*Research Article*

**The Vector-Valued Functions Associated with Circular Cones**

**Jinchuan Zhou**

^{1}**and Jein-Shan Chen**

^{2,3}*1**Department of Mathematics, School of Science, Shandong University of Technology, Zibo, Shandong 255049, China*

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

*3**Mathematics Division, National Center for Theoretical Sciences, Taipei, Taiwan*

Correspondence should be addressed to Jein-Shan Chen; jschen@math.ntnu.edu.tw Received 6 April 2014; Accepted 15 May 2014; Published 22 June 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 J. Zhou and J.-S. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which
the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetL_{𝜃}
denote the circular cone inR^{𝑛}. For a function𝑓 from R to R, one can define a corresponding vector-valued function 𝑓^{L}^{𝜃}onR^{𝑛}
by applying𝑓 to the spectral values of the spectral decomposition of 𝑥 ∈ R^{𝑛}with respect toL_{𝜃}. In this paper, we study properties
that this vector-valued function inherits from𝑓, including H¨older continuity, 𝐵-subdifferentiability, 𝜌-order semismoothness, and
positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in
circular cone constraints.

**1. Introduction**

The circular cone is a pointed closed convex cone having
hyperspherical sections orthogonal to its axis of revolution
about which the cone is invariant to rotation, which includes
second-order cone as a special case when the rotation angle is
45 degrees. LetL_{𝜃}denote the circular cone inR^{𝑛}. Then, the
𝑛-dimensional circular cone L_{𝜃}is expressed as

L_{𝜃}:= {𝑥 = (𝑥_{1}, 𝑥_{2})^{𝑇}∈ R × R^{𝑛−1}| cos 𝜃 ‖𝑥‖ ≤ 𝑥_{1}} . (1)

The application ofL_{𝜃}lies in engineering field, for example,
optimal grasping manipulation for multigingered robots;

see [1].

In our previous work [2], we have explored some impor-
tant features about circular cone, such as characterizing its
tangent cone, normal cone, and second-order regularity. In
particular, the spectral decomposition associated with L_{𝜃}
was discovered; that is, for any𝑧 = (𝑧_{1}, 𝑧_{2}) ∈ R × R^{𝑛−1}, one
has

𝑧 = 𝜆_{1}(𝑧) 𝑢^{1}_{𝑧}+ 𝜆_{2}(𝑧) 𝑢^{2}_{𝑧}, (2)

where

𝜆_{1}(𝑧) = 𝑧_{1}− 𝑧^{2}ctan𝜃,
𝜆_{2}(𝑧) = 𝑧_{1}+ 𝑧^{2}tan𝜃,
𝑢^{1}_{𝑧}= 1

1 + ctan^{2}𝜃[1 0

0 ctan 𝜃 ⋅ 𝐼] [ 1

−𝑧_{2}] ,
𝑢^{2}_{𝑧}= 1

1 + tan^{2}𝜃[1 0
0 tan 𝜃 ⋅ 𝐼] [1

𝑧_{2}] ,

(3)

with𝑧_{2}:= 𝑧_{2}/‖𝑧_{2}‖ if 𝑧_{2} ̸= 0, and 𝑧_{2}being any vector𝑤 ∈ R^{𝑛−1}
satisfying‖𝑤‖ = 1 if 𝑧_{2}= 0. With this spectral decomposition
(2), analogous to the so-called SOC-function𝑓^{soc} (see [3–

5]) and SDP-function𝑓^{mat} (see [6, 7]), we define a vector-
valued function associated with circular cone as below. More
specifically, for𝑓 : R → R, we define 𝑓^{L}^{𝜃}: R^{𝑛} → R^{𝑛}as

𝑓^{L}^{𝜃}(𝑧) = 𝑓 (𝜆_{1}(𝑧)) 𝑢^{1}_{𝑧}+ 𝑓 (𝜆_{2}(𝑧)) 𝑢_{𝑧}^{2}. (4)
It is not hard to see that 𝑓^{L}^{𝜃} is well-defined for all𝑧. In
particular, if𝑧_{2}= 0, then

