(x; d, w), e.*

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

*x+ td +*1
2*t*^{2}*w*^{}=

*λ*1*(x) + t ˜d +*1
2*t*^{2}*˜w*^{}

*u*^{(1)}_{x}*+ λ*2*(x)u*^{(2)}*x*

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

*f*^{L}^{θ}

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

*= f*

*λ*1*(x) + t ˜d +*1
2*t*^{2}*˜w*^{}

*u*^{(1)}_{x}*+ f (λ*2*(x)) u*^{(2)}*x*

and*( f*^{L}^{θ}*)*^{}*(x; d) = f*^{}*(λ*1*(x); ˜d)u*^{(1)}*x* *. Thus,*
*f*

*λ*1*(x) + t ˜d +*^{1}_{2}*t*^{2}*˜w*^{}

*− f (λ*^{1}*(x)) − t f*^{}

*λ*1*(x); ˜d*

1
2*t*^{2}

=

1+ cot^{2}*θ f*^{L}^{θ}

*x+ td +* ^{1}_{2}*t*^{2}*w*^{}

*− f*^{L}^{θ}*(x) − t*
*f*^{L}^{θ}_{}

*(x; d)*

1

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

*,*

which says

*f*^{}

*λ*1*(x); ˜d, ˜w*

=

1+ cot^{2}*θ*
*f*^{L}^{θ}

*(x; d, w), u*^{(1)}*x*

*.*

*The similar arguments can be used for f atλ*2*(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*^{L}* ^{θ}* is just the projection operator associated with

*circular cone. For x*

*∈ L*

*, from [16], we know the tangent cone is given by*

_{θ}*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*_{L}^{}_{θ}*(x; d) = d*

*,* (8)

which, together with the formula of*Π*_{L}^{}* _{θ}*, yields

*T*_{L}_{θ}*(x) =*

⎧⎨

⎩

R^{n}*,* *if x∈ intL*_{θ}*,*

*L*_{θ}*,* *if x= 0,*

*d* *: d*_{2}^{T}*x*2*− d*1*x*1tan^{2}*θ ≤ 0*

*, if x ∈ bdL**θ**/{0}.*

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

*w ∈ R** ^{n}* : dist

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

*= o*
*t*_{n}^{2}

*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 ∈ T*_{L}_{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∈ intT**L**θ**(x),*
*T*_{L}_{θ}*(d),* *if x= 0,*

*w : w*_{2}^{T}*x*2cot*θ − w*1*x*1tan*θ ≤ d*_{1}^{2}tan*θ − d*2^{2}cot*θ*

*, otherwise.*

*Proof First, we note thatL*_{θ}*= {x : −λ*1*(x) ≤ 0}. With this, we have*
*w ∈ T*_{L}^{2}_{θ}*(x; d) ⇐⇒ −λ*1

*x+ td +*1

2*t*^{2}*w + o*
*t*^{2}

≤ 0

*⇐⇒ −λ*1*(x) − tλ*^{}1*(x; d) −*1

2*t*^{2}*λ*^{}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∈ intT*

_{L}

_{θ}*(x)*(corresponding to

*λ*1

*(x) = 0 and −λ*

^{}

_{1}

*(x; d) < 0) ensures that (9) holds for allw ∈ R*

*.*

^{n}*For the case x= 0 and d = 0, it follows from Theorem*3.2and (9) that
*w ∈ T*_{L}^{2}_{θ}*(x; d) ⇒ −w*1*+ w*2* cot θ ≤ 0 ⇐⇒ w ∈ L**θ**.*

Conversely, if*w ∈ L** _{θ}*, then dist(

^{1}

_{2}

*t*

^{2}

*w, L*

_{θ}*) = 0 due to L*

*is a cone, which implies*

_{θ}*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 ∈ bdT*_{L}_{θ}*(x)\{0} = bdL** _{θ}*\{0}, it follows from Theorem3.2
and (9) that

*w ∈ T*_{L}^{2}_{θ}*(x; d) ⇒ −w*1*d*1tan^{2}*θ + d*2^{T}*w*2*≤ 0 ⇐⇒ w ∈ T*_{L}_{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 ∈ bdT*_{L}_{θ}*(x), i.e., x*1 *= x*2* cot θ and*
*d*_{2}^{T}*x*2*= d*1*x*1tan^{2}*θ. Since x*2*= 0, −λ*1*is second-order differentiable at x. Hence, it*
follows from Theorem3.2that

*T*_{L}^{2}

*θ**(x; d) =*

*w : −λ*^{}1*(x; d, w) ≤ 0*

=

*w : −x*1*w*1tan*θ + x*2^{T}*w*2cot*θ + d*2^{2}cot*θ − d*1^{2}tan*θ ≤ 0*
*,*

where the last step is due to *¯x*_{2}^{T}*d*2*= d*1tan*θ.*

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*_{L}_{θ}*(x), we have*
*T*_{L}^{2}

*θ**(x, d) =*

*w : Π*_{L}^{}_{θ}*(x; d, w) = w*
*.*

*Proof The desired result follows from*

*T*_{L}^{2}_{θ}*(x, d) =*

*w : dist*

*x+ td +*1
2*t*^{2}*w, L*_{θ}

*= o*
*t*^{2}

*, t ≥ 0*

=

*w : Π*_{L}_{θ}

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

−

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

*= o*
*t*^{2}

*, t ≥ 0*

=

*w : Π*_{L}_{θ}

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

*− Π*_{L}_{θ}*(x) − tΠ*_{L}^{}_{θ}*(x; d) −*1
2*t*^{2}*w*

*= o*
*t*^{2}

*, 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*_{L}_{θ}*(x) by (8).*