to appear in Nonlinear Analysis: Theory, Methods and Applications, 2013

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

Yu-Lin Chang

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: ylchang@math.ntnu.edu.tw

Ching-Yu Yang Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: yangcy@math.ntnu.edu.tw

Jein-Shan Chen ^{1}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

January 8, 2013

Abstract. Let L_{θ} be the circular cone in IR^{n}which includes second-order cone as a spe-
cial case. For any function f from IR to IR, one can define a corresponding vector-valued
function f^{c}(x) on IR^{n} by applying f to the spectral values of the spectral decomposition
of x ∈ IR^{n} with respect to L_{θ}. We show that this vector-valued function inherits from
f the properties of continuity, Lipschitz continuity, directional differentiability, Fr´echet
differentiability, continuous differentiability, as well as semismoothness. These results
will play crucial role in designing solution methods for optimization problem associated
with circular cone.

Key words. Circular cone, vector-valued function, semismooth function, complemen- tarity, spectral decomposition.

AMS subject classifications. 26A27, 26B05, 26B35, 49J52, 90C33, 65K05

1Corresponding author. Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is supported by National Science Council of Taiwan

### 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,^{π}_{2}). Then, the n-dimensional circular cone denoted
by L_{θ} can be expressed as

L_{θ} := x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}| kxk cos θ ≤ x_{1}

(1)
:= x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k cot θ ≤ x_{1} .

See Figure 1 as below.

(a) 0 < θ < 45^{◦} (b) θ = 45^{◦} (c) 45^{◦}< θ < 90^{◦}

Figure 1: The graphs of circular cones.

When θ = 45^{◦}, the circular cone reduces to the well-known second-order cone (SOC,
also called Lorentz cone) given by

K^{n} := x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k ≤ x_{1}
:= (x_{1}, x_{2}) ∈ IR × IR^{n−1}

kxk cos 45^{◦} ≤ x_{1} .

With respect to SOC, for any x = (x1, x2) ∈ IR × IR^{n−1}, we can decompose x as

x = λ_{1}(x)u^{(1)}_{x} + λ_{2}(x)u^{(2)}_{x} , (2)
where λ_{1}(x), λ_{2}(x) and u^{(1)}x , u^{(2)}x are the spectral values and the associated spectral vectors
of x with respect to K^{n}, given by

λ_{i}(x) = x_{1}+ (−1)^{i}kx_{2}k,

u^{(i)}_{x} =

1 2

1, (−1)^{i} x_{2}
kx_{2}k

, if x_{2} 6= 0,

1 2

1, (−1)^{i}w

, if x_{2} = 0,

for i = 1, 2 with w being any vector in IR^{n−1} satisfying kwk = 1. If x_{2} 6= 0, the
decomposition (2) is unique. With this spectral decomposition (2), for any function

f : IR → IR, the following vector-valued function associated with K^{n}(n ≥ 1) is considered
(see [3, 4]):

f^{soc}(x) = f (λ_{1})u^{(1)}+ f (λ_{2})u^{(2)} ∀x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}. (3)
If f is defined only on a subset of IR, then f^{soc} is defined on the corresponding subset
of IR^{n}. The definition (3) is unambiguous whether x_{2} 6= 0 or x_{2} = 0. The above defini-
tion (3) is analogous to one associated with the semidefinite cone S^{n}, see [5, 6]. It was
shown [4] that the properties of continuity, strict continuity, Lipschitz continuity, direc-
tional differentiability, differentiability, continuous differentiability, and semismoothness
are each inherited by f^{soc} from f . These results are useful in the design and analysis
of smoothing 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 for optimal grasping manipulation for multi- fingered robots, the grasping force of i-th finger is subject to a contact friction constraint expressed as

(u_{i1}, u_{i3})

≤ µu_{i1} (4)

where µ is the friction coefficient, see Figure 2. Indeed, (4) is a circular cone constraint

Figure 2: The grasping force forms a circular cone where α = tan^{−1}µ < 45^{◦}.
corresponding to u_{i} = (u_{i1}, u_{i2}, u_{i3}) ∈ L_{θ} with θ = tan^{−1}µ < 45^{◦}. Note that the cir-
cular cone L_{θ} is a non-self-dual (or non-symmetric cone) and its related study is rather
limited. Nonetheless, motivated by the real world application regarding circular cone,
the structures and properties about L_{θ} are investigated in [2]. In particular, the spectral
factorization of z associated with circular cone is characterized in [2, Theorem 3.1]. For
convenience, we restate it as below.

Theorem 1.1. [2, Theorem 3.1] For any z = (z_{1}, z_{2}) ∈ IR × IR^{n−1}, one has

z = λ_{1}(z) · u^{(1)}_{z} + λ_{2}(z) · u^{(2)}_{z} (5)
where

λ_{1}(z) = z_{1}− kz_{2}k cot θ

λ2(z) = z1+ kz2k tan θ (6)

and

u^{(1)}z = 1
1 + cot^{2}θ

1 0 0 cot θ

1

−w

=

sin^{2}θ

−(sin θ cos θ)w

u^{(2)}z = 1
1 + tan^{2}θ

1 0

0 tan θ

1 w

=

cos^{2}θ
(sin θ cos θ)w

(7)

with w = z_{2}

kz_{2}k if z_{2} 6= 0, and any vector in IR^{n−1} satisfying kwk = 1 if z_{2} = 0.