𝑓^{L}^{𝜃}(𝑧) = [𝑓 (𝑧_{1})

0 ] . (5)

Volume 2014, Article ID 603542, 21 pages http://dx.doi.org/10.1155/2014/603542

Note that when𝜃 = 45^{∘},L_{𝜃}reduces to the second-order
cone (SOC) and the vector-valued function𝑓^{L}^{𝜃}defined as in
(4) corresponds to the SOC-function𝑓^{soc}given by

𝑓^{soc}(𝑥) = 𝑓 (𝜆_{1}(𝑥)) 𝑢^{(1)}_{𝑥} + 𝑓 (𝜆_{2}(𝑥)) 𝑢^{(2)}_{𝑥} ,

∀𝑥 = (𝑥_{1}, 𝑥_{2}) ∈ R × R^{𝑛−1}, (6)
where𝜆_{𝑖}(𝑥) = 𝑥_{1}+ (−1)^{𝑖}‖𝑥_{2}‖ and 𝑢^{(𝑖)}_{𝑥} = (1/2)(1, (−1)^{𝑖}𝑥_{2})^{𝑇}.

It is well known that the vector-valued function𝑓^{soc}asso-
ciated with second-order cone and matrix-valued function
𝑓^{mat} associated with positive semidefinite cone play crucial
role in the theory and numerical algorithm for second-
order cone programming and semidefinite programming,
respectively. In particular, many properties of𝑓^{soc}and𝑓^{mat}
are inherited from𝑓, such as continuity, strictly continuity,
directional differentiability, Fr´echet differentiability, continu-
ous differentiability, and semismoothness. It should be men-
tioned that, compared with second-order cone and positive
semidefinite cone, L_{𝜃} is a nonsymmetric cone. Hence a
natural question arises whether these properties are still true
for𝑓^{L}^{𝜃}. In [1], the authors answer the questions from the
following aspects:

(a)𝑓^{L}^{𝜃}is continuous at𝑧 ∈ R^{𝑛}if and only if𝑓 is contin-
uous at𝜆_{𝑖}(𝑧) for 𝑖 = 1, 2;

(b)𝑓^{L}^{𝜃} is directionally differentiable at𝑧 ∈ R^{𝑛} if and
only if 𝑓 is directionally differentiable at 𝜆_{𝑖}(𝑧) for
𝑖 = 1, 2;

(c)𝑓^{L}^{𝜃}is (Fr´echet) differentiable at𝑧 ∈ R^{𝑛}if and only if
𝑓 is (Fr´echet) differentiable at 𝜆_{𝑖}(𝑧) for 𝑖 = 1, 2;

(d)𝑓^{L}^{𝜃} is continuously differentiable at 𝑧 ∈ R^{𝑛} if and
only if𝑓 is continuously continuous at 𝜆_{𝑖}(𝑧) for 𝑖 =
1, 2;

(e)𝑓^{L}^{𝜃}is strictly continuous at𝑧 ∈ R^{𝑛}if and only if𝑓 is
strictly continuous at𝜆_{𝑖}(𝑧) for 𝑖 = 1, 2;

(f)𝑓^{L}^{𝜃}is Lipschitz continuous with constant𝑘 > 0 if and
only if𝑓 is Lipschitz continuous with constant 𝑘 > 0;

(g)𝑓^{L}^{𝜃}is semismooth at𝑧 if and only if 𝑓 is semismooth
at𝜆_{𝑖}(𝑧) for 𝑖 = 1, 2.

In this paper, we further study some other properties
associated with 𝑓^{L}^{𝜃}, such as H¨older continuity, 𝜌-order
semismoothness, directionally differentiability in the Hada-
mard sense, the characterization of B-subdifferential, positive
homogeneity, and boundedness. Of course, one may wonder
whether 𝑓^{soc} and 𝑓^{L}^{𝜃} always share the same properties.

Indeed, they do not. There exists some property that holds
for𝑓^{soc}and𝑓 but fails for 𝑓^{L}^{𝜃} and𝑓. A counterexample is
presented in the final section.

