Nonlinear Analysis

Nonlinear Analysis 85 (2013) 160–173

Contents lists available atSciVerse ScienceDirect

Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Smooth and nonsmooth analyses of vector-valued functions associated with circular cones

Yu-Lin Chang, Ching-Yu Yang, Jein-Shan Chen

^∗

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

a r t i c l e i n f o

Article history:

Received 10 January 2013 Accepted 29 January 2013 Communicated by S. Carl

MSC:

26A27 26B05 26B35 49J52 90C33 65K05 Keywords:

Circular cone Vector-valued function Semismooth function Complementarity Spectral decomposition

a b s t r a c t

LetL_θbe the circular cone in Rⁿwhich includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function f^c(^x) on Rⁿby applying f to the spectral values of the spectral decomposition of x ∈ _Rⁿwith respect toL_θ. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The circular cone [1,2] is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation. Let its half-aperture angle be

θ

^with

θ ∈ (

⁰

,

^π₂

)

. Then, the n-dimensional circular cone denoted byL_θ can be expressed as

L_θ

:= 

x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹

| ∥

x

∥

cos

θ ≤

^x1

 := 

x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹

| ∥

x₂

∥

cot

θ ≤

^x1

 .

⁽¹⁾

SeeFig. 1.

When

θ =

⁴⁵°, the circular cone reduces to the well-known second-order cone (SOC, also called Lorentz cone) given by Kⁿ

:= 

x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹

| ∥

x₂

∥ ≤

x₁

 := 

(

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹

| ∥

x

∥

cos 45°

≤

x₁

 .

∗Correspondence to: Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office, Taiwan. Tel.: +886 2 29325417; fax: +886 2 29332342.

E-mail addresses:[email protected](Y.-L. Chang),[email protected](C.-Y. Yang),[email protected],[email protected] (J.-S. Chen).

0362-546X/$ – see front matter©2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.na.2013.01.017

(a) 0< θ <⁴⁵°. (b)θ =⁴⁵°. (c) 45°< θ <⁹⁰°. Fig. 1. The graphs of circular cones.

Fig. 2. The grasping force forms a circular cone whereα =^tan−1µ <⁴⁵°^.

With respect to SOC, for any x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹, we can decompose x as

x

= λ

1

(

^x

)

^u(x¹⁾

+ λ

2

(

^x

)

^u(x²⁾

,

⁽²⁾

where

λ

1

(

^x

)

^,

λ

2

(

^x

)

^{and u(}x¹⁾, u⁽_x²⁾are the spectral values and the associated spectral vectors of x with respect toKⁿ, given by

λ

i

(

^x

) =

^x1

+ (−

¹

)

ⁱ

∥

x₂

∥ ,

u⁽_xⁱ⁾

=



 

 

1 2



1

, (−

¹

)

^{i x2}

∥

x₂

∥ ,

^{if x}2

̸=

0

,

1

2



1

, (−

¹

)

ⁱ

w,

^{if x}2

=

0

,

for i

=

1

,

^{2 with}

w

being any vector in Rⁿ⁻¹satisfying

∥ w∥ =

^{1. If x}2

̸=

0, the decomposition(2)is unique. With this spectral decomposition(2), for any function f

:

_R

→

R, the following vector-valued function associated withKⁿ(n

≥

1) is considered (see [3,4]):

f^soc

(

^x

) =

^f

(λ

1

)

^u(1)

+

f

(λ

2

)

^u(2)

∀

x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹

.

⁽³⁾ If f is defined only on a subset of R, then f^socis defined on the corresponding subset of Rⁿ. The definition(3)is unambiguous whether x₂

̸=

0 or x₂

=

0. The above definition(3)is analogous to the one associated with the semidefinite coneSⁿ; see [5,6]. It was shown [4] that the properties of continuity, strict continuity, Lipschitz continuity, directional differentiability, differentiability, continuous differentiability, and semismoothness are each inherited by f^socfrom f . These results are useful in the design and analysis of smooth and nonsmooth methods for solving second-order cone programs (SOCP) and second- order cone complementarity problem (SOCCP); see [3,4,7,8] and references therein.