Analogous to (3), with the spectral factorization (5), for any function f : IR → IR,
we consider the following vector-valued function associated with L_{θ} (n ≥ 1):

f^{c}(z) = f (λ_{1})u^{(1)}_{z} + f (λ_{2})u^{(2)}_{z} ∀z = (z_{1}, z_{2}) ∈ IR × IR^{n−1}. (8)
Can the properties of continuity, strict continuity, Lipschitz continuity, directional dif-
ferentiability, differentiability, continuous differentiability, and semismoothness be each
inherited by f^{c} from 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 Fr´echet sense) mapping F : IR^{n} → IR^{m}, we denote its Jacobian (not transposed)
at x ∈ IR^{n} by ∇F (x) ∈ IR^{m×n}, i.e., (F (x + u) − F (x) − ∇F (x)u)/kuk → 0 as u → 0.

“ := ” means “define”. We write z = O(α) (respectively, z = o(α)), with α ∈ IR and
z ∈ IR^{n}, to mean kzk/|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, differentiabil- ity, continuous differentiability, as well as semismoothness.

Suppose F : IR^{n} → IR^{m}. Then, F is continuous at x ∈ IR^{n} if F (y) → F (x) as y → x;

and F is continuous if F is continuous at every x ∈ IR^{n}. We say F is strictly continuous
(also called “locally Lipschitz continuous”) at x ∈ IR^{n} if there exist scalars κ > 0 and
δ > 0 such that

kF (y) − F (z)k ≤ κky − zk ∀y, z ∈ IR^{n} with ky − xk ≤ δ, kz − xk ≤ δ;

and F is strictly continuous if F is strictly continuous at every x ∈ IR^{n}. We say F is
directionally differentiable at x ∈ IR^{n} if

F^{0}(x; h) := lim

t→0^{+}

F (x + th) − F (x)

t exists ∀h ∈ IR^{n};

and F is directionally differentiable if F is directionally differentiable at every x ∈ IR^{n}.
F is differentiable (in the Fr´echet sense) at x ∈ IR^{n} if there exists a linear mapping

∇F (x) : IR^{n} → IR^{m} such that

F (x + h) − F (x) − ∇F (x)h = o(khk).

If F is differentiable at every x ∈ IR^{n} 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 differen-
tiable 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 ∂_{B}F (x), where

∂_{B}F (x) :=

lim

x^{j}→x∇F (x^{j})

F is differentiable at x^{j} ∈ IR^{n}

.

The notation ∂_{B} is 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 : IR^{n} → IR^{m} is 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) − V h = o(khk).

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) − V h = O(khk^{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^{c} by examining
only those points x ∈ IR^{n}where f^{c} is 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 section 4.

Lemma 2.1. Suppose F : IR^{n}→ IR^{n} is strictly continuous and directionally differentiable
in a neighborhood of x ∈ IR^{n}. 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) − vh = o(khk) (respectively, O(khk)^{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(khk) (respectively, O(khk)^{1+ρ}).

We say F is semismooth (respectively, ρ-order semismooth) if F is semismooth (re-
spectively, ρ-order semismooth) at every x ∈ IR^{n}. 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, 14, 15, 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 circular
cone and basic definitions before starting the proofs.

Lemma 3.1. Let λ_{1} ≤ λ_{2} be the spectral values of x ∈ IR^{n} and m_{1} ≤ m_{2} be the spectral
values of y ∈ IR^{n}. Then, we have

|λ_{1}− m_{1}|^{2}sin^{2}θ + |λ_{2}− m_{2}|^{2}cos^{2}θ = kx − yk^{2}, (9)
and hence, |λi− mi| ≤ c kx − yk, ∀i = 1, 2, where c = max{sec θ, csc θ}.

Proof. The proof follows from a direct computation. 2

Lemma 3.2. Let x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} and y = (y_{1}, y_{2}) ∈ IR × IR^{n−1}.
(a) If x2 6= 0, y2 6= 0, then we have

ku^{(i)}− v^{(i)}k ≤ 2 sin cos θ

kx_{2}k kx − yk, i = 1, 2, (10)
where u^{(i)}, v^{(i)} are the unique spectral vectors of x and y, respectively.

(b) If either x_{2} = 0 or y_{2} = 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)} = sin^{2}θ

1 , (−1) cot θ x_{2}
kx_{2}k

, v^{(1)} = sin^{2}θ

1 , (−1) cot θ y_{2}
ky_{2}k

,
where u^{(1)}, v^{(1)} are unique. This gives u^{(1)} − v^{(1)} = sin^{2}θ

0 , (−1) cot θ(_{kx}^{x}^{2}

2k − _{ky}^{y}^{2}

2k) . Then,

ku^{(1)}− v^{(1)}k = sin θ cos θ

x_{2}

kx_{2}k− y_{2}
ky_{2}k

= sin θ cos θ

x_{2}− y_{2}

kx_{2}k + (ky_{2}k − kx_{2}k)y_{2}
kx_{2}k · ky_{2}k

≤ sin θ cos θ

1

kx_{2}kkx2− y2k + 1

kx_{2}k|ky2k − kx2k|

≤ sin θ cos θ

1

kx_{2}kkx2− y2k + 1

kx_{2}kkx2 − y2k

≤ 2 sin θ cos θ

kx_{2}k kx − yk,

where the inequalities follow from the triangle inequality. Similar arguments apply for
ku^{(2)}− v^{(2)}k.

(b) We can choose the same spectral vectors for x and y from the spectral factorization
(5) since either x_{2} = 0 or y_{2} = 0. Then, it is obvious. 2

Lemma 3.3. For any w 6= 0 ∈ IR^{n}, we have ∇_{w}

w kwk

= 1

kwk

I − ww^{T}
kwk^{2}

. Proof. See [18, Lemma 3.3] or check it by direct computation. 2

Now, we are ready to present our first main result about continuity between f and f^{c}
Theorem 3.1. For any f : IR → IR, f^{c} is continuous at x ∈ IR^{n} with spectral values
λ_{1}, λ_{2} if 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 = (x_{1}, x_{2}) ∈ IR×IR^{n−1}and
y → x, let the spectral factorizations of x, y be x = λ_{1}u^{(1)}+λ_{2}u^{(2)}and y = m_{1}v^{(1)}+m_{2}v^{(2)},
respectively. Then, we discuss two cases.