To end the third section, we briefly review our notations
and some basic concepts which will be needed for subsequent
analysis. First, we denote byR^{𝑛}the space of𝑛-dimensional
real column vectors and let𝑒 = (1, 0, . . . , 0) ∈ R^{𝑛}. Given
𝑥, 𝑦 ∈ R^{𝑛}, the Euclidean inner product and norm are⟨𝑥, 𝑦⟩=

𝑥^{𝑇}𝑦 and ‖𝑥‖ = √𝑥^{𝑇}𝑥. For a linear mapping 𝐻 : R^{𝑛} → R^{𝑚},

its operator norm is‖𝐻‖ := max_{‖𝑥‖=1}‖𝐻𝑥‖. For 𝛼 ∈ R and
𝑠 ∈ R^{𝑛}, we write 𝑠 = 𝑂(𝛼) (resp., 𝑠 = 𝑜(𝛼)) to means

‖𝑠‖/|𝛼| is uniformly bounded (resp., tends to zero) as 𝛼 → 0.

In addition, given a function𝐹 : R^{𝑛} → R^{𝑚}, we say the
following:

(a)𝐹 is H¨older continuous with exponent 𝛼 ∈ (0, 1], if

[𝐹]_{𝛼}:= sup

𝑥 ̸= 𝑦

𝐹(𝑥) − 𝐹(𝑦)

𝑥 − 𝑦^{𝛼} < +∞; (7)
(b)𝐹 is directionally differentiable at 𝑥 ∈ R^{𝑛}in the Hada-
mard sense, if the directional derivative𝐹^{}(𝑥; 𝑑) exists
for all𝑑 ∈ R^{𝑛}and

𝐹^{}(𝑥; 𝑑) = lim

𝑑^{}→ 𝑑
𝑡↓0

𝐹 (𝑥 + 𝑡𝑑^{}) − 𝐹 (𝑥)

𝑡 ; (8)

(c)𝐹 is 𝐵-differentiable (Bouligand-differentiable) at 𝑥, if𝐹 is Lipschitz continuous near 𝑥 and directionally differentiable at𝑥;

(d) if𝐹 is strictly continuous (locally Lipschitz continu-
ous), the generalized Jacobian𝜕𝐹(𝑥) is the convex hull
of the𝜕_{𝐵}𝐹(𝑥), where

𝜕_{𝐵}𝐹 (𝑥) := { lim_{𝑧 → 𝑥}∇𝐹 (𝑧) | 𝑧 ∈ 𝐷_{𝐹}} , (9)

where𝐷_{𝐹}denotes the set of all differentiable points of
𝐹;

(e)𝐹 is semismooth at 𝑥, if 𝐹 is strictly continuous near 𝑥, directionally differentiable at 𝑥, and for any 𝑉 ∈

𝜕𝐹(𝑥 + ℎ),

𝐹 (𝑥 + ℎ) − 𝐹 (𝑥) − 𝑉ℎ = 𝑜 (‖ℎ‖) ; (10)

(f)𝐹 is 𝜌-order semismooth at 𝑥 (𝜌 > 0) if 𝐹 is semi- smooth at𝑥 and for any 𝑉 ∈ 𝜕𝐹(𝑥 + ℎ),

𝐹 (𝑥 + ℎ) − 𝐹 (𝑥) − 𝑉ℎ = 𝑂 (‖ℎ‖^{1+𝜌}) ; (11)

in particular, we say𝐹 is strongly semismooth if it is 1-order semismooth;

(g)𝐹 is positively homogeneous with exponent 𝛼 > 0, if
𝐹 (𝑘𝑥) = 𝑘^{𝛼}𝐹 (𝑥) , ∀𝑥 ∈ R^{𝑛}, 𝑘 ≥ 0; (12)

(h)𝐹 is bounded if there exists a positive scalar 𝑀 > 0 such that

‖𝐹 (𝑥)‖ ≤ 𝑀, ∀𝑥 ∈ R^{𝑛}. (13)

**2. Directional Differentiability,**

**Strict Continuity, Hölder Continuity,** **and** **𝐵-Differentiability**

This section is devoted to study the properties of directional
differentiability, strict continuity, and H¨older continuity. The
relationship of directional differentiability between𝑓^{L}^{𝜃} and
𝑓 has been given in [1, Theorem 3.2] without giving the
exact formula of directional differentiability. Nonetheless,
such formulas can be easily obtained from its proof. Here we
just list them as follows.

