(1)SMOOTH ANALYSIS ON CONE FUNCTION ASSOCIATED WITH ELLIPSOIDAL CONE YUE LU∗ AND JEIN-SHAN CHEN† Abstract

SMOOTH ANALYSIS ON CONE FUNCTION ASSOCIATED WITH ELLIPSOIDAL CONE

YUE LU^∗ AND JEIN-SHAN CHEN^†

Abstract. As an important prototype in closed convex cones, ellip- soidal cone covers several practical instances such as second-order cone, circular cone and elliptic cone. In virtue of a recent study on its de- composition expression, we present a symmetric type of ellipsoidal cone function and show that this vector-valued function inherits some smooth properties from its corresponding scalar function, particularly in conti- nuity, directional differentiability, differentiability and continuous differ- entiability. We believe that these results will play important roles on further analysis and study about conic programming problems associ- ated with ellipsoidal cone.

1. Introduction

Consider the ellipsoidal cone [2, 21, 22, 23] with the form (1.1) K_E :=x ∈ Rⁿ| x^TQx ≤ 0, u^T_nx ≥ 0 ,

where Q ∈ R^n×nis a real-valued nonsingular symmetric matrix and (λi, ui) ∈ R × Rⁿdenotes its i-th eigenpair. In addition, these pairs satisfy the under- line relations:

λ1≥ · · · ≥ λ_n−1> 0 > λn and

 u^T_i uj = 1, if i = j, u^T_i uj = 0, if i 6= j.

Under the standard Euclidean inner product h·, ·i and the norm k · k defined on Rⁿ, its dual cone (KE)^∗_h·,·ihas an explicit expression (due to [16, Theorem 2.1]) as follows:

(1.2) (KE)^∗_h·,·i:= {y ∈ Rⁿ | y^TQ⁻¹y ≤ 0, u^T_ny ≥ 0}.

It is not difficult to see that the ellipsoidal cone KE defined as in (1.1) can be viewed as a type of nonsymmetric cones.

During the past two decades, conic programming have been extensively studied [4, 6, 7, 8, 9, 12, 13, 14, 20], particularly in three types of closed

2010 Mathematics Subject Classification. 90C25.

Key words and phrases. Nonsymmetric cones, Ellipsoidal cones, Cone function.

∗The author’s work is supported by National Natural Science Foundation of China (Grant Number: 11601389), Doctoral Foundation of Tianjin Normal University (Grant Number: 52XB1513).

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

1

convex cones, i.e., the nonnegative octant Rⁿ+, second-order cone Kⁿ and positive semi-definite cone S₊ⁿ. All of these cones are fully addressed and some fundamental topics such as projection, spectral decomposition, cone function and cone-convexity have been studied. A fascinating feature among them is to unify all these results under the framework of Euclidean Jordan Algebra defined on symmetric cones, we refer to the monograph [10] for more details. A natural question is how to extend these observations on symmetric cones suitable for nonsymmetric counterparts? Recently, Miao, Lu and Chen [18] look into the first three items in the setting of some nonsymmetric cases such as circular cone, p-order cone, geometric cone, exponential cone and power cone, in which the lack of explicit projection formulae onto these cones (except for the circular cone case) become the main hurdle for non-symmetric cone optimization problems and cause some unpleasant consequences. For example, the classical Moreau decomposition in convex analysis cannot be used directly; the associated decomposition expressions and cone functions are correspondingly missing. These observations motivate us to focus on algebraic properties of nonsymmetric cones and to provide a systematical study on their analytic features.

As an important prototype, several famous instances can be generated from ellipsoidal cones by different choices of parameters (Q, un). For in- stances, let us take

Q =

 I_n−1 0

0 −1

 or

 I_n−1 0 0 − tan²θ

 or

 M^TM 0

0 −1



and u_n= e_n, where In−1 denotes the identity matrix of order n − 1, θ ∈ (0,^π₂), M is any nonsingular matrix of order n − 1 and e_n is the n-th column vector of I_n. In these cases, the ellipsoidal cone respectively reduces to the second-order cone [5, 8]:

Kⁿ:=(¯xn−1, xn) ∈ Rⁿ⁻¹× R | k¯xn−1k ≤ x_n , the circular cone [3, 24]:

L_θ:=(¯x_n−1, x_n) ∈ Rⁿ⁻¹× R | k¯x_n−1k ≤ x_ntan θ and the elliptic cone [1]:

Kⁿ_M :=(¯xn−1, xn) ∈ Rⁿ⁻¹× R | kM ¯xn−1k ≤ x_n .

Therefore, ellipsoidal cone is a natural generalization of second-order cone, circular cone and elliptic cone.

For algebraic properties of ellipsoidal cones, there have been several liter- atures in recent studies. More specifically, Lu and Chen [16] discuss its self- duality and positive homogeneity, in which the authors observe that ellip- soidal cone can become self-dual by introducing a new inner product and the associated automorphism group can be characterized as the similarity trans- formation of its special counterpart in the second-order cone setting with an appropriate nonsingular matrix. Furthermore, they also provide an investi- gation on its variational geometry, projection expression and decomposition,

see [17] for more details. In particular, the decomposition of the given point associated with the ellipsoidal cone is characterized in [17, Theorem 8]. For completeness, we restate it as follows. Due to the eigenvalue decomposition, we rewrite Q ∈ R^n×n as the form Q = U ΛU^T with an orthogonal matrix U ∈ R^n×n and a diagonal matrix Λ ∈ R^n×n, where U := [ ¯U_n−1, u_n] ∈ R^n×n, U¯n−1 := [u1, u2, · · · , un−1] ∈ R^n×(n−1) and Λ := diag( ¯Λn−1, λn) ∈ R^n×n, Λ¯_n−1 := diag (λ₁, λ₂, · · · , λ_n−1) ∈ R(n−1)×(n−1).

