DOI 10.1007/s10898-015-0312-5
The H-differentiability and calmness of circular cone functions
Jinchuan Zhou1 · Yu-Lin Chang2 · Jein-Shan Chen2
Received: 25 January 2014 / Accepted: 20 May 2015 / Published online: 28 May 2015
© Springer Science+Business Media New York 2015
Abstract LetLθbe the circular cone inRnwhich includes second-order cone as a special case. For any function f fromR to R, one can define a corresponding vector-valued function fLθonRnby applying f to the spectral values of the spectral decomposition of x∈ Rnwith respect toLθ. The main results of this paper are regarding the H -differentiability and calmness of circular cone function fLθ. Specifically, we investigate the relations of H -differentiability and calmness between f and fLθ. In addition, we propose a merit function approach for solving the circular cone complementarity problems under H -differentiability. These results are crucial to subsequent study regarding various analysis towards optimizations associated with circular cone.
Keywords Circular cone· H-differentiable · Calmness
Mathematics Subject Classification 26A27· 26B05 · 26B35 · 49J52 · 90C33 · 65K05
Jinchuan Zhou’s work is supported by National Natural Science Foundation of China (11101248, 11271233, 11171247) and Shandong Province Natural Science Foundation (ZR2012AM016).
Jein-Shan Chen’s work is supported by Ministry of Science and Technology, Taiwan.
B
Jein-Shan Chen jschen@math.ntnu.edu.tw Jinchuan Zhoujinchuanzhou@163.com Yu-Lin Chang
ylchang@math.ntnu.edu.tw
1 Department of Mathematics, School of Science, Shandong University of Technology, Zibo 255049, People’s Republic of China
2 Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
1 Introduction
Conic programming has drawn a lot of attention in the last decade. Generally speaking, the research works in this field are divided into two directions. One is on the general convex cone, see the excellent monograph written by Bonnans and Shapiro [3]; while the other focuses on some specific convex cones. In the latter case, much attention is paid to the so-called symmetric cone, which includes positive semi-definite matrices cone [26] and second-order cone [1] as special cases. The Jordan algebraic structure associated with symmetric cones allows us to deal with them in an unified way [13]. However, there exists a lot of cones which are convex but non-symmetric; for example, p-order cone [2], Lp-cone [15], and copositive cone [12], etc. For these non-symmetric cones, until now we don’t know how to tackle with them in an unified framework. In fact, it needs to investigate them one-by-one because their structures are rather different. In this paper, we focus on a special non-symmetric convex cone, called circular cone. More precisely, the circular cone [5,27,28] 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 beθ with θ ∈ (0,π2). Then, the n-dimensional circular cone denoted byLθcan be expressed as
Lθ := {x = (x1, x2)T ∈ R × Rn−1| cos θx ≤ x1}.
Note thatL45◦ corresponds the well-known second-order coneKn (SOC, for short), which is given by
Kn:= {x = (x1, x2)T ∈ R × Rn−1| x2 ≤ x1}.
For any x = (x1, x2) ∈ R × Rn−1, there is a spectral decomposition for x associated with circular cone case [27, Theorem 3.1], which is given as
x= λ1(x) · u(1)x + λ2(x) · u(2)x (1)
where
λ1(x) = x1− x2 cot θ
λ2(x) = x1+ x2 tan θ (2)
and ⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
u(1)x = 1 1+ cot2θ
1 0
0 cotθ I
1
− ¯x2
=
sin2θ
−(sin θ cos θ) ¯x2
u(2)x = 1 1+ tan2θ
1 0
0 tanθ I
1
¯x2
=
cos2θ (sin θ cos θ) ¯x2
(3)
with ¯x2 = x2/x2 if x2 = 0, and ¯x2 being any vectorw in Rn−1satisfyingw = 1 if x2 = 0. With this spectral factorization (1), for any given f : R → R, we can define the following vector-valued function associated with circular cone (which we call it “circular cone function” in general):
fLθ(x) := f (λ1(x)) u(1)x + f (λ2(x)) u(2)x . (4) Indeed, we can write out an explicit expression for (4) by plugging inλi(x) and u(i)x :
fLθ(x) =
⎡
⎢⎢
⎢⎣
f(x1− x2 cot θ)
1+ cot2θ + f(x1+ x2 tan θ) 1+ tan2θ
−f(x1− x2 cot θ) cot θ
1+ cot2θ + f(x1+ x2 tan θ) tan θ 1+ tan2θ
¯x2
⎤
⎥⎥
⎥⎦. (5)
In particular, the formula (1) and (4) reduce to the well-known spectral decomposition and spectral function associated with second-order cone programming; see [6,9,10] for more details. A natural question is what properties of fLθ are inhered from f and vice versa.