**Lemma 1. Let 𝑓 : R → R and 𝑓**^{L}^{𝜃} *be defined as in (4).*

*Then,*𝑓^{L}^{𝜃}*is directionally differentiable at𝑧 if and only if 𝑓 is*
*directionally differentiable at*𝜆_{𝑖}*(𝑧) for 𝑖 = 1, 2. Moreover, for*
*any*ℎ = (ℎ_{1}, ℎ_{2}) ∈ R × R^{𝑛−1}*, we have*

(𝑓^{L}^{𝜃})^{}(𝑧; ℎ) = [𝑓^{}(𝑧_{1}; ℎ_{1})

0 ] = 𝑓^{}(𝑧_{1}; ℎ_{1}) 𝑒, (14)
*when*𝑧_{2}*= 0 and ℎ*_{2}*= 0. Consider*

(𝑓^{L}^{𝜃})^{}(𝑧; ℎ) = 1

1 + ctan^{2}𝜃𝑓^{}(𝑧_{1}; ℎ_{1}− ℎ^{2} ctan 𝜃)

× [1 0

0 ctan 𝜃 ⋅ 𝐼][[ [

1

− ℎ_{2}

ℎ^{2}

]] ]

+ 1

1 + tan^{2}𝜃𝑓^{}(𝑧_{1}; ℎ_{1}+ ℎ^{2}tan𝜃)

× [1 0

0 tan 𝜃 ⋅ 𝐼][[ [

1
ℎ_{2}

ℎ^{2}

]] ] ,

(15)
*when*𝑧_{2}*= 0 and ℎ*_{2} *̸= 0; otherwise,*

(𝑓^{L}^{𝜃})^{}(𝑧; ℎ) = 1

1 + ctan^{2}𝜃𝑓^{}(𝜆_{1}(𝑧) ; ℎ_{1}−𝑧_{2}^{𝑇}ℎ_{2}

𝑧^{2} ctan 𝜃)

× [1 0

0 ctan 𝜃 ⋅ 𝐼][ [

1

− 𝑧_{2}

𝑧^{2}]
]

− ctan𝜃
1 + ctan^{2}𝜃

𝑓 (𝜆_{1}(𝑧))

𝑧^{2} 𝑀^{𝑧}^{2}ℎ

+ 1

1 + tan^{2}𝜃𝑓^{}(𝜆_{2}(𝑧) ; ℎ_{1}+𝑧^{𝑇}_{2}ℎ_{2}

𝑧^{2} tan𝜃)

× [1 0

0 tan 𝜃 ⋅ 𝐼][ [

𝑧1_{2}

𝑧^{2}]
]
+ tan𝜃

1 + tan^{2}𝜃

𝑓 (𝜆_{2}(𝑧))

𝑧^{2} 𝑀^{𝑧}^{2}ℎ,

(16)

*where*

𝑀_{𝑧}_{2} := [[
[

0 0

0 𝐼 − 𝑧_{2}𝑧^{𝑇}_{2}

𝑧^{2}^{2}
]]
]

. (17)

**Lemma 2. Let 𝑓 : R → R and 𝑓**^{L}^{𝜃}*be defined as in (4). Then,*
*the following hold.*

(a)𝑓^{L}^{𝜃}*is differentiable at𝑧 if and only if 𝑓 is differentiable*
*at*𝜆_{𝑖}*(𝑧) for 𝑖 = 1, 2. Moreover, if 𝑧*_{2}*= 0, then*

∇𝑓^{L}^{𝜃}(𝑧) = 𝑓^{}(𝑧_{1}) 𝐼; (18)
*otherwise,*

∇𝑓^{L}^{𝜃}(𝑧) =
[[
[[
[
[

𝜉 𝑧^{𝑇}_{2}

𝑧^{2}

𝑧_{2}

𝑧^{2} 𝑎𝐼 + (𝜂 − 𝑎) 𝑧_{2}𝑧^{𝑇}_{2}

𝑧^{2}^{2}
]]
]]
]
]