Recently, there have been found circular cone constraints involved in real engineering problems. For example, in the formulation of optimal grasping manipulation for multi-fingered robots, the grasping force of the i-th finger is subject to a contact friction constraint expressed as

| (

^ui1

,

^ui3

)| ≤ µ

^ui1 (4)

where

µ

is the friction coefficient; seeFig. 2.

Indeed,(4)is a circular cone constraint corresponding to u_i

= (

^ui1

,

^ui2

,

^ui3

) ∈

^Lθ with

θ =

^tan−1

µ <

⁴⁵°^{. Note} that the circular coneL_θ is a non-self-dual (or non-symmetric) cone and its related study is rather limited. Nonetheless,

162 Y.-L. Chang et al. / Nonlinear Analysis 85 (2013) 160–173

motivated by the real world application regarding circular cone, the structures and properties aboutL_θ are investigated in [2]. In particular, the spectral factorization of z associated with the circular cone is characterized in [2, Theorem 3.1]. For convenience, we restate it as follows.

Theorem 1.1 ([2, Theorem 3.1]). For any z

= (

^z1

,

^z2

) ∈

R

×

_Rⁿ⁻¹, one has

z

= λ

1

(

^z

) ·

^u(z¹⁾

+ λ

2

(

^z

) ·

^u(z²⁾ (5)

where

 λ

1

(

^z

) =

^z1

− ∥

z₂

∥

cot

θ

λ

2

(

^z

) =

^z1

+ ∥

z₂

∥

tan

θ

⁽⁶⁾

and



 



 



u⁽_z¹⁾

=

¹ 1

+

cot²

θ



1 0 0 cot

θ

 

1

− w



=



sin²

θ

− (

^sin

θ

^cos

θ)w



u⁽_z²⁾

=

¹ 1

+

tan²

θ



1 0 0 tan

θ

 

1

w



=



cos²

θ (

^sin

θ

^cos

θ)w



⁽⁷⁾

with

w =

∥^zz²2∥if z₂

̸=

0, and any vector in Rⁿ⁻¹satisfying

∥ w∥ =

^{1 if z}2

=

0.

Analogous to(3), with the spectral factorization(5), for any function f

:

_R

→

R, we consider the following vector-valued function associated withL_θ (n

≥

1):

f^c

(

^z

) =

^f

(λ

1

)

^u(z¹⁾

+

_f

(λ

2

)

^u(z²⁾

∀

_z

= (

^z1

,

^z2

) ∈

R

×

_Rⁿ⁻¹

.

⁽⁸⁾ Can the properties of continuity, strict continuity, Lipschitz continuity, directional differentiability, differentiability, continuous differentiability, and semismoothness be each inherited by f^cfrom f ? These are what we want to explore in this paper.

At last, we say a few words about notations. In what follows, for any differentiable (in the Fréchet sense) mapping F

:

_Rⁿ

→

_R^m, we denote its Jacobian (not transposed) at x

∈

_Rⁿby

∇

F

(

^x

) ∈

R^m×n, i.e.,

(

^F

(

^x

+

u

)−

^F

(

^x

)−∇

^F

(

^x

)

^u

)/∥

^u

∥ →

0 as u

→

0. ‘‘

:=

’’ means ‘‘define’’. We write z

=

O

(α)

(respectively, z

=

o

(α)

^{), with}

α ∈

R and z

∈

_Rⁿ, to mean

∥

z

∥ /|α|

^is uniformly bounded (respectively, tends to zero) as

α →

^0.

2. Preliminaries

In this section, we review some basic concepts regarding vector-valued functions. These contain continuity, (local) Lipschitz continuity, directional differentiability, differentiability, continuous differentiability, as well as semismoothness.