Theorem 1.1 (Decomposition). [17, Theorem 8] Let KE ∈ Rⁿ be an ellipsoidal cone defined as in (1.1) and (KE)^∗_h·,·i be its dual cone defined as in (1.2). For any given x ∈ Rⁿ, it has the following decomposition:

x = ( s⁽¹⁾_I

a(x) · v_I⁽¹⁾

a (x) + s⁽²⁾_I

a(x) · v_I⁽²⁾

a (x) if U¯_n−1^T x 6= 0, s⁽¹⁾_I

b (x) · v_I⁽¹⁾

b (x) + s⁽²⁾_I

b (x) · v_I⁽²⁾

b (x) if U¯_n−1^T x = 0, where s⁽¹⁾_I

a(x), s⁽²⁾_I

a(x), s⁽¹⁾_I

b (x), s⁽²⁾_I

b (x) and v⁽¹⁾_I

a (x), v_I⁽²⁾

a (x), v_I⁽¹⁾

b (x), v_I⁽²⁾

b (x) are respectively given by

s⁽¹⁾_I

a(x) := u^T_nx + k ¯M ¯U_n−1^T xk, v_I⁽¹⁾

a (x) := 1

2 · ¯U_n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk + u_n

!

∈ K_E,

s⁽²⁾_I

a(x) := u^T_nx − k ¯M ¯U_n−1^T xk, v_I⁽²⁾

a (x) := 1

2 · −U¯_n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk + u_n

!

∈ K_E, s⁽¹⁾_I

b (x) := u^T_nx, v_I⁽¹⁾

b (x) := 1 2 ·

U¯_n−1w k ¯M wk + u_n



∈ K_E, s⁽²⁾_I

b (x) := u^T_nx, v_I⁽²⁾

b (x) := 1 2 ·



−U¯_n−1w k ¯M wk + u_n



∈ K_E with any given nonzero vector w ∈ Rⁿ⁻¹ and a diagonal matrix ¯M looks like

(1.3)

M :=¯

" ¯U_n−1^T (Q − λnunu^T_n) ¯Un−1

(−λ_n)

#1/2

= diag s

λ1

(−λn), s

λ2

(−λn), · · · , s

λn−1

(−λn)

! . Theorem 1.1 indicates that by denoting

(1.4)



λ⁽¹⁾_I (x), λ⁽²⁾_I (x), u⁽¹⁾_I (x), u⁽²⁾_I (x)



:=







 s⁽¹⁾_I

a (x), s⁽²⁾_I

a (x), v_I⁽¹⁾

a (x), v⁽²⁾_I

a (x)

if ¯U_n−1^T x 6= 0,

 s⁽¹⁾_I

b (x), s⁽²⁾_I

b (x), v_I⁽¹⁾

b (x), v⁽²⁾_I

b (x)



if ¯U_n−1^T x = 0, the decomposition formula now can be rewritten as follows:

(1.5) x = λ⁽¹⁾_I (x) · u⁽¹⁾_I (x) + λ⁽²⁾_I (x) · u⁽²⁾_I (x), ∀x ∈ Rⁿ.

For any function f : R → R, the following vector-valued function associated with KE is considered:

(1.6) f_I^EC(x) = f (λ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) + f (λ⁽²⁾_I (x)) · u⁽²⁾_I (x), ∀x ∈ Rⁿ. If f is defined only on the subset of R, then fI^EC is defined on the cor- responding subset of Rⁿ. Notice that the expression (1.6) is well-defined whether ¯U_n−1^T x 6= 0 or ¯U_n−1^T x = 0. In the sequel, we call this function a symmetric type of ellipsoidal cone function, due to the fact that the vectors u⁽¹⁾_I (x), u⁽²⁾_I (x) in (1.4) are both contained in KE. For any given x ∈ Rⁿ, λ⁽ⁱ⁾_I (x), u⁽ⁱ⁾_I (x) (i = 1, 2) are the spectral values and the spectral vectors of x, respectively.

In this paper, we aim to study smooth properties of the vector-valued function f_I^EC, particularly in continuity, directional differentiability, dif- ferentiability, continuous differentiability inherited by f_I^EC from f . As a byproduct, we also establish these results for some special cases of ellip- soidal cone such as second-order cone, circular cone and elliptic cone.

The rest of this paper is organized as follows. In Section 2, we present some technical lemmas used in the sequel. The main conclusions will be established in Section 3. We next discuss some special examples in Section 4. Finally, some concluding remarks are drawn.

1.1. Notation and terminology. In what follows, we review some basic concepts about vector-valued functions. For the mapping F : Rⁿ→ R^m, we say F to be continuous at x ∈ Rⁿ if

F (y) → F (x) as y → x,

and F is continuous if F is continuous at every x ∈ Rⁿ. Similarly, we say F is directionally differentiable at x ∈ Rⁿ if

F⁰(x; h) = lim

t↓0

F (x + th) − F (x) t

exists for all h ∈ Rⁿ and F is directionally differentiable if F is directionally differentiable at every x ∈ Rⁿ. Moreover, F is differentiable (in the Fr´echet sense) at x ∈ Rⁿif there exists a linear mapping DF : Rⁿ→ R^m such that

F (x + h) − F (x) − DF (x)h = o(khk).