, (19)

*where*
𝑎 = tan𝜃

1 + tan^{2}𝜃

𝑓 (𝜆_{2}(𝑧))

𝑧^{2} − ctan𝜃
1 + ctan^{2}𝜃

𝑓 (𝜆_{1}(𝑧))

𝑧^{2}

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

𝜉 = 𝑓^{}(𝜆_{1}(𝑧))

1 + ctan^{2}𝜃+𝑓^{}(𝜆_{2}(𝑧))
1 + tan^{2}𝜃 ,
𝜂 = 𝜉 − ( ctan 𝜃 − tan 𝜃) ,

= − ctan𝜃

1 + ctan^{2}𝜃𝑓^{}(𝜆_{1}(𝑧)) + tan𝜃

1 + tan^{2}𝜃𝑓^{}(𝜆_{2}(𝑧)) .
(20)
(b)𝑓^{L}^{𝜃}*is continuously differentiable (smooth) at𝑧 if and*
*only if𝑓 is continuously differentiable (smooth) at 𝜆*_{𝑖}(𝑧)
*for𝑖 = 1, 2.*

Note that the formula of gradient ∇𝑓^{L}^{𝜃} given in [1,
Theorem 3.3] andLemma 2is the same by using the following
facts:

1

1 + ctan^{2}𝜃 = sin^{2}𝜃, 1

1 + tan^{2}𝜃 = cos^{2}𝜃,
ctan𝜃

1 + ctan^{2}𝜃 = tan𝜃

1 + tan^{2}𝜃 = sin 𝜃 cos 𝜃.

(21)

The following result indicating that𝜆_{𝑖}is Lipschitz contin-
uous onR^{𝑛}for𝑖 = 1, 2 will be used in proving the Lipschitz
continuity between𝑓^{L}^{𝜃}and𝑓.

**Lemma 3. Let 𝑧, 𝑦 ∈ R**^{𝑛} *with spectral values* 𝜆_{𝑖}*(𝑧), 𝜆*_{𝑖}*(𝑦),*
*respectively. Then, we have*

𝜆^{𝑖}(𝑧) − 𝜆_{𝑖}(𝑦) ≤ √2 max {tan 𝜃, ctan 𝜃}𝑧 − 𝑦,
𝑓𝑜𝑟 𝑖 = 1, 2. (22)

*Proof. First, we observe that*

𝜆^{1}(𝑧) − 𝜆_{1}(𝑦)

= 𝑧^{1}− 𝑧^{2}ctan𝜃 − 𝑦^{1}+ 𝑦^{2}ctan𝜃

≤ 𝑧^{1}− 𝑦_{2} + 𝑧^{2}− 𝑦_{2}ctan𝜃

≤ max {1, ctan 𝜃} (𝑧^{1}− 𝑦_{1} + 𝑧^{2}− 𝑦_{2})

≤ max {1, ctan 𝜃} √2√𝑧^{1}− 𝑦_{1}^{2}+ 𝑧^{2}− 𝑦_{2}^{2}

= max {1, ctan 𝜃} √2 𝑧 − 𝑦.

(23)

Applying the similar argument to𝜆_{2}yields

𝜆^{2}(𝑧) − 𝜆_{2}(𝑦) ≤ max {1, tan 𝜃} √2𝑧 − 𝑦. (24)
Then, the desired result follows from the fact that max{1,
ctan𝜃, tan 𝜃} max{ctan 𝜃, tan 𝜃}.