Suppose F

:

_Rⁿ

→

_R^m. Then, F is continuous at x

∈

_Rⁿif F

(

^y

) →

^F

(

^x

)

^{as y}

→

x, and F is continuous if F is continuous at every x

∈

_Rⁿ. We say F is strictly continuous (also called ‘‘locally Lipschitz continuous’’) at x

∈

_Rⁿif there exist scalars

κ >

^{0 and}

δ >

0 such that

∥

F

(

^y

) −

^F

(

^z

)∥ ≤ κ∥

^y

−

z

∥ ∀

y

,

^z

∈

_Rⁿwith

∥

y

−

x

∥ ≤ δ, ∥

^z

−

x

∥ ≤ δ;

and F is strictly continuous if F is strictly continuous at every x

∈

_Rⁿ. We say F is directionally differentiable at x

∈

_Rⁿif F^′

(

^x

;

h

) :=

^lim

t→0+

F

(

^x

+

th

) −

^F

(

^x

)

t exists

∀

h

∈

_Rⁿ

;

and F is directionally differentiable if F is directionally differentiable at every x

∈

_Rⁿ. F is differentiable (in the Fréchet sense) at x

∈

_Rⁿif there exists a linear mapping

∇

F

(

^x

) :

Rⁿ

→

_R^msuch that

F

(

^x

+

h

) −

^F

(

^x

) − ∇

^F

(

^x

)

^h

=

o

(∥

^h

∥ ).

If F is differentiable at every x

∈

_Rⁿ_and

∇

F is continuous, then F is continuously differentiable. We notice that, in the above expression about strict continuity of F , if

δ

can be taken to be

∞

, then F is called Lipschitz continuous with Lipschitz constant

κ

^.

It is well-known that if F is strictly continuous, then F is almost everywhere differentiable by Rademacher’s Theorem;

see [9] and [10, Section 9J]. In this case, the generalized Jacobian

∂

^F

(

^x

)

of F at x (in the Clarke sense) can be defined as the convex hull of the generalized Jacobian

∂

BF

(

^x

)

^{, where}

∂

BF

(

^x

) :=



lim

x^j→x

∇

F

(

^xj

) |

F is differentiable at x^j

∈

_Rⁿ



.

The notation

∂

Bis adopted from [11]. In [10, Chapter 9], the case of m

=

1 is considered and the notations ‘‘

∇ ¯

’’ and ‘‘

∂ ¯

^{’’ are} used instead of, respectively, ‘‘

∂

B’’ and ‘‘

∂

’’. Assume F

:

_Rⁿ

→

_R^mis strictly continuous, then F is said to be semismooth at x if F is directionally differentiable at x and, for any V

∈ ∂

^F

(

^x

+

h

)

^{, we have}

F

(

^x

+

h

) −

^F

(

^x

) −

^Vh

=

o

(∥

^h

∥ ).

Moreover, F is called

ρ

-order semismooth at x (0

< ρ < ∞

) if F is semismooth at x and, for any V

∈ ∂

^F

(

^x

+

h

)

^{, we have} F

(

^x

+

h

) −

^F

(

^x

) −

^Vh

=

O

(∥

^h

∥

^1+ρ

).

The following lemma, proven by Sun and Sun [5, Theorem 3.6] using the definition of generalized Jacobian, enables one to study the semismooth property of f^cby examining only those points x

∈

_Rⁿwhere f^cis differentiable and thus work only with the Jacobian of f^c, rather than the generalized Jacobian. It is a very useful working lemma for verifying semismoothness property in Section4.

Lemma 2.1. Suppose F

:

_Rⁿ

→

_Rⁿis strictly continuous and directionally differentiable in a neighborhood of x

∈

_Rⁿ. Then, for any 0

< ρ < ∞

, the following two statements are equivalent.

(a) For any

v ∈ ∂

^F

(

^x

+

h

)

^{and h}

→

0,

F

(

^x

+

h

) −

^F

(

^x

) − v

^h

=

o

(∥

^h

∥ ) (

respectively, O

(∥

^h

∥ )

^1+ρ

).

(b) For any h

→

0 such that F is differentiable at x

+

h,

F

(

^x

+

h

) −

^F

(

^x

) − ∇

^F

(

^x

+

h

)

^h

=

o

(∥

^h

∥ ) (

respectively

,

^O

(∥

^h

∥ )

^1+ρ

).