We call DF (x) the Jacobian of F at x ∈ Rⁿ. Furthermore, if F is differen- tiable at every x ∈ Rⁿ and DF (x) is also continuous, then F is continuous differentiable. For a differentiable mapping g : Rⁿ → R, the gradient of g with respect to the variable x ∈ Rⁿ is denoted by ∇xg.

2. Preliminaries

Before establishing smooth analytic properties of f_I^EC, we need the fol- lowing technical lemmas.

Lemma 2.1 (Perturbation of spectral values). Let λ⁽ⁱ⁾_I (x) (i = 1, 2) be the spectral values of x ∈ Rⁿ and m⁽ⁱ⁾_I (y) (i = 1, 2) be the spectral values of y ∈ Rⁿ. Then, we have

(2.1)  

λ⁽ⁱ⁾_I (x) − m⁽ⁱ⁾_I (y)  

≤ 1 + k ¯M ¯U_n−1^T k
· kx − yk, ∀i = 1, 2, where k ¯M ¯U_n−1^T k is the induced matrix norm on the space R^(n−1)×nof ¯M ¯U_n−1^T as follows:

k ¯M ¯U_n−1^T k := sup{k ¯M ¯U_n−1^T xk

kxk = 1, x ∈ Rⁿ}.

Proof. The proof can be obtained easily by simple calculation. Note that

|λ⁽ⁱ⁾_I (x) − m⁽ⁱ⁾_I (y)|

≤ 

u^T_n(x − y) +

k ¯M ¯U_n−1^T xk − k ¯M ¯U_n−1^T yk 

≤ ku_nk · kx − yk + k ¯M ¯U_n−1^T (x − y)k

≤ 1 + k ¯M ¯U_n−1^T k
· kx − yk, ∀i = 1, 2,

where the second inequality follows from the facts |a^Tb| ≤ kak · kbk, |kak −

kbk| ≤ ka − bk, ∀a, b ∈ Rⁿ. 

Lemma 2.2 (Perturbation of spectral vectors). Let u⁽ⁱ⁾_I (x) (i = 1, 2) be the spectral vectors of x ∈ Rⁿ and p⁽ⁱ⁾_I (y) (i = 1, 2) be the spectral vectors of y ∈ Rⁿ.

(a) If ¯U_n−1^T x 6= 0, ¯U_n−1^T y 6= 0, then we have (2.2)

u⁽ⁱ⁾_I (x) − p⁽ⁱ⁾_I (y) ≤ 1

2

k ¯U_n−1U¯_n−1^T k

k ¯M ¯U_n−1^T xk +k ¯U_n−1M¯⁻¹k · k ¯M ¯U_n−1^T k k ¯M ¯U_n−1^T xk

!

· kx − yk

for any i = 1, 2. In this case, u⁽ⁱ⁾_I (x) and p⁽ⁱ⁾_I (y) are the unique spectral vectors of x and y, respectively.

(b) If either ¯U_n−1^T x = 0 or ¯U_n−1^T y = 0, then we can choose u⁽ⁱ⁾_I (x), p⁽ⁱ⁾_I (y) such that the left hand side of the above inequality (2.2) is zero.

Proof. (a) If ¯U_n−1^T x 6= 0, ¯U_n−1^T y 6= 0, according to the decomposition in Theorem 1.1 and (1.4), we obtain

u⁽¹⁾_I (x) = 1

2· ¯Un−1U¯_n−1^T x k ¯M ¯U_n−1^T xk + un

!

, u⁽²⁾_I (x) = 1

2 · −U¯n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk + un

! , p⁽¹⁾_I (y) = 1

2· ¯Un−1U¯_n−1^T y k ¯M ¯U_n−1^T yk + un

!

, p⁽²⁾_I (y) = 1

2 · −U¯n−1U¯_n−1^T y k ¯M ¯U_n−1^T yk + un

! .

From the above, we see that u⁽ⁱ⁾_I (x), p⁽ⁱ⁾_I (y) (i = 1, 2) are unique. In addition, we have

ku⁽ⁱ⁾_I (x) − p⁽ⁱ⁾_I (y)k

≤ 1

2

U¯n−1U¯_n−1^T x

k ¯M ¯U_n−1^T xk −U¯n−1U¯_n−1^T y k ¯M ¯U_n−1^T yk

≤ 1

2

U¯_n−1U¯_n−1^T x − ¯U_n−1U¯_n−1^T y k ¯M ¯U_n−1^T xk +

U¯_n−1U¯_n−1^T y

k ¯M ¯U_n−1^T xk −U¯_n−1U¯_n−1^T y k ¯M ¯U_n−1^T yk

≤ 1

2

k ¯Un−1U¯_n−1^T k

k ¯M ¯U_n−1^T xk · kx − yk + k ¯Un−1M¯⁻¹k k ¯M ¯U_n−1^T xk ·

k ¯M ¯U_n−1^T yk − k ¯M ¯U_n−1^T xk 

!

≤ 1

2

k ¯U_n−1U¯_n−1^T k

k ¯M ¯U_n−1^T xk · kx − yk + k ¯U_n−1M¯⁻¹k

k ¯M ¯U_n−1^T xk · k ¯M ¯U_n−1^T (y − x)k

!