Case (i): If x_{2} 6= 0, then we have
f^{c}(y) − f^{c}(x)

= f (m_{1})v^{(1)}− u^{(1)} + [f (m_{1}) − f (λ_{1})] u^{(1)} (11)
+f (m_{2})v^{(2)}− u^{(2)} + [f (m2) − f (λ_{2})] u^{(2)}.

Since f is continuous at λ_{1}, λ_{2}, and from Lemma 3.1, |m_{i} − λ_{i}| ≤ c ky − xk, we know
f (m_{i}) −→ f (λ_{i}) as y → x. In addition, by Lemma 3.2, we have kv^{(i)}−u^{(i)}k −→ 0 as y →
x. Thus, equation (11) yields f^{c}(y) −→ f^{c}(x) as y → x because both f (m_{i}) and ku^{(i)}k
are bounded. Hence, f^{c} is continuous at x ∈ IR^{n}.

Case (ii): If x_{2} = 0, no matter y_{2} is zero or not, we can arrange that x, y have the same
spectral vectors. Thus, f^{c}(y) − f^{c}(x) = [f (m_{1}) − f (λ_{1})] u^{(1)}+ [f (m_{2}) − f (λ_{2})] u^{(2)}. Then,
f^{c} is continuous at x ∈ IR^{n} by similar arguments.

“⇒” The proof for this direction is straightforward or refer to similar arguments for [4, Prop. 2]. 2

Theorem 3.2. For any f : IR → IR, f^{c} is directionally differentiable at x ∈ IR^{n} with
spectral values λ_{1}, λ_{2} if and only if f is directionally differentiable at λ_{1}, λ_{2}.

Proof. “⇐” Suppose f is directionally differentiable at λ1, λ2. Fix any x = (x1, x2) ∈
IR × IR^{n−1}, then we discuss two cases as below.

Case (i): If x_{2} 6= 0, we have f^{c}(x) = f (λ_{1})u^{(1)}+f (λ_{2})u^{(2)}where λ_{i} = x_{1}+(−1)^{i}(tan θ)^{(−1)}^{i}kx_{2}k
and u^{(i)} = (−1)^{i}sin θ cos θ

(tan θ)^{(−1)}^{i},_{kx}^{x}^{T}^{2}

2k

for all i = 1, 2. From Lemma 3.3, we know
that u^{(i)} is Fr´echet-differentiable with respect to x, with

∇_{x}u^{(i)}= (−1)^{i}sin θ cos θ
kx2k

0 0

0 I − x_{2}x^{T}_{2}
kx2k^{2}

∀i = 1, 2. (12)

Also by the expression of λ_{i}, we know that λ_{i} is Fr´echet-differentiable with respect to x,
with

∇_{x}λ_{i} =

1 , (−1)^{i}tan^{(−1)}^{i}θ x^{T}_{2}
kx_{2}k

∀i = 1, 2. (13)

In general, we cannot apply chain rule, when functions are only directionally differen-
tiable. But, it works well for single-variable functions, that is, when single-variable func-
tions are composed with 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^{0}(λ_{1}; 1),
lim

t→0^{+}

f (λ1− t × 1) − f (λ1)

t = f^{0}(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) = x_{1}−cot θkx_{2}k is differentiable when x_{2} 6= 0,
which yields

λ_{1}(x + th) = λ_{1}(x) + t∇_{x}λ_{1}h + o(t).

Let y := ∇_{x}λ_{1}h + ^{o(t)}_{t} . For the case of ∇_{x}λ_{1}h < 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 (λ1(x) + ty) − f (λ1(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^{0}(λ_{1}(x); −1)(−∇_{x}λ_{1}h)

= f^{0}(λ_{1}(x); ∇_{x}λ_{1}h).

Here the positively homogeneous property of directionally differentiable functions is used
in the last equation. Similarly, for the other case of ∇_{x}λ_{1}h ≥ 0, we have

lim

t→0^{+}

f (λ_{1}(x + th)) − f (λ_{1}(x))

t = f^{0}(λ_{1}(x); ∇_{x}λ_{1}h).

In summary, the composite function f ◦ λ_{1}(·) is directionally differentiable at x. Now we
can apply chain rule and product rule on f^{c}(x) = f (λ_{1})u^{(1)}+ f (λ_{2})u^{(2)}. In other words,

(f^{c})^{0}(x; h)

= f (λ_{1})∇_{x}u^{(1)}h + f^{0}(λ_{1}; ∇_{x}λ_{1}h)u^{(1)}+ f (λ_{2})∇_{x}u^{(2)}h + f^{0}(λ_{2}; ∇_{x}λ_{2}h)u^{(2)}

= (A1, A2) ∈ IR × IR^{n−1},
where

A_{1} = f^{0}

λ_{1}; h_{1}− cot θx^{T}_{2}h_{2}
kx_{2}k

sin^{2}θ + f^{0}

λ_{2}; h_{1}+ tan θx^{T}_{2}h_{2}
kx_{2}k

cos^{2}θ (14)
and

A2 =

f^{0}

λ2; h1+ tan θx^{T}_{2}h_{2}
kx_{2}k

− f^{0}

λ1; h1− cot θx^{T}_{2}h_{2}
kx_{2}k

sin θ cos θ x_{2}

kx_{2}k (15)
+f (λ_{2}) − f (λ_{1})

λ_{2}− λ_{1}

I − x_{2}x^{T}_{2}
kx_{2}k^{2}

h_{2},
with h = (h_{1}, h_{2}) ∈ IR × IR^{n−1}.