We say F is semismooth (respectively,

ρ

-order semismooth) if F is semismooth (respectively,

ρ

-order semismooth) at every x

∈

_Rⁿ. We say F is strongly semismooth if it is 1-order semismooth. Convex functions and piecewise continuously differentiable functions are examples of semismooth functions. The composition of two (respectively,

ρ

-order) semismooth functions is also a (respectively,

ρ

-order) semismooth function. The property of semismoothness, as introduced by Mifflin [12] for functionals and scalar-valued functions and further extended by Qi and Sun [13] for vector-valued functions, is of particular interest due to the key role it plays in the superlinear convergence analysis of certain generalized Newton methods [11,13–16]. For extensive discussions of semismooth functions, see [12,13,17].

3. Properties of continuity and differentiability

In this section, we focus on the properties of continuity and differentiability between f and f^c. We need some technical lemmas which come from the simple structure of the circular cone and basic definitions before starting the proofs.

Lemma 3.1. Let

λ

1

≤ λ

2be the spectral values of x

∈

_Rⁿand m₁

≤

m₂be the spectral values of y

∈

_Rⁿ. Then, we have

| λ

1

−

m₁

|

²sin²

θ + |λ

2

−

m₂

|

²cos²

θ = ∥

^x

−

y

∥

²

,

⁽⁹⁾ and hence,

| λ

i

−

m_i

| ≤

c

∥

x

−

y

∥ , ∀

ⁱ

=

1

,

^{2, where c}

=

max

{

sec

θ,

^csc

θ}

^.

Proof. The proof follows from a direct computation. 

Lemma 3.2. Let x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹and y

= (

^y1

,

^y2

) ∈

R

×

_Rⁿ⁻¹. (a) If x₂

̸=

0

,

^y2

̸=

0, then we have

∥

u⁽ⁱ⁾

− v

⁽ⁱ⁾

∥ ≤

^{2 sin cos}

θ

∥

x₂

∥ ∥

x

−

y

∥ ,

ⁱ

=

1

,

²

,

⁽¹⁰⁾

where u⁽ⁱ⁾

, v

⁽ⁱ⁾are the unique spectral vectors of x and y, respectively.

(b) If either x₂

=

0 or y₂

=

0, then we can choose u⁽ⁱ⁾

, v

⁽ⁱ⁾such that the left hand side of inequality(10)is zero.

Proof. (a) From the spectral factorization(5), we know that u⁽¹⁾

=

sin²

θ



1

, (−

¹

)

^cot

θ ∥

^xx²₂

∥



, v

⁽¹⁾

=

sin²

θ



1

, (−

¹

)

^cot

θ ∥

^yy²₂

∥



,

164 Y.-L. Chang et al. / Nonlinear Analysis 85 (2013) 160–173

where u⁽¹⁾

, v

⁽¹⁾are unique. This gives u⁽¹⁾

− v

⁽¹⁾

=

sin²

θ 

0

, (−

¹

)

^cot

θ(

∥^xx²2∥

−

^y2

∥y2∥

) 

. Then,

∥

u⁽¹⁾

− v

⁽¹⁾

∥ =

sin

θ

^cos

θ









x₂

∥

x₂

∥ −

^y2

∥

y₂

∥









=

_sin

θ

^cos

θ









x₂

−

y₂

∥

x₂

∥ + (∥

^y2

∥ − ∥

x₂

∥ )

^y2

∥

x₂

∥ · ∥

y₂

∥









≤

sin

θ

^cos

θ



1

∥

x₂

∥ ∥

x₂

−

y₂

∥ +

¹

∥

x₂

∥ |∥

y₂

∥ − ∥

x₂

∥|



≤

sin

θ

^cos

θ



1

∥

_x2

∥ ∥

x₂

−

y₂

∥ +

¹

∥

_x2

∥ ∥

x₂

−

y₂

∥



≤

^{2 sin}

θ

^cos

θ

∥

x₂

∥ ∥

x

−

y

∥ ,

where the inequalities follow from the triangle inequality. Similar arguments apply for

∥

u⁽²⁾

− v

⁽²⁾

∥

.