≤ 1

2

k ¯Un−1U¯_n−1^T k

k ¯M ¯U_n−1^T xk + k ¯Un−1M¯⁻¹k · k ¯M ¯U_n−1^T k k ¯M ¯U_n−1^T xk

!

· kx − yk, ∀i = 1, 2.

(b) It is clear that we can choose the same spectral vectors for x and y from the relation (1.4), since either ¯U_n−1^T x = 0 or ¯U_n−1^T y = 0.  Lemma 2.3 (Gradients). Let A ∈ R^n×n, B ∈ R^s×n and x ∈ Rⁿ. If Bx 6= 0, then we have

∇_x

 Ax kBxk



= 1

kBxk



In−(B^TB)(xx^T) kBxk²

 A^T, (2.3)

∇_x(kBxk) = 1

kBxkB^TBx.

(2.4)

Proof. Let us rewrite

A =











a₁₁ a₁₂ · · · a_1n a₂₁ a₂₂ · · · a_2n ... ... . .. ... a_n1 a_n2 · · · a_nn











=









 a^T₁ a^T₂ ... a^T_n











∈ R^n×n,

B =











b11 b12 · · · b1n

b₂₁ b₂₂ · · · b_2n ... ... . .. ... b_s1 b_s2 · · · b_sn











=

B·,1 B·,2 · · · B·,n  ∈ R^s×n

with a_i∈ Rⁿ (i = 1, 2, · · · , n) and B·,j ∈ R^s (j = 1, 2, · · · , n). Therefore, we have

∇_x

 Ax kBxk



=



∇_x

 a^T₁x kBxk

 , ∇x

 a^T₂x kBxk



, · · · , ∇x

 a^T_nx kBxk



,

where ∇_x_aT ix kBxk



(i = 1, 2, · · · , n) is defined as follows:

∇_x

 a^T_i x kBxk



=























∂

∂x1

 a^T_i x kBxk



∂

∂x2

 a^T_i x kBxk

 ...

∂

∂xn

 a^T_i x kBxk

























∈ Rⁿ.

By direct calculation, we have

∂

∂x_j

 a^T_i x kBxk



= aij

1

kBxk− (a^T_i x)(Bx)^TB·,j

kBxk³ , j = 1, 2, · · · , n.

Consequently, we obtain

∇_x

 Ax kBxk



= 1

kBxkA^T − 1

kBxk³[(Ax)(Bx)^TB]^T

= 1

kBxk



I_n− (B^TB)(xx^T) kBxk²

 A^T,

which shows that Eq. (2.3) holds. On the other hand, similar to the first part, we obtain

∂

∂x_j(kBxk) = (Bx)^TB·,j

kBxk , j = 1, 2, · · · , n, and therefore the gradient of kBxk with respect to x is given by

∇_x(kBxk) =





















∂

∂x₁(kBxk)

∂

∂x2

(kBxk) ...

∂

∂x_n(kBxk)





















=























(Bx)^TB·,1

kBxk (Bx)^TB·,2

kBxk ... (Bx)^TB·,n

kBxk























= 1

kBxkB^TBx,

which implies that Eq. (2.4) is true. 

3. Smooth analysis

In this section, we aim to show the properties of continuity and differen- tiability between the scalar function f and its associated cone function f_I^EC. Now, after the above preparations, we are ready to present our first main result about continuity between f and f_I^EC.

Theorem 3.1 (Continuous). For any given function f : R → R, let f_I^EC be its corresponding cone function defined as in (1.6). Then, the following statements are equivalent to each other.

(a) f is continuous at λ⁽ⁱ⁾_I (x) (i = 1, 2).

(b) f_I^EC is continuous at x ∈ Rⁿ with spectral values λ⁽ⁱ⁾_I (x) (i = 1, 2).

Proof. (a) ⇒ (b) Suppose f is continuous at λ⁽ⁱ⁾_I (x) (i = 1, 2). For any fixed x ∈ Rⁿ and y → x, let x and y be decomposed as

x = λ⁽¹⁾_I (x)·u⁽¹⁾_I (x)+λ⁽²⁾_I (x)·u⁽²⁾_I (x), y = m⁽¹⁾_I (y)·p⁽¹⁾_I (y)+m⁽²⁾_I (y)·p⁽²⁾_I (y).

Then, we consider the following two cases:

Case (a): If ¯U_n−1^T x 6= 0, then we have f_I^EC(y) − f_I^EC(x)

= f (m⁽¹⁾_I (y)) · p⁽¹⁾_I (y) + f (m⁽²⁾_I (y)) · p⁽²⁾_I (y)

−f (λ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) − f (λ⁽²⁾_I (x)) · u⁽²⁾_I (x)

= f (m⁽¹⁾_I (y)) · (p⁽¹⁾_I (y) − u⁽¹⁾_I (x)) + (f (m⁽¹⁾_I (y)) − f (λ⁽¹⁾_I (x))) · u⁽¹⁾_I (x) +f (m⁽²⁾_I (y)) · (p⁽²⁾_I (y) − u⁽²⁾_I (x)) + (f (m⁽²⁾_I (y)) − f (λ⁽²⁾_I (x))) · u⁽²⁾_I (x).