Now, applying equations (12) and (13) and using the fact that λ_{2}− λ_{1} = sin θ cos θ^{kx}^{2}^{k} in the
A_{2} term, we see that (f^{c})^{0}(x; h) can be rewritten in a more compact form as below:

(f^{c})^{0}(x; h) = f^{0}

λ1; h1− cot θx^{T}_{2}h_{2}
kx_{2}k

u^{(1)}+ f^{0}

λ2; h1+ tan θx^{T}_{2}h_{2}
kx_{2}k

u^{(2)}
+f (λ_{2}) − f (λ_{1})

λ_{2}− λ_{1}

I − x_{2}x^{T}_{2}
kx_{2}k^{2}

h_{2}. (16)

Case (ii): If x_{2} = 0, we compute the directional derivative (f^{c})^{0}(x; h) at x for any
direction h by definition. Let h = (h_{1}, h_{2}) ∈ IR × IR^{n−1}. We have two subcases.

First, consider the subcase of h_{2} 6= 0. From the spectral factorization, we can choose
u^{(1)} =

sin^{2}θ , − sin θ cos θ_{kh}^{h}^{2}

2k

and u^{(2)} =

cos^{2}θ , sin θ cos θ_{kh}^{h}^{2}

2k

such that

f^{c}(x + th) = f (λ + 4λ1)u^{(1)}+ f (λ + 4λ2)u^{(2)}
f^{c}(x) = f (λ)u^{(1)}+ f (λ)u^{(2)}

where λ = x_{1} and 4λ_{i} = t

h_{1}+ (−1)^{i}tan^{(−1)}^{i}θkh_{2}k

for all i = 1, 2. Thus, we obtain
f^{c}(x + th) − f^{c}(x) = [f (λ + 4λ_{1}) − f (λ)] u^{(1)}+ [f (λ + 4λ_{2}) − f (λ)] u^{(2)}.
Using the following facts

lim

t→0^{+}

f (λ + 4λ_{1}) − f (λ)

t = lim

t→0^{+}

f (λ + t(h_{1}− cot θkh_{2}k)) − f (λ)

t = f^{0}(λ; h_{1}− cot θkh_{2}k)
lim

t→0^{+}

f (λ + 4λ_{2}) − f (λ)

t = lim

t→0^{+}

f (λ + t(h_{1}+ tan θkh_{2}k)) − f (λ)

t = f^{0}(λ; h_{1}+ tan θkh_{2}k)
yields

lim

t→0^{+}

f^{c}(x + th) − f^{c}(x)
t

= lim

t→0^{+}

f (λ + 4λ_{1}) − f (λ)

t u^{(1)}+ lim

t→0^{+}

f (λ + 4λ_{2}) − f (λ)

t u^{(2)}

= f^{0}(λ; h_{1}− cot θkh_{2}k)u^{(1)}+ f^{0}(λ; h_{1}+ tan θkh_{2}k)u^{(2)} (17)
which says (f^{c})^{0}(x; h) exists.

Secondly, for the subcase of h_{2} = 0, the same arguments apply except h_{2}/kh_{2}k is replaced
by any w ∈ IR^{n−1} with kwk = 1, i.e., choosing u^{(1)} = sin^{2}θ , − sin θ cos θw and u^{(2)} =
(cos^{2}θ , sin θ cos θw). Analogously, we obtain

lim

t→0^{+}

f^{c}(x + th) − f^{c}(x)

t = f^{0}(λ; h_{1})u^{(1)}+ f^{0}(λ; h_{1})u^{(2)}. (18)
which implies (f^{c})^{0}(x; h) exists with form of (18). From all the above, it shows that f^{c} is
directionally differentiable at x when x_{2} = 0 and its directional derivative (f^{c})^{0}(x; h) is
either in form of (17) or (18).

“⇒” Suppose f^{c} is directionally differentiable at x ∈ IR^{n} with spectral values λ_{1}, λ_{2}, we
will prove that f is directionally differentiable at λ_{1}, λ_{2}. For λ_{1} ∈ IR and any direction
d_{1} ∈ IR, let h := d_{1}u^{(1)}+ 0u^{(2)} where x = λ_{1}u^{(1)}+ λ_{2}u^{(2)}. Then, x + th = (λ_{1}+ td_{1})u^{(1)}+
λ_{2}u^{(2)} and

f^{c}(x + th) − f^{c}(x)

t = f (λ_{1}+ td_{1}) − f (λ_{1})

t u^{(1)}.

Since f^{c} is directionally differentiable at x, the above equation implies

f^{0}(λ_{1}; d_{1}) = lim

t→0^{+}

f (λ_{1}+ td_{1}) − f (λ_{1})

t exists.

This means f is directionally differentiable at λ_{1}. Similarly, f is also directionally differ-
entiable at λ_{2}. 2

Theorem 3.3. For any f : IR → IR, f^{c} is differentiable at x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}
with spectral values λ_{1}, λ_{2} if and only if f is differentiable at λ_{1}, λ_{2}. Moreover, for given
h = (h1, h2) ∈ IR × IR^{n−1}, we have

∇f^{c}(x)h =

b cx^{T}_{2}

kx_{2}k
cx_{2}

kx_{2}k aI + (¯b − a)x_{2}x^{T}_{2}
kx_{2}k^{2}

h_{1}
h_{2}

, when x_{2} 6= 0,

where

a = f (λ_{2}) − f (λ_{1})
λ_{2}− λ_{1} ,

b = f^{0}(λ1) sin^{2}θ + f^{0}(λ2) cos^{2}θ,

¯b = f^{0}(λ_{1}) cos^{2}θ + f^{0}(λ_{2}) sin^{2}θ,
c = [f^{0}(λ_{2}) − f^{0}(λ_{1})] sin θ cos θ.