Once this is done, then we can analyze the property of the vector-valued function fLθ by just study the single variable scalar function f . This reduces the difficulty of our analysis significantly, since f is a real valued function onR. In [5,28], the authors have answered this question in part, i.e., they show that the properties of continuity, strict continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, and (ρ-order) semismoothness are each inherited by fLθ from f , and vice versa. However, it should be noted that the properties of Fréchet differentiability, continuous differentiability, and (ρ-order) semismoothness are requiring the condition of the locally Lipschitz continuity in advance. Hence, we hope to further study the properties between of f and fLθ without imposing Lipschitz continuity. Inspired by these points, we study the properties of Calmness and H -differentiability. Moreover, the exact formula of calmness modulus and H -differential are also established between fLθ and f . In addition, we propose a merit function approach for solving the circular cone complementarity problems under H -differentiability.
It is well known that there exists a variety of definitions regarding nonsmoothness for extending the classical concept of differentiability. Why do we focus on the H - differentiability? We clarify our motivation as below. As indicated in [3, Chapter 4], the topic of studying nonsmoothness is a natural thing in optimization field; for example, for each fixed x∈ R consider the following optimization problem:
mint∈t x subject to := [−1, 1].
Clearly, the optimal value function is−|x|, which is not differentiable at the origin. Note that the data involved in the above problem is rather simple and is smoothing. An important concept in the field of nonsmooth analysis is the generalized Jacobian for locally Lipschitz functions; see [11]. How to deal with the non-Lipschitz function? By this motivation, the concept of “H -differentiability” is introduced. It is well known that the Fréchet derivative of a Fréchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitz continuous function, the Bouligand subdifferential of a semismooth function, and the C- differential of a C-differentiable function are all examples of H -differentials; see [22] for more detailed discussion.
In [28], the following important relationship between circular cone and second-order cone is discovered. In particular, there holds thatLθ= A−1Knwhere
A=
tanθ 0
0 I
.
This simple and basic relation helps us to study the normal cone, tangent cone, second-order tangent cone, second-order regularity of circular cone by using the corresponding results in second-order cone [27]. It however does not means that the extension of the results from second-order cone to circular cone is trivial. Indeed, the following two cases are possible:
(i) one category of results is independent of the angle (i.e., still holds in the framework of circular cone); (ii) the second category is dependent of the angle, for example, for x, y ∈Lθ, the inequality
det(e + x + y) ≤ det(e + x) det(e + y),
where det(x) := λ1(x)λ2(x) and e := (1, 0, . . . , 0) ∈ Rn, holds in the second-order cone setting. But, for circular cone setting, we show that this inequality fails forθ ∈ (0, 45◦), but
holds forθ ∈ [45◦, 90◦); see [29] for more information. In addition, it is surprised that a necessary condition for f to be theLθ-convexity is thatθ ∈ [45◦, 90◦). Moreover, the exact formula of various derivative of projection over the circular coneLθcannot be obtained by simple using the above basic relationship between the circular cone and second-order cone;
see [30]. The aforementioned facts and observations give us new insight on circular cone and attract our attention to figuring out what role played by the angleθ in different settings.
To end this section, we say a few words about notations used in this paper. Define the ball of radiusδ > 0 centered at x asB(x, δ) := {v ∈ Rn| v − x ≤ δ}. For convenience of notation, the unit ball at origin is written asB, i.e.,B:=B(0, 1) = {v ∈ Rn| v ≤ 1}. For a vector x∈ Rnand a matrix M∈ Rn×n, letx stand for the Euclidean norm and M for the norm induced by · , i.e., x =n
i=1xi2andM = max
x=1Mx.
2 Preliminaries
In this section, we review some basic concepts and materials about H -differentiability and calmness that will be used in subsequent analysis. We start with the concept of calmness.
Definition 2.1 A mapping F: Rn → Rmis called to be calm at ¯x if there exist δ > 0 and L> 0 such that
F(x) − F( ¯x) ≤ Lx − ¯x, ∀x ∈B( ¯x, δ). (6) This is equivalent to saying
cam(F)( ¯x) := lim sup
x→ ¯x x= ¯x
F(x) − F( ¯x)
x − ¯x < +∞. (7)
Here we call cam(F)( ¯x) the calm modulus of F at ¯x.
Note that (7) means that for any > 0 there exists δ > 0 such that
F(x) − F( ¯x) ≤ (cam(F)( ¯x) + ) x − ¯x, ∀x ∈B( ¯x, δ). (8) Recall from [21, Chapter 9] that F is locally Lipschitz at¯x if and only if lip(F)( ¯x) < +∞, where
lip(F)( ¯x) := lim sup
x,x→ ¯x x=x
F(x) − F(x)
x − x .
