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

## Full text

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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 Lθ be the circular cone in IRnwhich 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 fc(x) on IRn by applying f to the spectral values of the spectral decomposition of x ∈ IRn with respect to Lθ. 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

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### 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 by Lθ can be expressed as

Lθ := x = (x1, x2) ∈ IR × IRn−1| kxk cos θ ≤ x1

(1) := x = (x1, x2) ∈ IR × IRn−1| kx2k cot θ ≤ x1 .

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

Kn := x = (x1, x2) ∈ IR × IRn−1| kx2k ≤ x1 := (x1, x2) ∈ IR × IRn−1

kxk cos 45 ≤ x1 .

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

x = λ1(x)u(1)x + λ2(x)u(2)x , (2) where λ1(x), λ2(x) and u(1)x , u(2)x are the spectral values and the associated spectral vectors of x with respect to Kn, given by

λi(x) = x1+ (−1)ikx2k,

u(i)x =





1 2



1, (−1)i x2 kx2k



, if x2 6= 0,

1 2



1, (−1)iw

, if x2 = 0,

for i = 1, 2 with w being any vector in IRn−1 satisfying kwk = 1. If x2 6= 0, the decomposition (2) is unique. With this spectral decomposition (2), for any function

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

fsoc(x) = f (λ1)u(1)+ f (λ2)u(2) ∀x = (x1, x2) ∈ IR × IRn−1. (3) If f is defined only on a subset of IR, then fsoc is defined on the corresponding subset of IRn. The definition (3) is unambiguous whether x2 6= 0 or x2 = 0. The above defini- tion (3) is analogous to one associated with the semidefinite cone Sn, 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 fsoc 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

(ui1, ui3)

≤ µui1 (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−1µ < 45. corresponding to ui = (ui1, ui2, ui3) ∈ Lθ with θ = tan−1µ < 45. Note that the cir- cular cone Lθ 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 Lθ 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.

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Theorem 1.1. [2, Theorem 3.1] For any z = (z1, z2) ∈ IR × IRn−1, one has

z = λ1(z) · u(1)z + λ2(z) · u(2)z (5) where

 λ1(z) = z1− kz2k cot θ

λ2(z) = z1+ kz2k tan θ (6)

and





u(1)z = 1 1 + cot2θ

1 0 0 cot θ

  1

−w



=

 sin2θ

−(sin θ cos θ)w



u(2)z = 1 1 + tan2θ

1 0

0 tan θ

  1 w



=

 cos2θ (sin θ cos θ)w

 (7)

with w = z2

kz2k if z2 6= 0, and any vector in IRn−1 satisfying kwk = 1 if z2 = 0.

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

fc(z) = f (λ1)u(1)z + f (λ2)u(2)z ∀z = (z1, z2) ∈ IR × IRn−1. (8) Can the properties of continuity, strict continuity, Lipschitz continuity, directional dif- ferentiability, differentiability, continuous differentiability, and semismoothness be each inherited by fc 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 : IRn → IRm, we denote its Jacobian (not transposed) at x ∈ IRn by ∇F (x) ∈ IRm×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 ∈ IRn, 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 : IRn → IRm. Then, F is continuous at x ∈ IRn if F (y) → F (x) as y → x;

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

kF (y) − F (z)k ≤ κky − zk ∀y, z ∈ IRn with ky − xk ≤ δ, kz − xk ≤ δ;

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and F is strictly continuous if F is strictly continuous at every x ∈ IRn. We say F is directionally differentiable at x ∈ IRn if

F0(x; h) := lim

t→0+

F (x + th) − F (x)

t exists ∀h ∈ IRn;

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

∇F (x) : IRn → IRm such that

F (x + h) − F (x) − ∇F (x)h = o(khk).

If F is differentiable at every x ∈ IRn 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 ∂BF (x), where

BF (x) :=

 lim

xj→x∇F (xj)

F is differentiable at xj ∈ IRn

 .