When x_{2} = 0, ∇f^{c}(x) = f^{0}(λ)I with λ = x_{1}.

Proof. “⇐” The proof of this direction is identical to the proof shown as in Theorem 3.2, in which only “directionally differentiable” needs to be replaced by “differentiable”.

Since f is differentiable at λ_{1} and λ_{2}, we have that f^{0}(λ_{1}; ·) and f^{0}(λ_{2}; ·) are linear, which
means f^{0}(λ_{i}; a + b) = f^{0}(λ_{i})a + f^{0}(λ_{i})b. This together with equations (14) and (15) yield

A_{1} = f^{0}

λ_{1}; h_{1}− cot θx^{T}_{2}h_{2}
kx2k

sin^{2}θ + f^{0}

λ_{2}; h_{1}+ tan θx^{T}_{2}h_{2}
kx2k

cos^{2}θ

= f^{0}(λ_{1})h_{1}sin^{2}θ − f^{0}(λ_{1}) cot θx^{T}_{2}h_{2}

kx_{2}ksin^{2}θ + f^{0}(λ_{2})h_{1}cos^{2}θ + f^{0}(λ_{2}) tan θx^{T}_{2}h_{2}
kx_{2}k cos^{2}θ

= f^{0}(λ1) sin^{2}θ + f^{0}(λ2) cos^{2}θ h1+ [f^{0}(λ2) − f^{0}(λ1)] sin θ cos θ x^{T}_{2}
kx_{2}kh2

and
A_{2} =

f^{0}

λ_{2}; h_{1}+ tan θx^{T}_{2}h_{2}
kx_{2}k

− f^{0}

λ_{1}; h_{1}− cot θx^{T}_{2}h_{2}
kx_{2}k

sin θ cos θ x_{2}
kx_{2}k
+f (λ_{2}) − f (λ_{1})

λ_{2}− λ_{1} (I − x_{2}x^{T}_{2}

kx_{2}k^{2})h_{2} (19)

=

f^{0}(λ_{2})h_{1}− f^{0}(λ_{1})h_{1}+ f^{0}(λ_{2}) tan θx^{T}_{2}h_{2}

kx_{2}k + f^{0}(λ_{1}) cot θx^{T}_{2}h_{2}
kx_{2}k

sin θ cos θ x_{2}
kx_{2}k
+f (λ_{2}) − f (λ_{1})

λ2− λ1

I − x_{2}x^{T}_{2}
kx2k^{2}

h_{2}

= [f^{0}(λ_{2}) − f^{0}(λ_{1})] sin θ cos θ x_{2}
kx_{2}kh_{1}
+f^{0}(λ_{2}) sin^{2}θ + f^{0}(λ_{1}) cos^{2}θ x2x^{T}_{2}

kx_{2}k^{2}h_{2}+f (λ2) − f (λ1)
λ_{2}− λ_{1}

I − x2x^{T}_{2}
kx_{2}k^{2}

h_{2}.
Thus, for x_{2} 6= 0, we have

∇f^{c}(x)h =

b cx^{T}_{2}

kx_{2}k
cx2

kx_{2}k aI + (¯b − a)x2x^{T}_{2}
kx_{2}k^{2}

h_{1}
h_{2}

(20)

with

a = f (λ_{2}) − f (λ_{1})
λ_{2}− λ_{1} ,

b = f^{0}(λ_{1}) sin^{2}θ + f^{0}(λ_{2}) cos^{2}θ,

¯b = f^{0}(λ_{1}) cos^{2}θ + f^{0}(λ_{2}) sin^{2}θ, (21)
c = [f^{0}(λ_{2}) − f^{0}(λ_{1})] sin θ cos θ.

From equation (16), ∇f^{c}(x)h can also be recast in a more compact form:

∇f^{c}(x)h = f^{0}(λ_{1})

h_{1}− cot θx^{T}_{2}h_{2}
kx_{2}k

u^{(1)}+ f^{0}(λ_{2})

h_{1}+ tan θx^{T}_{2}h_{2}
kx_{2}k

u^{(2)}
+f (λ2) − f (λ1)

λ_{2}− λ_{1}

I − x2x^{T}_{2}
kx_{2}k^{2}

h_{2}. (22)

For case of x_{2} = 0, with linearity of f^{0}(λ; ·) and equations (17) and (18), we have

∇f^{c}(x) = f^{0}(λ)I, (23)

where λ = λ_{1} = λ_{2} = x_{1}.