(b) We can choose the same spectral vectors for x and y from the spectral factorization(5)since either x₂

=

0 or y₂

=

0.

Then, it is obvious. _

Lemma 3.3. For any

w ̸=

⁰

∈

_Rⁿ, we have

∇

_w



w

∥w∥



=

¹

∥w∥



I

−

^wwT

∥w∥²



.

Proof. See [18, Lemma 3.3] or check it by direct computation. _

Now, we are ready to present our first main result about continuity between f and f^c.

Theorem 3.1. For any f

:

_R

→

_{R, f}^cis continuous at x

∈

_Rⁿwith spectral values

λ

1

, λ

2if and only if f is continuous at

λ

1

, λ

2. Proof. ‘‘

⇐

’’ Suppose f is continuous at

λ

1

, λ

2. For any fixed x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹and y

→

x, let the spectral factorizations of x

,

^{y be x}

= λ

1u⁽¹⁾

+ λ

2u⁽²⁾and y

=

m₁

v

⁽¹⁾

+

m₂

v

⁽²⁾, respectively. Then, we discuss two cases.

Case (i). If x₂

̸=

0, then we have

f^c

(

^y

) −

^fc

(

^x

) =

^f

(

^m1

) v

⁽¹⁾

−

u⁽¹⁾

 +

[f

(

^m1

) −

^f

(λ

1

)

^{] u(1)}

+

f

(

^m2

) v

⁽²⁾

−

u⁽²⁾

 +

[f

(

^m2

) −

^f

(λ

2

)

^{] u(2)}

.

⁽¹¹⁾ Since f is continuous at

λ

1

, λ

2, and fromLemma 3.1,

|

m_i

− λ

i

| ≤

c

∥

y

−

x

∥

, we know that f

(

^mi

) −→

^f

(λ

i

)

^{as y}

→

x. In addition, byLemma 3.2, we have

∥ v

⁽ⁱ⁾

−

u⁽ⁱ⁾

∥ −→

0 as y

→

x

.

^{Thus, Eq.(11)yields fc}

(

^y

) −→

^fc

(

^x

)

^{as y}

→

x because both f

(

^mi

)

^and

∥

u⁽ⁱ⁾

∥

are bounded. Hence, f^cis continuous at x

∈

_Rⁿ.

Case (ii). If x₂

=

0, no matter y₂is zero or not, we can arrange that x

,

y have the same spectral vectors. Thus, f^c

(

^y

) −

^fc

(

^x

) =

[f

(

^m1

) −

^f

(λ

1

)

^{] u(1)}

+

[f

(

^m2

) −

^f

(λ

2

)

^{] u(2)}

.

^{Then, fc}is continuous at x

∈

_Rⁿby similar arguments.

‘‘

⇒

’’ The proof for this direction is straightforward or refer to similar arguments for [4, Prop. 2]. _

Theorem 3.2. For any f

:

_R

→

_{R, f}^cis directionally differentiable at x

∈

_Rⁿwith spectral values

λ

1,

λ

2if and only if f is directionally differentiable at

λ

1,

λ

2.

Proof. ‘‘

⇐

’’ Suppose f is directionally differentiable at

λ

1

, λ

2. Fix any x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹, then we discuss two cases as below.

Case (i). If x₂

̸=

0, we have f^c

(

^x

) =

^f

(λ

1

)

^u(1)

+

_f

(λ

2

)

^{u(2) where}

λ

i

=

_x1

+ (−

¹

)

ⁱ

(

^tan

θ)

⁽⁻¹⁾ⁱ

∥

_x2

∥

_{and u}⁽ⁱ⁾

= (−

¹

)

^isin

θ

^cos

θ (

^tan

θ)

⁽⁻¹⁾ⁱ

,

∥^xx^T22∥



for all i

=

1

,

^{2. From}Lemma 3.3, we know that u⁽ⁱ⁾is Fréchet-differentiable with respect to x, with

∇

_xu⁽ⁱ⁾

= (−

¹

)

^isin

θ

^cos

θ

∥

x₂

∥





0 0

0 I

−

^x2x

T 2

∥

x₂

∥

²



 ∀

i

=

1

,

²

.