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 : IRn → IRm 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(khk1+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 fc by examining only those points x ∈ IRnwhere fc is differentiable and thus work only with the Jacobian of fc, rather than the generalized Jacobian. It is a very useful working lemma for verifying semismoothness property in section 4.

Lemma 2.1. Suppose F : IRn→ IRn is strictly continuous and directionally differentiable in a neighborhood of x ∈ IRn. Then, for any 0 < ρ < ∞, the following two statements are equivalent:

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(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 ∈ IRn. 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 fc. 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 λ1 ≤ λ2 be the spectral values of x ∈ IRn and m1 ≤ m2 be the spectral values of y ∈ IRn. Then, we have

1− m1|2sin2θ + |λ2− m2|2cos2θ = kx − yk2, (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 = (x1, x2) ∈ IR × IRn−1 and y = (y1, y2) ∈ IR × IRn−1. (a) If x2 6= 0, y2 6= 0, then we have

ku(i)− v(i)k ≤ 2 sin cos θ

kx2k kx − yk, i = 1, 2, (10) where u(i), v(i) are the unique spectral vectors of x and y, respectively.

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(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) = sin2θ



1 , (−1) cot θ x2 kx2k



, v(1) = sin2θ



1 , (−1) cot θ y2 ky2k

 , where u(1), v(1) are unique. This gives u(1) − v(1) = sin2θ

0 , (−1) cot θ(kxx2

2kkyy2

2k) . Then,

ku(1)− v(1)k = sin θ cos θ

x2

kx2k− y2 ky2k

= sin θ cos θ

x2− y2

kx2k + (ky2k − kx2k)y2 kx2k · ky2k

≤ sin θ cos θ

 1

kx2kkx2− y2k + 1

kx2k|ky2k − kx2k|



≤ sin θ cos θ

 1

kx2kkx2− y2k + 1

kx2kkx2 − y2k



≤ 2 sin θ cos θ

kx2k 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 x2 = 0 or y2 = 0. Then, it is obvious. 2

Lemma 3.3. For any w 6= 0 ∈ IRn, we have ∇w

 w kwk



= 1

kwk



I − wwT kwk2

 . 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 fc Theorem 3.1. For any f : IR → IR, fc is continuous at x ∈ IRn with spectral values λ1, λ2 if and only if f is continuous at λ1, λ2.

Proof. “⇐” Suppose f is continuous at λ1, λ2. For any fixed x = (x1, x2) ∈ IR×IRn−1and y → x, let the spectral factorizations of x, y be x = λ1u(1)2u(2)and y = m1v(1)+m2v(2), respectively. Then, we discuss two cases.

Case (i): If x2 6= 0, then we have fc(y) − fc(x)

= f (m1)v(1)− u(1) + [f (m1) − f (λ1)] u(1) (11) +f (m2)v(2)− u(2) + [f (m2) − f (λ2)] u(2).

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Since f is continuous at λ1, λ2, and from Lemma 3.1, |mi − λi| ≤ c ky − xk, we know f (mi) −→ f (λi) as y → x. In addition, by Lemma 3.2, we have kv(i)−u(i)k −→ 0 as y → x. Thus, equation (11) yields fc(y) −→ fc(x) as y → x because both f (mi) and ku(i)k are bounded. Hence, fc is continuous at x ∈ IRn.

Case (ii): If x2 = 0, no matter y2 is zero or not, we can arrange that x, y have the same spectral vectors. Thus, fc(y) − fc(x) = [f (m1) − f (λ1)] u(1)+ [f (m2) − f (λ2)] u(2). Then, fc is continuous at x ∈ IRn 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, fc is directionally differentiable at x ∈ IRn with spectral values λ1, λ2 if and only if f is directionally differentiable at λ1, λ2.