“⇒” Let f^{c} be Fr´echet-differentiable at x ∈ IR^{n} with spectral eigenvalues λ_{1}, λ_{2}, we will
show that f is Fr´echet-differentiable at λ_{1}, λ_{2}. Suppose not, then f is not Fr´echet-
differentiable at λ_{i} for some i ∈ {1, 2}. Thus, either the right- and left-directional

derivatives of f at λ_{i} are unequal or one of them does not exist. In either case, this implies
that there exist two sequences of non-zero scalars t^{ν} and τ^{ν}, ν = 1, 2, . . . , converging to
zero such that the limits

ν→∞lim

f (λ_{i}+ t^{ν}) − f (λ_{i})

t^{ν} , lim

ν→∞

f (λ_{i}+ τ^{ν}) − f (λ_{i})
τ^{ν}

either are unequal or one of them does not exist. Now for any x = λ_{1}u^{(1)} + λ_{2}u^{(2)},
let h := 1 · u^{(1)} + 0 · u^{(2)} = u^{(1)}. Then, we know x + th = (λ1 + t)u^{(1)} + λ2u^{(2)} and
f^{c}(x + th) = f (λ_{1}+ t)u^{(1)}+ f (λ_{2})u^{(2)}, which give

ν→∞lim

f^{c}(x + t^{ν}h) − f^{c}(x)

t^{ν} = lim

ν→∞

f (λ_{1}+ t^{ν}) − f (λ_{1})
t^{ν} u^{(1)}

ν→∞lim

f^{c}(x + τ^{ν}h) − f^{c}(x)

τ^{ν} = lim

ν→∞

f (λ_{1}+ τ^{ν}) − f (λ_{1})
τ^{ν} u^{(1)}.

It follows that these two limits either are unequal or one of them does not exist. This
implies that f^{c} is not Fr´echet-differentiable at x, which is a contradiction. 2

Theorem 3.4. For any f : IR → IR, f^{c} is continuously differentiable (smooth) at x ∈ IR^{n}
with spectral values λ_{1}, λ_{2} if and only if f is continuously differentiable (smooth) at λ_{1},
λ_{2}.

Proof. “⇐” Suppose f is continuously differentiable at x ∈ IR^{n}. From equation (20),
it can been seen that ∇f^{c} is continuous at every x with x_{2} 6= 0. It remains to show
that ∇f^{c} is continuous at every x with x_{2} = 0. Fix any x = (x_{1}, 0) ∈ IR^{n}, which says
λ_{1} = λ_{2} = x_{1}. Let y^{ν} = (y_{1}^{ν}, y_{2}^{ν}) ∈ IR × IR^{n−1} be any sequence converging to x. For
those y^{ν}_{2} = 0, applying equation (23) gives ∇f^{c}(y^{ν}) = f^{0}(λ(y^{ν}))I. Suppose y_{2}^{ν} 6= 0, from
equation (21), we have

lim

y^{ν}→x,y_{2}^{ν}6=0a = lim

y^{ν}→x,y_{2}^{ν}6=0

f (λ_{2}(y^{ν})) − f (λ_{1}(y^{ν}))

λ2(y^{ν}) − λ1(y^{ν}) = f^{0}(x_{1}),
lim

y^{ν}→x,y_{2}^{ν}6=0b = lim

y^{ν}→x,y_{2}^{ν}6=0 f^{0}(λ_{1}(y^{ν})) sin^{2}θ + f^{0}(λ_{2}(y^{ν})) cos^{2}θ = f^{0}(x_{1}),

y^{ν}→x,ylim^{ν}_{2}6=0c y_{2}^{ν}

ky_{2}^{ν}k = lim

y^{ν}→x,y_{2}^{ν}6=0sin θ cos θ [ f^{0}(λ_{2}(y^{ν})) − f^{0}(λ_{1}(y^{ν})) ] y^{ν}_{2}
ky^{ν}_{2}k = 0,

y^{ν}→x,ylim^{ν}_{2}6=0(¯b − a)y_{2}^{ν}y_{2}^{νT}

ky_{2}^{ν}k^{2} = lim

y^{ν}→x,y_{2}^{ν}6=0

f^{0}(λ_{1}(y^{ν})) cos^{2}θ + f^{0}(λ_{2}(y^{ν})) sin^{2}θ

−f (λ_{2}(y^{ν})) − f (λ_{1}(y^{ν}))
λ_{2}(y^{ν}) − λ_{1}(y^{ν})

y_{2}^{ν}y_{2}^{νT}
ky_{2}^{ν}k^{2} = 0.

Using the facts that both _{ky}^{y}^{ν}^{2}ν

2k and ^{y}_{ky}^{ν}^{2}^{y}ν^{2}^{νT}

2k^{2} are bounded by 1 and then taking the limit
in (20) as y → x yield lim

y→x∇f^{c}(y) = f^{0}(x_{1})I = ∇f^{c}(x). This says ∇f^{c} is continuous at
every x ∈ IR^{n} .

“⇒” The proof for this direction is similar to the one for [4, Prop. 5], so we omit it. 2

Next, we move to property of (locally) Lipschitz continuity. To this end, we need the following result, which is from [10, Theorem 9.67].

Lemma 3.4. [10, Theorem 9.67] Suppose f : IR^{n} → IR is strictly continuous. Then,
there exist continuously differentiable functions f^{ν} : IR^{n} → IR, ν = 1, 2, · · · , converging
uniformly to f on any compact set C in IR^{n} and satisfying

k∇f^{ν}(x)k ≤ sup

y∈C

Lipf (y) ∀x ∈ C, ν = 1, 2, 3, · · ·

where Lipf (x) := lim sup

y,z→x,y6=z

kf (y) − f (z)k ky − zk .

Theorem 3.5. For any f : IR → IR, the following results hold:

(a) f^{c} is strictly continuous at x ∈ IR^{n} with spectral values λ1, λ2 if and only if f is
strictly continuous at λ_{1}, λ_{2}.

(b) f^{c} is Lipschitz continuous (with respect to k · k) with constant κ if and only if f is
Lipschitz continuous with constant κ.

Proof. (a) “⇐” Fix any x ∈ IR^{n} with spectral values λ_{1} and λ_{2} given by (6). Suppose
f is strictly continuous at λ_{1} and λ_{2}. Then, there exist κ_{i} > 0 and δ_{i} > 0 for i = 1, 2
such that

|f (b) − f (a)| ≤ κ_{i}|b − a|, ∀ a, b ∈ [λ_{i}− δ_{i}, λ_{i}+ δ_{i}] i = 1, 2.