It is clear that cam(F)( ¯x) ≤ lip(F)( ¯x). However, the inequality can be strict. To see this, we check the function
f(x) =
⎧⎨
⎩ x sin
1 x
if x= 0,
0 if x= 0,
from which we see cam( f )(0) = 1 < +∞ = lip( f )(0). As mentioned in [21, page 351], the inequality (6) only involves comparisons between ¯x and nearpoint x, not between all possible pairs of points x and xin some neighborhood of ¯x. Indeed, the locally Lipschitz continuity can be viewed as “locally uniform calmness”, while Lipschitz continuity is viewed as “uniform calmness”.
Next, we talk about the concepts of H -differentiability and H -differential of a function, which were first proposed by Gowda and Ravindran in [17]. Their motivation was to study a generalization (to nonsmooth case) of a result of Gale and Nikaido [14] which asserts that if the Jacobian matrix of a differentiable function f from a closed rectangle K ⊆ RnintoRn is a P-matrix at each point of K , then f is one-to-one on K .
Definition 2.2 Given a function F : ⊆ Rn → Rm, where is an open set in Rn and ¯x ∈ , we say that a nonempty subset T ( ¯x), also denoted by TF( ¯x), of Rm×n is an H -differential of F at ¯x if for every sequence {xk} ⊆ converging to ¯x, there exist a subsequence{xkj} and a matrix A ∈ T ( ¯x) such that
F(xkj) − F( ¯x) − A(xkj − ¯x) = o(xkj− ¯x). (9) We say that F is H -differentiable at ¯x if the H-differential of F at ¯x is nonempty.
As remarked in [16,17,22–25], the Fréchet derivative of a Fréchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitz continuous function, the Bouligand subdifferential of a semismooth function, and the C-differential of an C-differentiable func- tion are all examples of H -differentials. In addition, any superset of an H -differential is an H -differential, H -differentiability implies continuity, and H -differentials satisfy simple sum, product, and chain rules. The class of H -differentiable functions is wider than the class of semismooth functions, since the former is not required to be locally Lipschitz continuous or directionally differentiable.
Now we point out that there is a useful equivalent expression for condition (9). For sim- plicity, let “” denote the convergence in the sense of taking some subsequence. With this notation, we see that condition (9) can be equivalently described as follows: For every sequence{x + tkdk} with tk↓ 0 and dk = 1 for all k, there exist tkj ↓ 0 and dkj → d and A∈ TF( ¯x) such that
F( ¯x + tkjdkj) − F( ¯x)
tkj → Ad, (10)
i.e.,
F( ¯x + tkdk) − F( ¯x)
tk Ad.
Below are summaries of some well-known facts about H -differentiability, for more details, please refer to [16,17,22–25].
Remark 2.1 (i) Any superset of an H -differential is an H -differential.
(ii) H -differentiability implies continuity.
(iii) If a function F : ⊆ Rn→ Rmis H -differentiable at a point ¯x, then F is calm at ¯x.
Note that the set TF( ¯x) plays an important role in the definition of H-differentiability. For example, the converse statement of Remark2.1(iii) holds by taking TF( ¯x) := Rm×n [25, page 281]. For completeness, we provide a simple proof for this claim as follows. Suppose that F is H -differentiable at ¯x with TF( ¯x) = Rm×n. If (6) fails to hold, then we can find a sequence{xk} converging to ¯x such that
F(xk) − F( ¯x)
xk− ¯x → +∞. (11)
For this sequence{xk}, by the definition of H-differentiability of F at ¯x, there exists a sequence{xkj} such that
F(xkj) − F( ¯x) − A(xkj − ¯x) = o(xkj − ¯x).
This implies
F(xkj) − F( ¯x)
xkj − ¯x = A(xkj− ¯x) + o(xkj − ¯x)
xkj − ¯x
≤ A +o(xkj − ¯x)
xkj − ¯x
→ A,
which contradicts (11). To see the converse, we take an arbitrary sequence{xk} satisfying xk= ¯x and xk→ ¯x. Since
F(xk)−F( ¯x)
xk− ¯x
is bounded by (6) and
xk− ¯x
xk− ¯x
is also bounded, there exists a subsequence,ξ ∈ Rm, and d∈ Rnsuch that
F(xkj) − F( ¯x)
xkj − ¯x → ξ and xkj− ¯x
xkj− ¯x → d.