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

Case (i): If x2 6= 0, we have fc(x) = f (λ1)u(1)+f (λ2)u(2)where λi = x1+(−1)i(tan θ)(−1)ikx2k and u(i) = (−1)isin θ cos θ

(tan θ)(−1)i,kxxT2

2k



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

xu(i)= (−1)isin θ cos θ kx2k

0 0

0 I − x2xT2 kx2k2

 ∀i = 1, 2. (12)

Also by the expression of λi, we know that λi is Fr´echet-differentiable with respect to x, with

xλi =



1 , (−1)itan(−1)iθ xT2 kx2k



∀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 λ1, then it is easy to compute

lim

t→0+

f (λ1 + t × 1) − f (λ1)

t = f01; 1), lim

t→0+

f (λ1− t × 1) − f (λ1)

t = f01; −1), lim

t→0+

f (λ1+ o(t)) − f (λ1)

t = 0.

Note that the spectral value function λ1(x) = x1−cot θkx2k is differentiable when x2 6= 0, which yields

λ1(x + th) = λ1(x) + t∇xλ1h + o(t).

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Let y := ∇xλ1h + o(t)t . For the case of ∇xλ1h < 0, we know y < 0 as t is small. Thus, lim

t→0+

f (λ1(x + th)) − f (λ1(x)) t

= lim

t→0+

f (λ1(x) + ty) − f (λ1(x)) t

= lim

t→0+

f (λ1(x) − (−ty)) − f (λ1(x))

−ty (−y)

= lim

−ty→0+

f (λ1(x) − (−ty)) − f (λ1(x))

−ty lim

t→0+(−y)

= f01(x); −1)(−∇xλ1h)

= f01(x); ∇xλ1h).

Here the positively homogeneous property of directionally differentiable functions is used in the last equation. Similarly, for the other case of ∇xλ1h ≥ 0, we have

lim

t→0+

f (λ1(x + th)) − f (λ1(x))

t = f01(x); ∇xλ1h).

In summary, the composite function f ◦ λ1(·) is directionally differentiable at x. Now we can apply chain rule and product rule on fc(x) = f (λ1)u(1)+ f (λ2)u(2). In other words,

(fc)0(x; h)

= f (λ1)∇xu(1)h + f01; ∇xλ1h)u(1)+ f (λ2)∇xu(2)h + f02; ∇xλ2h)u(2)

= (A1, A2) ∈ IR × IRn−1, where

A1 = f0



λ1; h1− cot θxT2h2 kx2k



sin2θ + f0



λ2; h1+ tan θxT2h2 kx2k



cos2θ (14) and

A2 =

 f0



λ2; h1+ tan θxT2h2 kx2k



− f0



λ1; h1− cot θxT2h2 kx2k



sin θ cos θ x2

kx2k (15) +f (λ2) − f (λ1)

λ2− λ1



I − x2xT2 kx2k2

 h2, with h = (h1, h2) ∈ IR × IRn−1.

Now, applying equations (12) and (13) and using the fact that λ2− λ1 = sin θ cos θkx2k in the A2 term, we see that (fc)0(x; h) can be rewritten in a more compact form as below:

(fc)0(x; h) = f0



λ1; h1− cot θxT2h2 kx2k



u(1)+ f0



λ2; h1+ tan θxT2h2 kx2k

 u(2) +f (λ2) − f (λ1)

λ2− λ1



I − x2xT2 kx2k2



h2. (16)

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Case (ii): If x2 = 0, we compute the directional derivative (fc)0(x; h) at x for any direction h by definition. Let h = (h1, h2) ∈ IR × IRn−1. We have two subcases.

First, consider the subcase of h2 6= 0. From the spectral factorization, we can choose u(1) =

sin2θ , − sin θ cos θkhh2

2k



and u(2) =

cos2θ , sin θ cos θkhh2

2k



such that

 fc(x + th) = f (λ + 4λ1)u(1)+ f (λ + 4λ2)u(2) fc(x) = f (λ)u(1)+ f (λ)u(2)

where λ = x1 and 4λi = t

h1+ (−1)itan(−1)iθkh2k

for all i = 1, 2. Thus, we obtain fc(x + th) − fc(x) = [f (λ + 4λ1) − f (λ)] u(1)+ [f (λ + 4λ2) − f (λ)] u(2). Using the following facts

lim

t→0+

f (λ + 4λ1) − f (λ)

t = lim

t→0+

f (λ + t(h1− cot θkh2k)) − f (λ)

t = f0(λ; h1− cot θkh2k) lim

t→0+

f (λ + 4λ2) − f (λ)

t = lim

t→0+

f (λ + t(h1+ tan θkh2k)) − f (λ)

t = f0(λ; h1+ tan θkh2k) yields

lim

t→0+

fc(x + th) − fc(x) t

= lim

t→0+

f (λ + 4λ1) − f (λ)

t u(1)+ lim

t→0+

f (λ + 4λ2) − f (λ)

t u(2)