Let ¯δ := min{δ_{1}, δ_{2}} and C := [λ_{1} − ¯δ_{1}, λ_{1} + ¯δ] ∪ [λ_{2} − ¯δ, λ_{2} + ¯δ]. Define a real-valued
function ¯f : IR → IR as

f (a) =¯

f (a) if a ∈ C,

(1 − t)f (λ_{1}+ ¯δ) if λ_{1}+ ¯δ < λ_{2}− ¯δ and, for some t ∈ (0, 1),
+tf (λ_{2}− ¯δ) a = (1 − t)(λ_{1}+ ¯δ) + t(λ_{2}− ¯δ),

f (λ_{1}− ¯δ) if a < λ_{1}− ¯δ,
f (λ_{2}+ ¯δ) if a > λ_{2}+ ¯δ.

From the above, we know that ¯f is Lipschitz continuous, which means there exists a
scalar κ > 0 such that Lip ¯f (a) ≤ κ for all a ∈ IR. Since C is compact, by Lemma 3.4,
there exist continuously differentiable functions f^{ν} : IR → IR, ν = 1, 2, · · · , converging
uniformly to ¯f and satisfying

|(f^{ν})^{0}(a)| ≤ κ, ∀ a ∈ C, ∀ ν.

On the other hand, from Lemma 3.1, there exists a δ such that C contains all spectral
values of w ∈ B(x, δ). Moreover, for any w ∈ B(x, δ) with spectral factorization w =
µ_{1}u^{(1)}+ µ_{2}u^{(2)}, by direct computation, we have

(f^{ν})^{c}(w) − f^{c}(w)

2 = sin^{2}θ|f^{ν}(µ_{1}) − f (µ_{2})|^{2}+ cos^{2}θ|f^{ν}(µ_{2}) − f (µ_{2})|^{2}.

This together with f^{ν} converging uniformly to f on C implies that (f^{ν})^{c} converges
uniformly to f^{c} on B(x, δ).

Next, we explain that k∇(f^{ν})^{c}(w)k is uniformly bounded. Indeed, for w_{2} = 0, from
equation (23) we have k∇(f^{ν})^{c}(w)k = |(f^{ν})^{0}(w_{1})| ≤ κ. For general w_{2} 6= 0, it is not
hard to check k∇(f^{ν})^{c}(w)k ≤ M for some uniform bound M ≥ κ on the set C by using
equation (22).

Fix any y, z ∈ B(x, δ). Since (f^{ν})^{c} converges uniformly to f^{c}, for any > 0 there exists
an integer ν_{0} such that for all ν ≥ ν_{0} we have

k(f^{ν})^{c}(w) − f^{c}(w)k ≤ ky − zk ∀w ∈ B(x, δ).

Note that f^{ν} is continuously differentiable, Theorem 3.4 implies (f^{ν})^{c}is also continuously
differentiable. Then, by the fact that k∇(f^{ν})^{c}(w)k is uniform bounded by M and the
Mean Value Theorem for continuously differentiable functions, we obtain

f^{c}(y) − f^{c}(z)

=

f^{c}(y) − (f^{ν})^{c}(y) + (f^{ν})^{c}(y) − (f^{ν})^{c}(z) + (f^{ν})^{c}(z) − f^{c}(z)

≤

f^{c}(y) − (f^{ν})^{c}(y)

+ k(f^{ν})^{c}(y) − (f^{ν})^{c}(z)k +

(f^{ν})^{c}(z) − f^{c}(z)

≤ 2ky − zk +

Z 1 0

∇(f^{ν})^{c}(z + t(y − z))(y − z)dt

≤ (M + 2)ky − zk.

This shows that f^{c} is strictly continuous at x.

“⇒” Suppose that f^{c} is strictly continuous at x with eigenvalues λ_{1} and λ_{2} and spectral
vectors u^{(1)} and u^{(2)}. This means there exist δ and M such that for y, z ∈ B(x, δ), we
have

f^{c}(y) − f^{c}(z)

≤ M ky − zk.

For any i ∈ {1, 2} and any a, b ∈ [λ_{i}− δ, λ_{i}+ δ], denote

y := x + (a − λ_{i})u^{(i)}, z := x + (b − λ_{i})u^{(i)}.

Then, ky − xk = |a − λi|ku^{(i)}k ≤ δ and kz − xk = |b − λi|ku^{(i)}k ≤ δ. Thus,

|f (b) − f (a)| ·
u^{(i)}

=

f^{c}(y) − f^{c}(z)

≤ M ky − zk.

which says that f is strictly continuous at λ_{1} and λ_{2} because ku^{(1)}k = sin θ and
u^{(2)}

= cos θ.

(b) This is immediate consequence of part (a). 2

### 4 Semismoothness Property

This section is devoted to presenting semismooth property between f and f^{c}. As men-
tioned earlier, Lemma 2.1 will be employed frequently in our analysis.

Theorem 4.1. For any f : IR → IR, f^{c} is semismooth at x ∈ IR^{n} with spectral values
λ_{1}, λ_{2} if and only if f is semismooth at λ_{1}, λ_{2}.

Proof. “⇒” Suppose f^{c} is semismooth, then f^{c} is strictly continuous and directionally
differentiable. By Theorem 3.2 and Theorem 3.5, f is strictly continuous and directionally
differentiable. Now, for any α ∈ IR and any η ∈ IR such that f is differentiable at α + η,
Theorem 3.2 yields that f^{c} is differentiable at x + h, where x := (α, 0) ∈ IR × IR^{n−1}
and h := (η, 0) ∈ IR × IR^{n−1}. Hence, we can choose the same spectral vectors for
x + h = (α + η, 0) and x = (α, 0) such that

f^{c}(x + h) = f (α + η)u^{(1)}+ f (α + η)u^{(2)},
f^{c}(x) = f (α)u^{(1)}+ f (α)u^{(2)}.