Now, take a matrix A∈ TF(x) = Rm×nsuch thatξ = Ad. Note that such matrix always exists because A has mn variables. Hence,
F(xkj) − F( ¯x) − A(xkj− ¯x)
xkj − ¯x → ξ − Ad = 0, i.e.,
F(xkj) − F( ¯x) − A(xkj − ¯x) = o(xkj − ¯x).
Remark2.1(i) says that any superset of an H -differential is also an H -differential. This indicates that if a function g is H -differentiable at x with Tg(x), then g is also H-differentiable at x with Tg(x) whenever Tg(x) ⊆ Tg(x). However, for an arbitrary function g, it is not guaranteed to become an H -differentiable function by simply taking a larger set. To see this, we present below that there exists a function that is not H -differentiable even if Tg(x) takes the whole space. For example, consider the function
g(t) = |t|p with p∈ (0, 1).
Let d = 1 and tk ↓ 0. Then, (g(tk) − g(0))/tk = |tk|p−1 → +∞, since p < 1. Hence, Tg(0) = ∅, which implies g is not H-differentiable at 0. Indeed, g(t) = |t|pis not calm at 0.
3 Calmness
This section is devoted to properties of calmness. First, we explore some basic properties about composite function and then establish the calmness relation between fLθand f .
For mappings F, G : Rn → Rm and S : Rl → Rn, we define (F · G)(x) :=
F(x)TG(x), (F ◦ S)(x) := F(S(x)), and (y ◦ F)(x) := yTF(x) =m
i=1yiFi(x) where y= (y1, · · · , ym)T ∈ Rmand Fiis the component function of F. Inspired by [21, Chapter 9], we obtain the following results.
Proposition 3.1 Given three mappings F, G : Rn → Rm, S : Rl → Rn, ¯x ∈ Rn, and
¯z ∈ Rl. Suppose that S is continuous at¯z. Then,
(a) cam(β F)( ¯x) = |β|cam(F)( ¯x) for all β ∈ R;
(b) cam(F + G)( ¯x) ≤ cam(F)( ¯x) + cam(G)( ¯x);
(c) cam(F · G)( ¯x) ≤ cam(F)( ¯x) cam(G)( ¯x) + F( ¯x) cam(G)( ¯x) + G( ¯x) cam(F)( ¯x);
(d) cam(F ◦ S)(¯z) ≤ cam(F)(S(¯z)) cam(S)(¯z);
(e) If cam(F)( ¯x) < +∞, then cam(F)( ¯x) = max
y∈B(y ◦ F)( ¯x) = max
y=1(y ◦ F)( ¯x);
(f) cam(Fi)( ¯x) ≤ cam(F)( ¯x) for i = 1, · · · , m and
cam(F)( ¯x) ≤ (cam(F1)( ¯x), cam(F2)( ¯x), · · · , cam(Fm)( ¯x)) . (12)
Proof (a) The result follows from (7) because
cam(β F)( ¯x) = lim sup
x→ ¯x x= ¯x
β F(x) − β F( ¯x)
x − ¯x = |β| lim sup
x→ ¯x x= ¯x
F(x) − F( ¯x)
x − ¯x = |β|cam(F)( ¯x).
(b) The result follows from
cam(F + G)( ¯x) = lim sup
x→ ¯x x= ¯x
(F + G)(x) − (F + G)( ¯x)
x − ¯x
≤ lim sup
x→ ¯x x= ¯x
F(x) − F( ¯x)
x − ¯x +G(x) − G( ¯x)
x − ¯x
≤ lim sup
x→ ¯x x= ¯x
F(x) − F( ¯x)
x − ¯x + lim sup
x→ ¯x x= ¯x
G(x) − G( ¯x)
x − ¯x
= cam(F)( ¯x) + cam(G)( ¯x).
(c) It is trivial if F is not calm at¯x, since cam(F)( ¯x) = +∞ in this case. If F is calm at ¯x, then for any > 0, there exists δ > 0 such that
F(x) − F( ¯x) ≤ (cam(F)( ¯x) + ) x − ¯x, ∀x ∈B( ¯x, δ).
For any x∈B( ¯x, ˆδ) with ˆδ := min{δ, 1}, we have
F(x) ≤ F(x) − F( ¯x) + F( ¯x) ≤ (cam(F)( ¯x) + ) x − ¯x + F( ¯x)
≤ cam(F)( ¯x) + F( ¯x) + ,
which means that F is locally bounded at ¯x. Thus,
cam(F · G)( ¯x) = lim sup
x→ ¯x x= ¯x
(F · G)(x) − (F · G)( ¯x)
x − ¯x
= lim sup
x→ ¯x x= ¯x
F(x)TG(x) − F( ¯x)TG( ¯x)
x − ¯x
≤ lim sup
x→ ¯x x= ¯x
F(x)TG(x)−F(x)TG( ¯x)
x − ¯x +F(x)TG( ¯x)−F( ¯x)TG( ¯x)
x − ¯x
≤ lim sup
x→ ¯x x= ¯x
F(x)G(x) − G( ¯x)
x − ¯x +G( ¯x)F(x) − F( ¯x)
x − ¯x
≤ (cam(F)( ¯x) + F( ¯x) + ) cam(G)( ¯x) + G( ¯x) cam(F)( ¯x).