= f0(λ; h1− cot θkh2k)u(1)+ f0(λ; h1+ tan θkh2k)u(2) (17) which says (fc)0(x; h) exists.

Secondly, for the subcase of h2 = 0, the same arguments apply except h2/kh2k is replaced by any w ∈ IRn−1 with kwk = 1, i.e., choosing u(1) = sin2θ , − sin θ cos θw and u(2) = (cos2θ , sin θ cos θw). Analogously, we obtain

lim

t→0+

fc(x + th) − fc(x)

t = f0(λ; h1)u(1)+ f0(λ; h1)u(2). (18) which implies (fc)0(x; h) exists with form of (18). From all the above, it shows that fc is directionally differentiable at x when x2 = 0 and its directional derivative (fc)0(x; h) is either in form of (17) or (18).

“⇒” Suppose fc is directionally differentiable at x ∈ IRn with spectral values λ1, λ2, we will prove that f is directionally differentiable at λ1, λ2. For λ1 ∈ IR and any direction d1 ∈ IR, let h := d1u(1)+ 0u(2) where x = λ1u(1)+ λ2u(2). Then, x + th = (λ1+ td1)u(1)+ λ2u(2) and

fc(x + th) − fc(x)

t = f (λ1+ td1) − f (λ1)

t u(1).

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Since fc is directionally differentiable at x, the above equation implies

f01; d1) = lim

t→0+

f (λ1+ td1) − f (λ1)

t exists.

This means f is directionally differentiable at λ1. Similarly, f is also directionally differ- entiable at λ2. 2

Theorem 3.3. For any f : IR → IR, fc is differentiable at x = (x1, x2) ∈ IR × IRn−1 with spectral values λ1, λ2 if and only if f is differentiable at λ1, λ2. Moreover, for given h = (h1, h2) ∈ IR × IRn−1, we have

∇fc(x)h =

b cxT2

kx2k cx2

kx2k aI + (¯b − a)x2xT2 kx2k2

 h1 h2



, when x2 6= 0,

where

a = f (λ2) − f (λ1) λ2− λ1 ,

b = f01) sin2θ + f02) cos2θ,

¯b = f01) cos2θ + f02) sin2θ, c = [f02) − f01)] sin θ cos θ.

When x2 = 0, ∇fc(x) = f0(λ)I with λ = x1.

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 λ1 and λ2, we have that f01; ·) and f02; ·) are linear, which means f0i; a + b) = f0i)a + f0i)b. This together with equations (14) and (15) yield

A1 = f0



λ1; h1− cot θxT2h2 kx2k



sin2θ + f0



λ2; h1+ tan θxT2h2 kx2k

 cos2θ

= f01)h1sin2θ − f01) cot θxT2h2

kx2ksin2θ + f02)h1cos2θ + f02) tan θxT2h2 kx2k cos2θ

= f01) sin2θ + f02) cos2θ h1+ [f02) − f01)] sin θ cos θ xT2 kx2kh2

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and A2 =

 f0



λ2; h1+ tan θxT2h2 kx2k



− f0



λ1; h1− cot θxT2h2 kx2k



sin θ cos θ x2 kx2k +f (λ2) − f (λ1)

λ2− λ1 (I − x2xT2

kx2k2)h2 (19)

=



f02)h1− f01)h1+ f02) tan θxT2h2

kx2k + f01) cot θxT2h2 kx2k



sin θ cos θ x2 kx2k +f (λ2) − f (λ1)