(3.1)

Since f is continuous at λ⁽ⁱ⁾_I (x) (i = 1, 2) and the inequality (2.1) in Lemma 2.1, we obtain

f (m⁽ⁱ⁾_I (y)) → f (λ⁽ⁱ⁾_I (x)) (i = 1, 2) as y → x.

According to the relation (2.2) in Lemma 2.2, we also know kp⁽ⁱ⁾_I (y) − u⁽ⁱ⁾_I (x)k → 0 (i = 1, 2) as y → x.

Moreover, the equation (3.1) and the boundedness of f (m⁽ⁱ⁾_I (y)), u⁽ⁱ⁾_I (x) yield that

f_I^EC(y) → f_I^EC(x) as y → x.

Therefore, f_I^EC is continuous at x ∈ Rⁿ.

Case (b): If ¯U_n−1^T x = 0, we can arrange that x, y have the same vector parts, regardless of ¯U_n−1^T y is equal to zero or not. At the same time, we obtain

f_I^EC(y) − f_I^EC(x) = (f (m⁽¹⁾_I ) − f (λ⁽¹⁾_I )) · u⁽¹⁾_I + (f (m⁽²⁾_I ) − f (λ⁽²⁾_I )) · u⁽²⁾_I . By similar arguments as Case (a), we know that f_I^EC is continuous at x ∈ Rⁿ. (b) ⇒ (a) The proof for this direction is straightforward and has a similar

arguments for [5, Proposition 2]. 

Theorem 3.2 (Directionally Differentiable). For any given function f : R → R, let f_I^EC be its corresponding cone function defined as in (1.6).

Then, the following statements are equivalent to each other.

(a) f is directionally differentiable at λ⁽ⁱ⁾_I (x) (i = 1, 2).

(b) f_I^EC is directionally differentiable at x ∈ Rⁿ with spectral values λ⁽ⁱ⁾_I (x) (i = 1, 2).

Proof. (a) ⇒ (b) Suppose f is directionally differentiable at λ⁽ⁱ⁾_I (x) (i = 1, 2). We divide our proof into the following two cases:

Case (a): If ¯U_n−1^T x 6= 0, then we have

f (x) = f (λ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) + f (λ⁽²⁾_I (x)) · u⁽²⁾_I (x),

where the scalars λ⁽¹⁾_I (x), λ⁽²⁾_I (x) and the vectors u⁽¹⁾_I (x), u⁽²⁾_I (x) are given by

λ⁽¹⁾_I (x) = u^T_nx + k ¯M U_n−1^T xk, λ⁽²⁾_I (x) = u^T_nx − k ¯M U_n−1^T xk, u⁽¹⁾_I (x) = 1

2· ¯U_n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk + u_n

!

, u⁽²⁾_I (x) = 1

2 · −U¯_n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk + u_n

! . Due to the nonsingularity of ¯M defined as in (1.3), we obtain ¯M ¯U_n−1^T x 6=

0. From Lemma 2.3, we know that λⁱ_I(x), u⁽ⁱ⁾_I (x) (i = 1, 2) are Fr´echet- differentiable with respect to the variable x, i.e.,

∇_xλ⁽¹⁾_I (x) = u_n+ 1

k ¯M ¯U_n−1^T xk( ¯U_n−1M¯^TM ¯¯U_n−1^T )x,

∇_xλ⁽²⁾_I (x) = un− 1

k ¯M ¯U_n−1^T xk( ¯Un−1M¯^TM ¯¯U_n−1^T )x,

∇_xu⁽¹⁾_I (x) = 1 2k ¯M ¯U_n−1^T xk

"

I_n−( ¯U_n−1M¯^TM ¯¯U_n−1^T )(xx^T) k ¯M ¯U_n−1^T xk²

#

( ¯U_n−1U¯_n−1^T ),

∇_xu⁽²⁾_I (x) = − 1 2k ¯M ¯U_n−1^T xk

"

I_n− ( ¯U_n−1M¯^TM ¯¯U_n−1^T )(xx^T) k ¯M ¯U_n−1^T xk²

#

( ¯U_n−1U¯_n−1^T ).

To show that f_I^EC is directionally differentiable at x ∈ Rⁿ with spectral values λ⁽ⁱ⁾_I (x) (i = 1, 2), we only need to verify the directional differentiabil- ity of the composition functions f (λ⁽ⁱ⁾_I (x)) (i = 1, 2) with respect to x ∈ Rⁿ and then use the product rule and the chain rule on f_I^EC.

Since f is directionally differentiable at λ⁽¹⁾_I (x), then it is easy to see that lim

t→0⁺

f (λ⁽¹⁾_I (x) + t · 1) − f (λ⁽¹⁾_I (x))

t = f⁰(λ⁽¹⁾_I (x); 1), lim

t→0⁺

f (λ⁽¹⁾_I (x) − t · 1) − f (λ⁽¹⁾_I (x))

t = f⁰(λ⁽¹⁾_I (x); −1), lim

t→0⁺

f (λ⁽¹⁾_I (x) + o(t)) − f (λ⁽¹⁾_I (x))

t = 0.

Using the fact that λ⁽¹⁾_I (x) is Fr´echet-differentiable at x, we obtain λ⁽¹⁾_I (x + th) = λ⁽¹⁾_I (x) + t · h^T∇_xλ⁽¹⁾_I (x) + o(t).