Since > 0 can be taken sufficiently small, the desired result follows.
(d) Notice that
cam(F ◦ S)(¯z) = lim sup
z→¯zz=¯z
(F ◦ S)(z) − (F ◦ S)(¯z)
z − ¯z
= lim sup
z→¯z z=¯z
F(S(z)) − F(S(¯z))
S(z) − S(¯z)
S(z) − S(¯z)
z − ¯z
≤ lim sup
z→¯zz=¯z
F(S(z)) − F(S(¯z))
S(z) − S(¯z) lim sup
z→¯zz=¯z
S(z) − S(¯z)
z − ¯z
≤ cam(F)(S(¯z)) cam(S)(¯z).
(e) Given y∈B, for the linear mapping y: Rn → R defined as y(v) := y, v, it is clear that cam(y)(v) = y for all v ∈ Rn, since
cam(y)(v) = lim sup
u→vu=v
y(u) − y(v)
u − v
= lim sup
u→v u=v
y, u − v
u − v
= y,
where the “limsup” can be attained by taking u= v + ty with t > 0. Hence cam(y ◦ F)( ¯x) = lim sup
x→ ¯x x= ¯x
(y ◦ F)(x) − (y ◦ F)( ¯x)
x − ¯x
= lim sup
x→ ¯x x= ¯x
y, F(x) − F( ¯x)
x − ¯x
≤ y lim sup
x→ ¯x x= ¯x
F(x) − F( ¯x)
x − ¯x
= y cam(F)( ¯x)
≤ cam(F)( ¯x),
where the last step is due to the facty ≤ 1 since y ∈B. Hence, cam(F)( ¯x) ≥ max
y∈Bcam(y ◦ F)( ¯x). (13)
Conversely,
cam(F)( ¯x)=lim sup
x→ ¯x x= ¯x
F(x)−F( ¯x)
x − ¯x = lim
xk→ ¯x xk= ¯x
F(xk)−F( ¯x)
xk− ¯x = lim
xk→ ¯x xk= ¯x
yk, F(xk) − F( ¯x)
xk− ¯x
(14) where the last step comes from the fact
F(xk) − F( ¯x) = max
y∈By, F(xk) − F( ¯x) = yk, F(xk) − F( ¯x)
for some yk withyk = 1. Since {yk} is bounded, we assume yk converges to ¯y with
¯y = 1. Thus, it follows from (14) that cam(F)( ¯x) = lim
xk→ ¯x xk= ¯x
¯y, F(xk) − F( ¯x)
xk− ¯x + lim
xk→ ¯x xk= ¯x
yk− ¯y, F(xk) − F( ¯x)
xk− ¯x
= lim
xk→ ¯x xk= ¯x
¯y, F(xk) − F( ¯x)
xk− ¯x
= lim
xk→ ¯x xk= ¯x
( ¯y ◦ F)(xk) − ( ¯y ◦ F)( ¯x)
xk− ¯x
≤ lim sup
x→ ¯x x= ¯x
( ¯y ◦ F)(x) − ( ¯y ◦ F)( ¯x)
x − ¯x
= cam( ¯y ◦ F)( ¯x) ≤ max
y=1cam(y ◦ F)( ¯x) ≤ max
y∈Bcam(y ◦ F)( ¯x), where the second equality is due to the fact cam(F)( ¯x) < +∞ and yk → ¯y. This together with (13) yields the desired result.
(f) Notice that
cam(Fi)( ¯x) = lim sup
x→ ¯x x= ¯x
|Fi(x) − Fi( ¯x)|
x − ¯x ≤ lim sup
x→ ¯x x= ¯x
F(x) − F( ¯x)
x − ¯x
= cam(F)( ¯x), ∀i = 1, . . . , m.
The relation (12) holds trivially if Fiis not calm at¯x for some i = 1, . . . , m. Hence we need to show that (12) holds when Fiis calm at¯x for all i = 1, . . . , m. In this case, for any > 0 we get from (8) that
|Fi(x) − Fi( ¯x)| ≤ (cam(Fi)( ¯x) + ) x − ¯x, ∀x ∈B( ¯x, δi), i = 1, . . . , m.