λ2− λ1



I − x2xT2 kx2k2

 h2

= [f02) − f01)] sin θ cos θ x2 kx2kh1 +f02) sin2θ + f01) cos2θ x2xT2

kx2k2h2+f (λ2) − f (λ1) λ2− λ1



I − x2xT2 kx2k2

 h2. Thus, for x2 6= 0, we have

∇fc(x)h =

b cxT2

kx2k cx2

kx2k aI + (¯b − a)x2xT2 kx2k2

 h1 h2



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with

a = f (λ2) − f (λ1) λ2− λ1 ,

b = f01) sin2θ + f02) cos2θ,

¯b = f01) cos2θ + f02) sin2θ, (21) c = [f02) − f01)] sin θ cos θ.

From equation (16), ∇fc(x)h can also be recast in a more compact form:

∇fc(x)h = f01)



h1− cot θxT2h2 kx2k



u(1)+ f02)



h1+ tan θxT2h2 kx2k

 u(2) +f (λ2) − f (λ1)

λ2− λ1



I − x2xT2 kx2k2



h2. (22)

For case of x2 = 0, with linearity of f0(λ; ·) and equations (17) and (18), we have

∇fc(x) = f0(λ)I, (23)

where λ = λ1 = λ2 = x1.

“⇒” Let fc be Fr´echet-differentiable at x ∈ IRn with spectral eigenvalues λ1, λ2, we will show that f is Fr´echet-differentiable at λ1, λ2. Suppose not, then f is not Fr´echet- differentiable at λi for some i ∈ {1, 2}. Thus, either the right- and left-directional

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derivatives of f at λi are unequal or one of them does not exist. In either case, this implies that there exist two sequences of non-zero scalars tν and τν, ν = 1, 2, . . . , converging to zero such that the limits

ν→∞lim

f (λi+ tν) − f (λi)

tν , lim

ν→∞

f (λi+ τν) − f (λi) τν

either are unequal or one of them does not exist. Now for any x = λ1u(1) + λ2u(2), let h := 1 · u(1) + 0 · u(2) = u(1). Then, we know x + th = (λ1 + t)u(1) + λ2u(2) and fc(x + th) = f (λ1+ t)u(1)+ f (λ2)u(2), which give

ν→∞lim

fc(x + tνh) − fc(x)

tν = lim

ν→∞

f (λ1+ tν) − f (λ1) tν u(1)

ν→∞lim

fc(x + τνh) − fc(x)

τν = lim

ν→∞

f (λ1+ τν) − f (λ1) τν u(1).

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

Theorem 3.4. For any f : IR → IR, fc is continuously differentiable (smooth) at x ∈ IRn with spectral values λ1, λ2 if and only if f is continuously differentiable (smooth) at λ1, λ2.

Proof. “⇐” Suppose f is continuously differentiable at x ∈ IRn. From equation (20), it can been seen that ∇fc is continuous at every x with x2 6= 0. It remains to show that ∇fc is continuous at every x with x2 = 0. Fix any x = (x1, 0) ∈ IRn, which says λ1 = λ2 = x1. Let yν = (y1ν, y2ν) ∈ IR × IRn−1 be any sequence converging to x. For those yν2 = 0, applying equation (23) gives ∇fc(yν) = f0(λ(yν))I. Suppose y2ν 6= 0, from equation (21), we have

lim

yν→x,y2ν6=0a = lim

yν→x,y2ν6=0

f (λ2(yν)) − f (λ1(yν))

λ2(yν) − λ1(yν) = f0(x1), lim

yν→x,y2ν6=0b = lim

yν→x,y2ν6=0 f01(yν)) sin2θ + f02(yν)) cos2θ  = f0(x1),

yν→x,ylimν26=0c y2ν

ky2νk = lim

yν→x,y2ν6=0sin θ cos θ [ f02(yν)) − f01(yν)) ] yν2 kyν2k = 0,

yν→x,ylimν26=0(¯b − a)y2νy2νT

ky2νk2 = lim

yν→x,y2ν6=0



f01(yν)) cos2θ + f02(yν)) sin2θ

−f (λ2(yν)) − f (λ1(yν)) λ2(yν) − λ1(yν)

 y2νy2νT ky2νk2 = 0.