Let y = h^T∇_xλ⁽¹⁾_I (x)+^o(t)_t , then y → h^T∇_xλ⁽¹⁾_I (x) as t → 0⁺. If h^T∇_xλ⁽¹⁾_I (x) <

0, then y < 0 as t is sufficiently close to 0 and we obtain

lim

t→0⁺

f (λ⁽¹⁾_I (x + th)) − f (λ⁽¹⁾_I (x)) t

= lim

t→0⁺

f (λ⁽¹⁾_I (x) + ty) − f (λ⁽¹⁾_I (x)) t

= lim

t→0⁺

f (λ⁽¹⁾_I (x) − (−ty) · 1) − f (λ⁽¹⁾_I (x))

(−ty) · (−y)

= lim

t→0⁺

f (λ⁽¹⁾_I (x) − (−ty) · 1) − f (λ⁽¹⁾_I (x))

(−ty) · lim

t→0⁺(−y)

= f⁰(λ⁽¹⁾_I (x); −1) · (−h^T∇_xλ⁽¹⁾_I (x))

= f⁰(λ⁽¹⁾_I (x); h^T∇_xλ⁽¹⁾_I (x)),

where the last equation follows from the positive homogeneous property of directionally differentiable functions. On the other hand, we can also deduce a similar result

lim

t→0⁺

f (λ⁽¹⁾_I (x + th)) − f (λ⁽¹⁾_I (x))

t = f⁰(λ⁽¹⁾_I (x); h^T∇_xλ⁽¹⁾_I (x))

under the case h^T∇_xλ⁽¹⁾_I (x) ≥ 0. Therefore, the directional differentiability of f (λ⁽¹⁾_I (x)) with respect to x ∈ Rⁿ is fulfilled and so does f (λ⁽²⁾_I (x)) by repeating the above procedure. Consequently, we obtain

(f_I^EC)⁰(x; h)

= f⁰(λ⁽¹⁾_I (x); h^T∇_xλ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) + f⁰(λ⁽²⁾_I (x); h^T∇_xλ⁽²⁾_I (x)) · u⁽²⁾_I (x) +f (λ⁽¹⁾_I (x)) · (∇xu⁽¹⁾_I (x))^Th + f (λ⁽²⁾_I (x)) · (∇xu⁽²⁾_I (x))^Th,

(3.2)

where the terms h^T∇_xλ⁽ⁱ⁾_I (x), (∇_xu⁽ⁱ⁾_I (x))^Th, (i = 1, 2) are defined as fol- lows:

h^T∇_xλ⁽¹⁾_I (x) = u^T_nh + 1

k ¯M ¯U_n−1^T xkx^T( ¯Un−1M¯^TM ¯¯U_n−1^T )h, h^T∇_xλ⁽²⁾_I (x) = u^T_nh − 1

k ¯M ¯U_n−1^T xkx^T( ¯U_n−1M¯^TM ¯¯U_n−1^T )h, (∇xu⁽¹⁾_I (x))^Th = 1

2k ¯M ¯U_n−1^T xk( ¯Un−1U¯_n−1^T )

"

In−(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) k ¯M ¯U_n−1^T xk²

# h, (∇xu⁽²⁾_I (x))^Th = − 1

2k ¯M ¯U_n−1^T xk( ¯Un−1U¯_n−1^T )

"

In−(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) k ¯M ¯U_n−1^T xk²

# h.

Hence, we obtain

f⁰(λ⁽¹⁾_I (x); h^T∇_xλ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) + f⁰(λ⁽²⁾_I (x); h^T∇_xλ⁽²⁾_I (x)) · u⁽²⁾_I (x)

= 1

2 · f⁰(λ⁽¹⁾_I (x); u^T_nh + 1

k ¯M ¯U_n−1^T xkx^T( ¯Un−1M¯^TM ¯¯U_n−1^T )h)

− f⁰(λ⁽²⁾_I (x); u^T_nh − 1

k ¯M ¯U_n−1^T xkx^T( ¯U_n−1M¯^TM ¯¯U_n−1^T )h)

!

·U¯_n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk +1

2 · f⁰(λ⁽¹⁾_I (x); u^T_nh + 1

k ¯M ¯U_n−1^T xkx^T( ¯Un−1M¯^TM ¯¯U_n−1^T )h) + f⁰(λ⁽²⁾_I (x); u^T_nh − 1

k ¯M ¯U_n−1^T xkx^T( ¯U_n−1M¯^TM ¯¯U_n−1^T )h)

!

· u_n (3.3)

and

f (λ⁽¹⁾_I (x)) · (∇_xu⁽¹⁾_I (x))^Th + f (λ⁽²⁾_I (x)) · (∇_xu⁽²⁾_I (x))^Th

= f (λ⁽¹⁾_I (x)) − f (λ⁽²⁾_I (x))

2k ¯M ¯U_n−1^T xk · ( ¯Un−1U¯_n−1^T )

"

In−(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) k ¯M ¯U_n−1^T xk²

# h

= f (λ⁽¹⁾_I (x)) − f (λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

· ( ¯Un−1U¯_n−1^T )

"

In−(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) k ¯M ¯U_n−1^T xk²

# h, (3.4)

where the last equation uses the relation λ⁽¹⁾_I (x) − λ⁽²⁾_I (x) = 2k ¯M ¯U_n−1^T xk.