For x∈B( ¯x, δ) with δ := min{δ1, . . . , δm}, we have
F(x) − F( ¯x) =
m
i=1
|Fi(x) − Fi( ¯x)|2
1
2
≤
m
i=1
(cam(Fi)( ¯x) + )2
1
2
x − ¯x.
Hence
cam(F)( ¯x) ≤ (cam(F1)( ¯x) + , cam(F2)( ¯x) + , · · · , cam(Fm)( ¯x) + ) . Since > 0 can be taken sufficiently small, the desired result follows.
Note that Prop.3.1(a) means that cam(F)( ¯x) is positive homogeneous on F, and Prop.
3.1(b) indicates cam(F)( ¯x) is sublinear on F. These two facts imply that cam(F)( ¯x) is convex in F. As mentioned above, we know that F is calm at ¯x if and only if cam(F)( ¯x) < +∞.
Hence, the above results further indicate the following statements.
Remark 3.1 (a) If F and G are calm at¯x, then F + G and β F for β ∈ R are calm at ¯x, i.e., the set of all functions being calm at ¯x constitutes a linear subspace.
(b) If F and G are calm at ¯x, then F · G is calm at ¯x.
(c) F is calm if and only if Fiis calm for i = 1, 2, · · · , m.
(d) If F is calm at S(¯z) and S is calm at ¯z, then F ◦ S is calm at ¯z.
The relation of calmness and calm modulus between fLθ and f are given below.
Theorem 3.1 Let f : R → R be a real-valued function and fLθ be defined as in (4).
Suppose x has spectral factorization given as in (1–3). Then,
(a) fLθis calm at x if and only if f is calm atλi(x) for i = 1, 2. Moreover, if fLθ is calm at x, then
cam( f )(λi(x)) ≤ cam( fLθ)(x), ∀ i = 1, 2;
if f is calm atλi(x) for i = 1, 2, then
cam( fLθ)(x) ≤
√2 max{tan θ, cot θ}(tan θ + cot θ + 2)
tanθ + cot θ cam( f )(x1), (15)
when x2= 0; otherwise cam( fLθ)(x)
≤
√2 max{tan θ, cot θ}
tanθ + cot θ {(1 + tan θ)cam( f )(λ1(x)) + (1 + cot θ)cam( f )(λ2(x))}
+| f (λ2(x)) − f (λ1(x))|
λ2(x) − λ1(x) (16)
(b) fLθis calm overRnif and only if f is calm overR.
Proof (a) “⇒” Suppose that fLθ is calm at x. To proceed the arguments, we discuss two cases.
Case 1 x2= 0. Note that
cam( f )(x1) = lim sup
t→0
| f (x1+ t) − f (x1)|
|t|
= lim sup
t→0
fLθ(x + te) − fLθ(x)
|t|
≤ cam( fLθ)(x).
This says that f is calm atλi(x) = x1with cam( f )(λi(x)) ≤ cam( fLθ)(x) for i = 1, 2.
Case 2 x2 = 0. Let y = x + tu(1)x for t∈ (λ1(x) − λ2(x), λ2(x) − λ1(x)). Then, λ1(y) = λ1(x) + t, λ2(y) = λ2(x), u(i)y = u(i)x for i= 1, 2. Note that
y − x = |t|u(1)x and fLθ(y) − fLθ(x) = |f(λ1(x) + t) − f (λ1(x))| · u(1)x .
Hence
cam( f )(λ1(x)) = lim sup
t→0
| f (λ1(x) + t) − f (λ1(x))|
|t|
= lim sup
y=x+tu(1)x t→0
fLθ(y) − fLθ(x)
y − x
≤ cam( fLθ)(x).
Hence, f is calm atλ1(x) with cam( f )(λ1(x)) ≤ cam( fLθ)(x). By following the same argu- ments, we readily obtain the calmness of f atλ2(x) with cam( f )(λ2(x)) ≤ cam( fLθ)(x).
“⇐” Suppose that f is calm at λi(x) for i = 1, 2. Consider the following two cases.