Using the facts that both kyyν2ν

2k and ykyν2yν2νT

2k2 are bounded by 1 and then taking the limit in (20) as y → x yield lim

y→x∇fc(y) = f0(x1)I = ∇fc(x). This says ∇fc is continuous at every x ∈ IRn .

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“⇒” 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 : IRn → IR is strictly continuous. Then, there exist continuously differentiable functions fν : IRn → IR, ν = 1, 2, · · · , converging uniformly to f on any compact set C in IRn and satisfying

k∇fν(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) fc is strictly continuous at x ∈ IRn with spectral values λ1, λ2 if and only if f is strictly continuous at λ1, λ2.

(b) fc 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 ∈ IRn with spectral values λ1 and λ2 given by (6). Suppose f is strictly continuous at λ1 and λ2. Then, there exist κi > 0 and δi > 0 for i = 1, 2 such that

|f (b) − f (a)| ≤ κi|b − a|, ∀ a, b ∈ [λi− δi, λi+ δi] i = 1, 2.

Let ¯δ := min{δ1, δ2} and C := [λ1 − ¯δ1, λ1 + ¯δ] ∪ [λ2 − ¯δ, λ2 + ¯δ]. Define a real-valued function ¯f : IR → IR as

f (a) =¯









f (a) if a ∈ C,

(1 − t)f (λ1+ ¯δ) if λ1+ ¯δ < λ2− ¯δ and, for some t ∈ (0, 1), +tf (λ2− ¯δ) a = (1 − t)(λ1+ ¯δ) + t(λ2− ¯δ),

f (λ1− ¯δ) if a < λ1− ¯δ, f (λ2+ ¯δ) if a > λ2+ ¯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 fν : IR → IR, ν = 1, 2, · · · , converging uniformly to ¯f and satisfying

|(fν)0(a)| ≤ κ, ∀ a ∈ C, ∀ ν.

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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 = µ1u(1)+ µ2u(2), by direct computation, we have

(fν)c(w) − fc(w)

2 = sin2θ|fν1) − f (µ2)|2+ cos2θ|fν2) − f (µ2)|2.

This together with fν converging uniformly to f on C implies that (fν)c converges uniformly to fc on B(x, δ).

Next, we explain that k∇(fν)c(w)k is uniformly bounded. Indeed, for w2 = 0, from equation (23) we have k∇(fν)c(w)k = |(fν)0(w1)| ≤ κ. For general w2 6= 0, it is not hard to check k∇(fν)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 (fν)c converges uniformly to fc, for any  > 0 there exists an integer ν0 such that for all ν ≥ ν0 we have

k(fν)c(w) − fc(w)k ≤ ky − zk ∀w ∈ B(x, δ).

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

fc(y) − fc(z)

=

fc(y) − (fν)c(y) + (fν)c(y) − (fν)c(z) + (fν)c(z) − fc(z)

fc(y) − (fν)c(y)

+ k(fν)c(y) − (fν)c(z)k +

(fν)c(z) − fc(z)

≤ 2ky − zk +

Z 1 0

∇(fν)c(z + t(y − z))(y − z)dt

≤ (M + 2)ky − zk.

This shows that fc is strictly continuous at x.

“⇒” Suppose that fc is strictly continuous at x with eigenvalues λ1 and λ2 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

fc(y) − fc(z)

≤ M ky − zk.

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

y := x + (a − λi)u(i), z := x + (b − λi)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)

=

fc(y) − fc(z)

≤ M ky − zk.

which says that f is strictly continuous at λ1 and λ2 because ku(1)k = sin θ and u(2)

= cos θ.

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

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### 4 Semismoothness Property

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

Theorem 4.1. For any f : IR → IR, fc is semismooth at x ∈ IRn with spectral values λ1, λ2 if and only if f is semismooth at λ1, λ2.