From (3.2), we can rewrite (f_I^EC)⁰(x; h) in a more compact form as below:

(f_I^EC)⁰(x; h)

= f⁰(λ⁽¹⁾_I (x); h^T∇_xλ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) + f⁰(λ⁽²⁾_I (x); h^T∇_xλ⁽²⁾_I (x)) · u⁽²⁾_I (x) +f (λ⁽¹⁾_I (x)) − f (λ⁽²⁾_I (x))

λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

· ( ¯U_n−1U¯_n−1^T )

"

I_n−(xx^T)( ¯U_n−1M¯^TM ¯¯U_n−1^T ) k ¯M ¯U_n−1^T xk²

# h.

(3.5)

Case (b): If ¯U_n−1^T x = 0, then we have

f_I^EC(x) = f (λ⁽¹⁾_I (x)) · u⁽¹⁾_I (x) + f (λ⁽²⁾_I (x)) · u⁽²⁾_I (x),

where the scalars λ⁽¹⁾_I (x), λ⁽²⁾_I (x) and the vectors u⁽¹⁾_I (x), u⁽²⁾_I (x) are given by

λ⁽¹⁾_I (x) = u^T_nx, λ⁽²⁾_I (x) = u^T_nx, u⁽¹⁾_I (x) = ¹₂·_U¯

n−1w

k ¯M wk + u_n

, u⁽²⁾_I (x) = ¹₂ ·

−^U¯_{k ¯}^n−1w

M wk + u_n ,

where w is any given nonzero vector in Rⁿ⁻¹. If ¯U_n−1^T h 6= 0, then ¯U_n−1^T (x + th) 6= 0 and f_I^EC(x + th) has the following decomposition

f_I^EC(x + th) = f (λ⁽¹⁾_I (x + th)) · u⁽¹⁾_I (x + th) + f (λ⁽²⁾_I (x + th)) · u⁽²⁾_I (x + th),

where the scalars λ⁽¹⁾_I (x+th), λ⁽²⁾_I (x+th) and the vectors u⁽¹⁾_I (x+th), u⁽²⁾_I (x+

th) satisfy the following relations

λ⁽¹⁾_I (x + th) = u^T_n(x + th) + k ¯M ¯U_n−1^T (x + th)k = λ⁽¹⁾_I (x) + tλ⁽¹⁾_I (h), λ⁽²⁾_I (x + th) = u^T_n(x + th) − k ¯M ¯U_n−1^T (x + th)k = λ⁽²⁾_I (x) + tλ⁽²⁾_I (h), u⁽¹⁾_I (x + th) = 1

2· ¯Un−1U¯_n−1^T (x + th) k ¯M ¯U_n−1^T (x + th)k + un

!

:= u⁽¹⁾_I (h), u⁽¹⁾_I (x + th) = 1

2· −U¯_n−1U¯_n−1^T (x + th) k ¯M ¯U_n−1^T (x + th)k + u_n

!

:= u⁽²⁾_I (h).

In addition, if we choose w = ¯U_n−1^T h, then u⁽¹⁾_I (x) = u⁽¹⁾_I (x + th) = u⁽¹⁾_I (h) and u⁽²⁾_I (x) = u⁽²⁾_I (x + th) = u⁽²⁾_I (h), which show that

f_I^EC(x + th) − f_I^EC(x) t

= f (λ⁽¹⁾_I (x) + tλ⁽¹⁾_I (h)) − f (λ⁽¹⁾_I (x))

t · u⁽¹⁾_I (h)

+f (λ⁽²⁾_I (x) + tλ⁽²⁾_I (h)) − f (λ⁽²⁾_I (x))

t · u⁽²⁾_I (h).

Therefore, the following relation is fulfilled under the directionally differen- tiability of f at λ⁽ⁱ⁾_I (x) (i = 1, 2):

(3.6)

(f_I^EC)⁰(x; h) = f⁰(λ⁽¹⁾_I (x); λ⁽¹⁾_I (h)) · u⁽¹⁾_I (h) + f⁰(λ⁽²⁾_I (x); λ⁽²⁾_I (h)) · u⁽²⁾_I (h).

On the other hand, if ¯U_n−1^T h = 0, then ¯U_n−1^T (x + th) = 0. In this case, we know

f_I^EC(x + th) = f (λ⁽¹⁾_I (x + th)) · u⁽¹⁾_I (x + th) + f (λ⁽²⁾_I (x + th)) · u⁽²⁾_I (x + th), where the scalars λ⁽¹⁾_I (x+th), λ⁽²⁾_I (x+th) and the vectors u⁽¹⁾_I (x+th), u⁽²⁾_I (x+

th) now can be rewritten as follows:

λ⁽¹⁾_I (x + th) = u^T_n(x + th) = λ⁽¹⁾_I (x) + tλ⁽¹⁾_I (h), λ⁽²⁾_I (x + th) = u^T_n(x + th) = λ⁽²⁾_I (x) + tλ⁽²⁾_I (h), u⁽¹⁾_I (x + th) = 1

2 ·

U¯_n−1η k ¯M ηk + u_n



:= u⁽¹⁾_I , u⁽¹⁾_I (x + th) = 1

2 ·



−U¯_n−1η k ¯M ηk + u_n



:= u⁽²⁾_I ,

where η is any given nonzero vector in Rⁿ⁻¹. In addition, if we choose w = η 6= 0, then u⁽¹⁾_I (x) = u⁽¹⁾_I (x + th) = u⁽¹⁾_I , u⁽²⁾_I (x) = u⁽²⁾_I (x + th) = u⁽²⁾_I . Similarly, we have

(3.7) (f_I^EC)⁰(x; h) = f⁰(λ⁽¹⁾_I (x); λ⁽¹⁾_I (h)) · u⁽¹⁾_I + f⁰(λ⁽²⁾_I (x); λ⁽²⁾_I (h)) · u⁽²⁾_I .