Case 1 x2= 0. Let φ(z2) = ¯z2= zz22for z2= 0. Since x2= 0, then φ(z2) is continuously differentiable near x2with∇φ(z2) = z12(I − ¯z2¯zT2). According to [10, Lemma 1]
I− ¯z2¯zT2 = (u1, . . . , un−2)diag[1, 1, . . . , 1](u1, . . . , un−2)T
where{u1, . . . , un−2} is any orthonormal set of vectors that spans the subspace of Rn−1 orthogonal to ¯z2. This implies∇φ(z2) = z1
2I − ¯z2¯zT2 = z1
2. For any given ∈
(0, x2), we have z2 ≥ x2 − as z2 sufficiently close x2, and hence∇φ(z2) ≤ 1/(x2 − ). Thus, as y is sufficiently close to x we have
φ(y2) − φ(x2) =
1
0 ∇φ (x2+ t(y2− x2)) (y2− x2)dt
≤
1
0 ∇φ (x2+ t(y2− x2)) y2− x2dt
≤ 1
x2 − y2− x2
≤ 1
x2 − y − x. (17)
Then, it follows from (5) that
fLθ(y) − fLθ(x)
= 1
tanθ + cot θ
⎡
⎣tanθ [ f (λ1(y)) − f (λ1(x))] + cot θ [ f (λ2(y)) − f (λ2(x))]
([ f (λ1(x)) − f (λ1(y))] + [ f (λ2(y)) − f (λ2(x))]) φ(y2) + [ f (λ2(x)) − f (λ1(x))] (φ(y2) − φ(x2))
⎤
⎦
≤ 1
tanθ + cot θ
tanθ [ f (λ1(y)) − f (λ1(x))] + cot θ [ f (λ2(y)) − f (λ2(x))]
([ f (λ1(x)) − f (λ1(y))] + [ f (λ2(y)) − f (λ2(x))]) φ(y2)
+ 1
tanθ + cot θ| f (λ2(x)) − f (λ1(x))|φ(y2) − φ(x2) (18)
≤ 1
tanθ + cot θ {|tan θ [ f (λ1(y)) − f (λ1(x))] + cot θ [ f (λ2(y)) − f (λ2(x))]|
+ |[ f (λ1(x)) − f (λ1(y))] + [ f (λ2(y)) − f (λ2(x))]|}
+ | f (λ2(x)) − f (λ1(x))|
(tan θ + cot θ)(x2 − )y − x
≤ 1
tanθ +cot θ{(1+tan θ) | f (λ1(y))− f (λ1(x))|+(1+cot θ)| f (λ2(y))− f (λ2(x)) |}
+| f (λ2(x)) − f (λ1(x))|
λ2(x) − λ1(x)
x2
x2 − y − x,
where the second inequality is due to (17) and the last step comes from the fact
| f (λ2(x)) − f (λ1(x))|
(tan θ + cot θ)(x2 − ) = | f (λ2(x)) − f (λ1(x))|
(tan θ + cot θ)x2
x2
x2 −
= | f (λ2(x)) − f (λ1(x))|
λ2(x) − λ1(x)
x2
x2 − due toλ2(x) − λ1(x) = (tan θ + cot θ)x2. Hence
fLθ(y) − fLθ(x)
y − x
≤ 1
tanθ + cot θ
×
(1 + tan θ)| f (λ1(y)) − f (λ1(x))|
y − x + (1 + cot θ)| f (λ2(y)) − f (λ2(x))|
y − x
+| f (λ2(x)) − f (λ1(x))|
λ2(x) − λ1(x)
x2
x2 −
≤
√2 max{tan θ, cot θ}
tanθ + cot θ
×
(1 + tan θ)| f (λ1(y)) − f (λ1(x))|
|λ1(y) − λ1(x)| + (1 + cot θ)| f (λ2(y)) − f (λ2(x))|
|λ2(y) − λ2(x)|
+| f (λ2(x)) − f (λ1(x))|
λ2(x) − λ1(x)
x2
x2 − , (19)
where the last step comes from
| f (λi(y)) − f (λi(x))|
y − x = | f (λi(y)) − f (λi(x))|
|λi(y) − λi(x)|
|λi(y) − λi(x)|
y − x
≤√
2 max{tan θ, cot θ}| f (λi(y)) − f (λi(x))|
|λi(y) − λi(x)|
becauseλi(y)−λi(x) ≤√
2 max{tan θ, cot θ}y − x for i = 1, 2 by [28]. Taking limsup on both sides of (19) and using the fact that > 0 can be sufficiently small, it follows that
fLθ is calm at x with the upper bound of cam( fLθ)(x) given as in (16).
Case 2 x2 = 0. In this case, take ¯x2 = ¯y2 (i.e.,φ(x2) = φ(y2)). Following the similar argument as (18) and (19), we have
fLθ(y) − fLθ(x)
y − x
≤
√2 max{tan θ, cot θ}
tanθ + cot θ {(1 + tan θ)cam( f )(λ1(x)) + (1 + cot θ)cam( f )(λ2(x))}
=
√2 max{tan θ, cot θ}(tan θ + cot θ + 2)
tanθ + cot θ cam( f )(x1),
where the last step follows from the fact cam( f )(λi) = cam( f )(x1) since λi(x) = x1for i= 1, 2. Hence, fLθ is calm at x with the upper bound of cam( fLθ)(x) given as in (15).