Proof. “⇒” Suppose fc is semismooth, then fc 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 fc is differentiable at x + h, where x := (α, 0) ∈ IR × IRn−1 and h := (η, 0) ∈ IR × IRn−1. Hence, we can choose the same spectral vectors for x + h = (α + η, 0) and x = (α, 0) such that

 fc(x + h) = f (α + η)u(1)+ f (α + η)u(2), fc(x) = f (α)u(1)+ f (α)u(2).

Since fc is semismooth, by Lemma 2.1, we know

fc(x + h) − fc(x) − ∇fc(x + h)h = o(khk). (24) On the other hand, equation (23) yields ∇fc(x + h)h = f0(α + η)Ih = (f0(α + η)η, 0) . Plugging this into equation (24) yields f (α + η) − f (α) − f0(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, fc is strictly continuous and directionally differentiable. For any x = (x1, x2) ∈ IR × IRn and h = (h1, h2) ∈ IR × IRn such that fc is differentiable at x + h, we will verify that

fc(x + h) − fc(x) − ∇fc(x + h)h = o(khk).

Case (i): If x2 6= 0, let λi 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 mi be the spectral values of z with the associated spectral vectors v(i). Hence, we have

 fc(x) = f (λ1)u(1)+ f (λ2)u(2), fc(x + h) = f (m1)v(1)+ f (m2)v(2). Suppose now fc is differentiable at z. From (20), we know

∇fc(x + h) =

b cz2T

kz2k cz2

kz2k aI + (¯b − a) z2z2T kz2k2

 ,

(17)

where

a = f (m2) − f (m1) m2− m1

,

b = f0(m1) sin2θ + f0(m2) cos2θ,

¯b = f0(m1) cos2θ + f0(m2) sin2θ, c = [f0(m2) − f0(m1)] sin θ cos θ.

With this, we can write out fc(x + h) − fc(x) − ∇fc(x + h)h := (Ξ1, Ξ2) where Ξ1 ∈ IR and Ξ2 ∈ IRn−1. Since the expansion is very long, for simplicity, we denote Ξ1 be the first component and Ξ2 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:

Ξ1 = sin2θ



f (m1) − f (λ1) − f0(m1)(h1− cot θz2Th2 kz2k)



+ cos2θ



f (m2) − f (λ2) − f0(m2)(h1+ tan θz2Th2 kz2k)



= sin2θ {f (m1) − f (λ1) − f0(m1) (h1− cot θ(kz2k − kx2k)) + o(khk)}

+ cos2θ {f (m2) − f (λ2) − f0(m2) (h1+ tan θ(kz2k − kx2k)) + o(khk)}

= o (h1− (kz2k − kx2k)) + o(khk) + o (h1+ (kz2k − kx2k)) + o(khk).

In the above expression of Ξ1, the third equality holds since the following:

z2Th2

kz2k = z2T(z2− x2)

kz2k = kz2k −kz2kkx2k kz2k cos α

= kz2k − kx2k 1 + O(α2) = kz2k − kx2k 1 + O(khk2)

= kz2k − kx2k (1 + o(khk))

where α is the angle between x2 and z2and note that z2−x2 = h2gives O(α2) = O(khk2).

In addition, the last equality in expression of Ξ1 holds because f is semismooth and mi− λi = h1+ (−1)i(tan θ)(−1)i(kz2k − kx2k).

On the other hand, due to

h1+ (−1)i(tan θ)(−1)i(kz2k − kx2k)

≤ |h1| + M kz2− x2k ≤ M (|h1| + kh2k) where M = max{tan θ, cot θ} ≥ 1. Then, we observe that when khk → 0,

|h1| + (−1)i(tan θ)(−1)i(kz2k − kx2k) → 0

|h1| + (−1)i(tan θ)(−1)i(kz2k − kx2k) = O(khk).

Thus, we obtain o

h1 + (−1)i(tan θ)(−1)i(kz2k − kx2k)

= o(khk), which implies that the first component Ξ1 is o(khk).

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