⁽¹²⁾

Also by the expression of

λ

i, we know that

λ

iis Fréchet-differentiable with respect to x, with

∇

_x

λ

i

=



1

, (−

¹

)

^itan(−1)i

θ ∥

^xx^T2₂

∥



∀

i

=

1

,

²

.

⁽¹³⁾

In general, we cannot apply the chain rule, when functions are only directionally differentiable. But, it works well for single- variable functions, that is, when single-variable functions are composed of a differentiable function. From the hypothesis, f

is directionally differentiable at

λ

1, then it is easy to compute lim

t→0+

f

(λ

1

+

t

×

1

) −

^f

(λ

1

)

t

=

f^′

(λ

1

;

1

),

lim

t→0+

f

(λ

1

−

t

×

1

) −

^f

(λ

1

)

t

=

f^′

(λ

1

; −

1

),

lim

t→₀₊

f

(λ

1

+

o

(

^t

)) −

^f

(λ

1

)

t

=

0

.

Note that the spectral value function

λ

1

(

^x

) =

^x1

−

cot

θ∥

^x2

∥

is differentiable when x₂

̸=

0, which yields

λ

1

(

^x

+

th

) = λ

1

(

^x

) +

^t

∇

_x

λ

1h

+

o

(

^t

).

Let y

:= ∇

_x

λ

1h

+

^o(t)

t . For the case of

∇

_x

λ

1h

<

0, we know y

<

0 as t is small. Thus, lim

t→0+

f

(λ

1

(

^x

+

th

)) −

^f

(λ

1

(

^x

))

t

=

lim

t→0+

f

(λ

1

(

^x

) +

^ty

) −

^f

(λ

1

(

^x

))

t

=

lim

t→₀₊

f

(λ

1

(

^x

) − (−

^ty

)) −

^f

(λ

1

(

^x

))

−

ty

(−

^y

) =

^lim

−_ty→₀₊

f

(λ

1

(

^x

) − (−

^ty

)) −

^f

(λ

1

(

^x

))

−

ty lim

t→₀₊

(−

^y

)

=

f^′

(λ

1

(

^x

); −

¹

)(−∇

x

λ

1h

) =

^f′

(λ

1

(

^x

); ∇

x

λ

1h

).

Here the positively homogeneous property of directionally differentiable functions is used in the last equation. Similarly, for the other case of

∇

_x

λ

1h

≥

0, we have

lim

t→0+

f

(λ

1

(

^x

+

th

)) −

^f

(λ

1

(

^x

))

t

=

f^′

(λ

1

(

^x

); ∇

x

λ

1h

).

In summary, the composite function f

◦ λ

1

(·)

is directionally differentiable at x. Now we can apply the chain rule and the product rule on f^c

(

^x

) =

^f

(λ

1

)

^u(1)

+

f

(λ

2

)

^u(2). In other words,

(

^fc

)

^′

(

^x

;

h

) =

^f

(λ

1

)∇

xu⁽¹⁾h

+

f^′

(λ

1

; ∇

_x

λ

1h

)

^u(1)

+

f

(λ

2

)∇

xu⁽²⁾h

+

f^′

(λ

2

; ∇

_x

λ

2h

)

^u(2)

= (

^A1

,

^A2

) ∈

R

×

_Rⁿ⁻¹

,

where

A₁

=

f^′



λ

1

;

h₁

−

cot

θ

^x

∥

^T2x^h₂

∥

²



sin²

θ +

^f′



λ

2

;

h₁

+

tan

θ ∥

^xT2x^h₂

∥

²



cos²

θ

⁽¹⁴⁾

and A₂

=



f^′



λ

2

;

h₁

+

tan

θ ∥

^xT2x^h₂

∥

²



−

f^′



λ

1

;

h₁

−

cot

θ ∥

^xT2x^h₂

∥

²



sin

θ

^cos

θ ∥

^xx²₂

∥

+

^f

(λ

2

) −

^f

(λ

1

) λ

2

− λ

1



I

−

^x2x

T 2

∥

x₂

∥

²



h₂

,

⁽¹⁵⁾

with h

= (

^h1

,

^h2

) ∈

R

×

_Rⁿ⁻¹.