(b) This is an immediate consequence of part (a).
4 H-differentiability
In this section, we answer the question about whether, as like the properties of continuity, strict continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, contin- uous differentiability, and (ρ-order) semismoothness (see [5,28]), the H -differentiability of fLθ can be inherited by that of f and vise versa? In addition, whether there exists some relationship between Tf and TfLθ? The following theorem provides an affirmative answer.
Theorem 4.1 Let f : R → R be a real-valued function and fLθ be defined as in (4).
Suppose x has spectral factorization given as in (1–3). Then, the following hold.
(a) If f is H -differentiable atλi(x) with Tf(λi(x)) as the H-differential for i = 1, 2, then fLθis H -differentiable at x with
TfLθ(x)
=
⎧⎪
⎪⎨
⎪⎪
⎩
⎡
⎢⎢
⎣ 1
1+ cot2θa1+ 1 1+ tan2θa2
− cotθ
1+ cot2θa1+ tanθ 1+ tan2θa2
wT
− cotθ
1+ cot2θa1+ tanθ 1+ tan2θa2
w
cot2θ
1+ cot2θa1+ tan2θ 1+ tan2θa2
I
⎤
⎥⎥
⎦
ai∈ Tf(x1) i= 1, 2
w = 1
⎫⎪
⎪⎬
⎪⎪
⎭
(20)
when x2 = 0; otherwise
TfLθ(x)
=
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩
⎡
⎢⎢
⎢⎢
⎢⎣
a1
1+ cot2θ + a2 1+ tan2θ
− cotθ
1+ cot2θa1+ tanθ 1+ tan2θa2
¯x2T
− cotθ
1+ cot2θa1+ tanθ 1+ tan2θa2
¯x2
cot2θ
1+ cot2θa1+ tan2θ 1+ tan2θa2
¯x2¯x2T
+f(λ2(x)) − f (λ1(x))
λ2(x) − λ1(x) I− ¯x2¯x2T
!
⎤
⎥⎥
⎥⎥
⎥⎦
ai∈ Tf(λi(x)) i= 1, 2
⎫⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎭
(21) (b) If fLθ is H -differentiable at x with TfLθ(x) as the H-differential, then f is H-
differentiable atλi(x) with
Tf(λi(x)) =
"
1
u(i)x 2(u(i)x )TAu(i)x A∈ TfLθ(x)
#
, i= 1, 2
when x2= 0; otherwise
Tf(λi(x)) =
eTAe| A ∈ TfLθ(x)
, i = 1, 2.
Proof (a) Let tk ↓ 0 and dk → d with dk = 1. We proceed the arguments by discussing two cases.
Case 1 For x2= 0, we know x + tkdk= (x1+ tkd1k, tkd2k)T. Hence, fLθ(x + tkdk) − fLθ(x)
=
⎡
⎢⎢
⎢⎣
f(x1+ tkd1k− tkd2k cot θ)
1+ cot2θ + f(x1+ tkd1k+ tkd2k tan θ) 1+ tan2θ
−f(x1+ tkd1k− tkd2k cot θ) cot θ
1+ cot2θ + f(x1+ tkd1k+ tkd2k tan θ) tan θ 1+ tan2θ
¯d2k
⎤
⎥⎥
⎥⎦
−
f(x1) 0
=
⎡
⎢⎢
⎢⎣
f(x1+ tkd1k− tkd2k cot θ)
1+ cot2θ + f(x1+ tkd1k+ tkd2k tan θ)
1+ tan2θ − f (x1)
−f(x1+ tkd1k− tkd2k cot θ) cot θ
1+ cot2θ + f(x1+ tkd1k+ tkd2k tan θ) tan θ 1+ tan2θ
¯d2k
⎤
⎥⎥
⎥⎦.
For$
fLθ(x + tkdk) − fLθ(x)%
/tk, the first component is 1
tk
&
f(x1+ tkd1k− tkd2k cot θ)
1+ cot2θ + f(x1+ tkd1k+ tkd2k tan θ)
1+ tan2θ − f (x1) '
= 1 tk
&
f(x1+ tk(d1k− d2k cot θ)) − f (x1)
1+ cot2θ + f(x1+ tk(d1k+ d2k tan θ)) − f (x1) 1+ tan2θ
'
a1(d1− d2 cot θ)
1+ cot2θ +a2(d1+ d2 tan θ)
1+ tan2θ as tk↓ 0, (22)