Since f^{c} is semismooth, by Lemma 2.1, we know

f^{c}(x + h) − f^{c}(x) − ∇f^{c}(x + h)h = o(khk). (24)
On the other hand, equation (23) yields ∇f^{c}(x + h)h = f^{0}(α + η)Ih = (f^{0}(α + η)η, 0) .
Plugging this into equation (24) yields f (α + η) − f (α) − f^{0}(α + η)η = o(|η|). Thus,
by Lemma 2.1 again, it follows that f is semismooth at α. Since α is arbitrary, f is
semismooth.

“⇐” Suppose f is semismooth, then f is strictly continuous and directionally differen-
tiable. By Theorem 3.2 and Theorem 3.5, f^{c} is strictly continuous and directionally
differentiable. For any x = (x1, x2) ∈ IR × IR^{n} and h = (h1, h2) ∈ IR × IR^{n} such that f^{c}
is differentiable at x + h, we will verify that

f^{c}(x + h) − f^{c}(x) − ∇f^{c}(x + h)h = o(khk).

Case (i): If x_{2} 6= 0, let λ_{i} be the spectral values of x and u^{(i)} be the associated spectral
vectors. We denote x + h by z for convenience, i.e., z := x + h and let m_{i} be the spectral
values of z with the associated spectral vectors v^{(i)}. Hence, we have

f^{c}(x) = f (λ_{1})u^{(1)}+ f (λ_{2})u^{(2)},
f^{c}(x + h) = f (m_{1})v^{(1)}+ f (m_{2})v^{(2)}.
Suppose now f^{c} is differentiable at z. From (20), we know

∇f^{c}(x + h) =

b cz_{2}^{T}

kz_{2}k
cz_{2}

kz_{2}k aI + (¯b − a) z_{2}z_{2}^{T}
kz_{2}k^{2}

,

where

a = f (m_{2}) − f (m_{1})
m2− m1

,

b = f^{0}(m_{1}) sin^{2}θ + f^{0}(m_{2}) cos^{2}θ,

¯b = f^{0}(m1) cos^{2}θ + f^{0}(m2) sin^{2}θ,
c = [f^{0}(m_{2}) − f^{0}(m_{1})] sin θ cos θ.

With this, we can write out f^{c}(x + h) − f^{c}(x) − ∇f^{c}(x + h)h := (Ξ_{1}, Ξ_{2}) where Ξ_{1} ∈ IR
and Ξ_{2} ∈ IR^{n−1}. Since the expansion is very long, for simplicity, we denote Ξ_{1} be the
first component and Ξ_{2} be the second component of the expansion. We will show that
Ξ1 and Ξ2 are both o(khk). First, we compute the first component Ξ1:

Ξ_{1} = sin^{2}θ

f (m_{1}) − f (λ_{1}) − f^{0}(m_{1})(h_{1}− cot θz_{2}^{T}h_{2}
kz_{2}k)

+ cos^{2}θ

f (m_{2}) − f (λ_{2}) − f^{0}(m_{2})(h_{1}+ tan θz_{2}^{T}h_{2}
kz_{2}k)

= sin^{2}θ {f (m_{1}) − f (λ_{1}) − f^{0}(m_{1}) (h_{1}− cot θ(kz_{2}k − kx_{2}k)) + o(khk)}

+ cos^{2}θ {f (m_{2}) − f (λ_{2}) − f^{0}(m_{2}) (h_{1}+ tan θ(kz_{2}k − kx_{2}k)) + o(khk)}

= o (h_{1}− (kz_{2}k − kx_{2}k)) + o(khk) + o (h_{1}+ (kz_{2}k − kx_{2}k)) + o(khk).

In the above expression of Ξ_{1}, the third equality holds since the following:

z_{2}^{T}h_{2}

kz_{2}k = z_{2}^{T}(z_{2}− x_{2})

kz_{2}k = kz2k −kz_{2}kkx_{2}k
kz_{2}k cos α

= kz2k − kx2k 1 + O(α^{2}) = kz2k − kx2k 1 + O(khk^{2})

= kz_{2}k − kx_{2}k (1 + o(khk))

where α is the angle between x_{2} and z_{2}and note that z_{2}−x_{2} = h_{2}gives O(α^{2}) = O(khk^{2}).

In addition, the last equality in expression of Ξ_{1} holds because f is semismooth and
mi− λi = h1+ (−1)^{i}(tan θ)^{(−1)}^{i}(kz2k − kx2k).

On the other hand, due to

h_{1}+ (−1)^{i}(tan θ)^{(−1)}^{i}(kz_{2}k − kx_{2}k)

≤ |h_{1}| + M kz_{2}− x_{2}k ≤ M (|h_{1}| + kh_{2}k)
where M = max{tan θ, cot θ} ≥ 1. Then, we observe that when khk → 0,

|h_{1}| + (−1)^{i}(tan θ)^{(−1)}^{i}(kz_{2}k − kx_{2}k) → 0

|h_{1}| + (−1)^{i}(tan θ)^{(−1)}^{i}(kz_{2}k − kx_{2}k) = O(khk).

Thus, we obtain o

h_{1} + (−1)^{i}(tan θ)^{(−1)}^{i}(kz_{2}k − kx_{2}k)

= o(khk), which implies that
the first component Ξ_{1} is o(khk).