In summary, we show that f_I^EC is directionally differentiable at x ∈ Rⁿwith spectral values λ⁽ⁱ⁾_I (x) (i = 1, 2).

(b) ⇒ (a) The proof for this direction is trivial by adapting the arguments

for [5, Proposition 3]. 

Theorem 3.3 (Differentiable). For any given function f : R → R, let f_I^EC be its corresponding cone function defined as in (1.6). Then, the fol- lowing statements are equivalent to each other.

(a) f is differentiable at λ⁽ⁱ⁾_I (x) (i = 1, 2).

(b) f_I^EC is differentiable at x ∈ Rⁿwith spectral values λ⁽ⁱ⁾_I (x) (i = 1, 2).

Moreover, the corresponding Jacobian of f_I^EC at x is defined as follows:

Df_I^EC(x)

= ( ¯Un−1U¯_n−1^T )

"

f⁰(λ⁽¹⁾_I (x)) − f⁰(λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

(xu^T_n)

+2 · f⁰(λ⁽¹⁾_I (x)) + f⁰(λ⁽²⁾_I (x)) (λ⁽¹⁾_I (x) − λ⁽²⁾_I (x))²

(xx^T)( ¯U_n−1M¯^TM ¯¯U_n−1^T )

+f (λ⁽¹⁾_I (x)) − f (λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

In− 4 ·(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) (λ⁽¹⁾_I (x) − λ⁽²⁾_I (x))²

!#

+(u_nu^T_n)

"

f⁰(λ⁽¹⁾_I (x)) − f⁰(λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

(u_nx^T)( ¯U_n−1M¯^TM ¯¯U_n−1^T )

+f⁰(λ⁽¹⁾_I (x)) + f⁰(λ⁽²⁾_I (x))

2 In

# (3.8)

if ¯U_n−1^T x 6= 0; otherwise,

(3.9) Df_I^EC(x) = f⁰(u^T_nx)In.

Proof. (a) ⇒ (b) The proof for this direction can be adapted from Theo- rem 3.2, in which we only need to use “differentiable” to replace “direction- ally differentiable”. At the same time, we know that f⁰(λ⁽ⁱ⁾_I (x), ·) (i = 1, 2) are linear, in other words,

(3.10) f⁰(λ⁽ⁱ⁾_I (x), a + b) = f⁰(λ⁽ⁱ⁾_I (x))a + f⁰(λ⁽ⁱ⁾_I (x))b, ∀a, b ∈ R, since f is differentiable at λ⁽ⁱ⁾_I (x) (i = 1, 2).

Next, the remaining part will be verified under the following two cases:

Case (a): If ¯U_n−1^T x 6= 0, according to the relations (3.3)-(3.5) and (3.10), then we have

(f_I^EC)⁰(x; h)

= 1

2 h

(f⁰(λ⁽¹⁾_I (x)) − f⁰(λ⁽²⁾_I (x)))u^T_nh +f⁰(λ⁽¹⁾_I (x)) + f⁰(λ⁽²⁾_I (x))

k ¯M ¯U_n−1^T xk x^T( ¯Un−1M¯^TM ¯¯U_n−1^T )h

#U¯n−1U¯_n−1^T x k ¯M ¯U_n−1^T xk +1

2 h

(f⁰(λ⁽¹⁾_I (x)) + f⁰(λ⁽²⁾_I (x)))u^T_nh +f⁰(λ⁽¹⁾_I (x)) − f⁰(λ⁽²⁾_I (x))

k ¯M ¯U_n−1^T xk x^T( ¯Un−1M¯^TM ¯¯U_n−1^T )h

# un

+f (λ⁽¹⁾_I (x)) − f (λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

· ( ¯Un−1U¯_n−1^T )

"

In−(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) k ¯M ¯U_n−1^T xk²

# h

= ( ¯U_n−1U¯_n−1^T )

"

f⁰(λ⁽¹⁾_I (x)) − f⁰(λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

(xu^T_n)

+2 · f⁰(λ⁽¹⁾_I (x)) + f⁰(λ⁽²⁾_I (x))

(λ⁽¹⁾_I (x) − λ⁽²⁾_I (x))² (xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) +f (λ⁽¹⁾_I (x)) − f (λ⁽²⁾_I (x))

λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

I_n− 4 ·(xx^T)( ¯Un−1M¯^TM ¯¯U_n−1^T ) (λ⁽¹⁾_I (x) − λ⁽²⁾_I (x))²

!#

h

+(u_nu^T_n)

"

f⁰(λ⁽¹⁾_I (x)) − f⁰(λ⁽²⁾_I (x)) λ⁽¹⁾_I (x) − λ⁽²⁾_I (x)

(u_nx^T)( ¯U_n−1M¯^TM ¯¯U_n−1^T )

+f⁰(λ⁽¹⁾_I (x)) + f⁰(λ⁽²⁾_I (x))

2 In

# h,

where the last equation follows from the fact λ⁽¹⁾_I (x)−λ⁽²⁾_I (x) = 2k ¯M ¯U_n−1^T xk.

The above relation shows that (f_I^EC)⁰(x; h) = Df_I^EC(x)h, where Df_I^EC(x) is defined as in (3.8).