Now, applying Eqs.(12)and(13)and using the fact that

λ

2

− λ

1

=

^∥x2∥

sinθcosθ in the A₂term, we see that

(

^fc

)

^′

(

^x

;

h

)

^{can be} rewritten in a more compact form as below:

(

^fc

)

^′

(

^x

;

h

) =

^f′



λ

1

;

h₁

−

cot

θ ∥

^xT2x^h₂

∥

²



u⁽¹⁾

+

f^′



λ

2

;

h₁

+

tan

θ ∥

^xT2x^h₂

∥

²



u⁽²⁾

+

^f

(λ

2

) −

^f

(λ

1

) λ

2

− λ

1



I

−

^x2x

T 2

∥

x₂

∥

²



h₂

.

⁽¹⁶⁾ Case (ii). If x₂

=

0, we compute the directional derivative

(

^fc

)

^′

(

^x

;

h

)

at x for any direction h by definition. Let h

= (

^h1

,

^h2

) ∈

R

×

_Rⁿ⁻¹. We have two subcases. First, consider the subcase of h₂

̸=

0. From the spectral factorization, we can choose u⁽¹⁾

=



sin²

θ, −

^sin

θ

^cos

θ

∥^hh²2∥



and u⁽²⁾

=



cos²

θ,

^sin

θ

^cos

θ

∥^hh²2∥



such that



f^c

(

^x

+

th

) =

^f

(λ + △λ

1

)

^u(1)

+

f

(λ + △λ

2

)

^u(2) f^c

(

^x

) =

^f

(λ)

^u(1)

+

f

(λ)

^u(2)

where

λ =

^x1and

△ λ

i

=

t



h₁

+ (−

¹

)

^itan(−1)i

θ∥

^h2

∥



for all i

=

1

,

2. Thus, we obtain f^c

(

^x

+

th

) −

^fc

(

^x

) =

^[f

(λ + △λ

1

) −

^f

(λ)

^{] u(1)}

+

[f

(λ + △λ

2

) −

^f

(λ)

^{] u(2)}

.

166 Y.-L. Chang et al. / Nonlinear Analysis 85 (2013) 160–173

Using the following facts lim

t→0+

f

(λ + △λ

1

) −

^f

(λ)

t

=

lim

t→0+

f

(λ +

^t

(

^h1

−

cot

θ∥

^h2

∥ )) −

^f

(λ)

t

=

f^′

(λ;

^h1

−

cot

θ∥

^h2

∥ )

lim

t→0+

f

(λ + △λ

2

) −

^f

(λ)

t

=

_lim

t→0+

f

(λ +

^t

(

^h1

+

tan

θ∥

^h2

∥ )) −

^f

(λ)

t

=

_f^′

(λ;

^h1

+

_tan

θ∥

^h2

∥ )

yields

lim

t→0+

f^c

(

^x

+

_th

) −

^fc

(

^x

)

t

=

lim

t→0+

f

(λ + △λ

1

) −

^f

(λ)

t u⁽¹⁾

+

lim

t→0+

f

(λ + △λ

2

) −

^f

(λ)

t u⁽²⁾

=

f^′

(λ;

^h1

−

cot

θ∥

^h2

∥ )

^u(1)

+

f^′

(λ;

^h1

+

tan

θ∥

^h2

∥ )

^{u(2) (17)} which says

(

^fc

)

^′

(

^x

;

h

)

^exists.

Second, for the subcase of h₂

=

0, the same arguments apply except h₂

/∥

^h2

∥

is replaced by any

w ∈

Rⁿ⁻¹with

∥ w∥ =

^1, i.e., choosing u⁽¹⁾

= 

sin²

θ, −

^sin

θ

^cos

θw

^{and u(2)}

= 

cos²

θ,

^sin

θ

^cos

θw

. Analogously, we obtain lim

t→0+

f^c

(

^x

+

th

) −

^fc

(

^x

)

t

=

f^′

(λ;

^h1

)

^u(1)

+

f^′

(λ;

^h1

)

^{u(2) (18)}

which implies

(

^fc

)

^′

(

^x

;

h

)

exists in the form of(18). From all the above, it shows that f^cis directionally differentiable at x when x₂

=

0 and its directional derivative

(

^fc

)

^′

(

^x

;

h

)

is either in the form of(17)or(18).

‘‘

⇒

’’ Suppose f^cis directionally differentiable at x

∈

_Rⁿwith spectral values

λ

1

, λ

2, we will prove that f is directionally differentiable at

λ

1

, λ

2. For

λ

1

∈

R and any direction d1

∈

_{R, let h}

:=

d₁u⁽¹⁾

+

0u⁽²⁾where x

= λ

1u⁽¹⁾

+ λ

2u⁽²⁾. Then, x

+

th

= (λ

1

+

td₁

)

^u(1)

+ λ

2u⁽²⁾and

f^c

(

^x

+

th

) −

^fc

(

^x

)

t

=

^f

(λ

1

+

td₁

) −

^f

(λ

1

)

t u⁽¹⁾

.

Since f^cis directionally differentiable at x, the above equation implies f^′

(λ

1

;

d₁

) =

^lim

t→0+

f

(λ

1

+

td₁

) −

^f

(λ

1

)

t exists

.

This means f is directionally differentiable at

λ

1. Similarly, f is also directionally differentiable at

λ

2. 

Theorem 3.3. For any f

:

_R

→

_{R, f}^cis differentiable at x

= (

^x1

,

^x2

) ∈

R

×

_Rⁿ⁻¹with spectral values

λ

1,

λ

2if and only if f is differentiable at

λ

1,

λ

2. Moreover, for given h

= (

^h1

,

^h2

) ∈

R

×

_Rⁿ⁻¹, we have

∇

f^c

(

^x

)

^h

=









b cx^T₂

∥

x₂

∥

cx₂

∥

x₂

∥

^aI

+ (¯

^b

−

a

) ∥

^xx²₂^x

∥

^T22











h₁ h₂



,

^{when x}2

̸=

0

,

where

a

=

^f

(λ

2

) −

^f

(λ

1

) λ

2

− λ

1

,

b

=

f^′

(λ

1

)

^sin2

θ +

^f′

(λ

2

)

^cos2

θ,

b

¯ =

f^′

(λ

1

)

^cos2

θ +

^f′

(λ

2

)

^sin2

θ,

c

= [

f^′

(λ

2

) −

^f′

(λ

1

)]

^sin

θ

^cos

θ.

When x₂

=

0,

∇

f^c

(

^x

) =

^f′

(λ)

^{I with}

λ =

^x1.

Proof. ‘‘

⇐

’’ The proof of this direction is identical to the proof shown as inTheorem 3.2, in which only ‘‘directionally differentiable’’ needs to be replaced by ‘‘differentiable’’. Since f is differentiable at

λ

1and

λ

2, we have that f^′

(λ

1

; · )

^and f^′

(λ

2

; · )

are linear, which means f^′

(λ

i

;

a

+

b

) =

^f′

(λ

i

)

^a

+

f^′

(λ

i

)

b. This together with Eqs.(14)and(15)yield

A₁

=

f^′



λ

1

;

h₁

−

cot

θ

^x

∥

^T2x^h₂

∥

²



sin²

θ +

^f′



λ

2

;

h₁

+

tan

θ ∥

^xT2x^h₂

∥

²



cos²

θ

=

f^′

(λ

1

)

^h1sin²

θ −

^f′

(λ

1

)

^cot

θ

^xT2h2

∥

x₂

∥

^sin

2

θ +

^f′

(λ

2

)

^h1cos²

θ +

^f′

(λ

2

)

^tan

θ

^xT2h2

∥

x₂

∥

^cos

2

θ

= 

f^′

(λ

1

)

^sin2

θ +

^f′

(λ

2

)

^cos2

θ

^h1

+ 

f^′

(λ

2

) −

^f′

(λ

1

)

^sin

θ

^cos

θ

^xT2

∥

x₂

∥

^h2