**Theorem 4. Let 𝑓 : R → R and 𝑓**^{L}^{𝜃} *be defined as in (4).*

*Then,*𝑓^{L}^{𝜃}*is strictly continuous (local Lipschitz continuity) at*𝑧
*if and only if𝑓 is strictly continuous (local Lipschitz continuity)*
*at*𝜆_{𝑖}*(𝑧) for 𝑖 = 1, 2.*

*Proof. “⇐” Suppose that 𝑓 is strictly continuous at 𝜆*_{𝑖}(𝑧), for
𝑖 = 1, 2; that is, there exist 𝑘_{𝑖}> 0 and 𝛿_{𝑖} > 0, for 𝑖 = 1, 2 such
that

𝑓(𝜏) − 𝑓(𝜁) ≤ 𝑘^{𝑖}𝜏 − 𝜁,

∀𝜏, 𝜁 ∈ [𝜆_{𝑖}(𝑧) − 𝛿_{𝑖}, 𝜆_{𝑖}(𝑧) + 𝛿_{𝑖}] , 𝑖 = 1, 2. (25)
Let𝛿 := min{𝛿_{1}, 𝛿_{2}} and 𝐶 := [𝜆_{1}(𝑧) − 𝛿, 𝜆_{1}(𝑧) + 𝛿] ∪ [𝜆_{2}(𝑧) −
𝛿, 𝜆_{2}(𝑧) + 𝛿]. Define

𝑓 (𝜏)̃

:=

{{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑓 (𝜏) if𝜏 ∈ 𝐶,

(1 − 𝑡) 𝑓 (𝜆_{1}(𝑧) + 𝛿)

+ 𝑡𝑓 (𝜆_{2}(𝑧) − 𝛿) if𝜆_{1}(𝑧) + 𝛿 < 𝜆_{2}(𝑧) − 𝛿,
𝜏 = (1 − 𝑡) (𝜆_{1}(𝑧) + 𝛿)

+ 𝑡 (𝜆_{2}(𝑧) − 𝛿)
with𝑡 ∈ (0, 1)
𝑓 (𝜆_{1}(𝑧) − 𝛿) if𝜏 < 𝜆_{1}(𝑧) − 𝛿
𝑓 (𝜆_{2}(𝑧) + 𝛿) if𝜏 > 𝜆_{2}(𝑧) + 𝛿.

(26)
Clearly, ̃𝑓 is Lipschitz continuous on R; that is, there exists
𝑘 > 0 such that lip ̃𝑓(𝜏) ≤ 𝑘, for all 𝜏 ∈ R. Since ̃𝐶 := conv(𝐶)
is compact, according to [6, Lemma 4.5] or [5, Lemma 3],
there exist continuously differentiable functions𝑓^{V}: R → R
forV = 1, 2, . . . converging uniformly to ̃𝑓 on ̃𝐶 such that

(𝑓^{V})^{}(𝜏) ≤ 𝑘 ∀𝜏 ∈𝐶, ∀V.̃ (27)

Now, let:= 𝛿/(√2 max{tan 𝜃, ctan 𝜃}). Then, fromLemma 3, we know that ̃𝐶 contains all spectral values of 𝑦 ∈ B(𝑧, 𝛿).

Therefore, for any𝑤 ∈ B(𝑧, 𝛿), we have 𝜆_{𝑖}(𝑤) ∈ ̃𝐶 for 𝑖 = 1, 2
and

(𝑓^{V})^{L}^{𝜃}(𝑤) − 𝑓^{L}^{𝜃}(𝑤)^{2}

= [𝑓^{V}(𝜆_{1}(𝑤)) − 𝑓 (𝜆_{1}(𝑤))] 𝑢^{1}_{𝑤}
+ [𝑓^{V}(𝜆_{2}(𝑤)) − 𝑓(𝜆_{2}(𝑤))] 𝑢^{2}_{𝑤}^{2}

= [𝑓^{V}(𝜆_{1}(𝑤)) − 𝑓 (𝜆_{1}(𝑤))]^{2}𝑢^{1}^{𝑤}^{2}
+ [𝑓^{V}(𝜆_{2}(𝑤)) − 𝑓 (𝜆_{2}(𝑤))]^{2}𝑢^{2}^{𝑤}^{2}

= 1

1 + ctan^{2}𝜃𝑓^{V}(𝜆_{1}(𝑤)) − 𝑓 (𝜆_{1}(𝑤))^{2}

+ 1

1 + tan^{2}𝜃𝑓^{V}(𝜆_{2}(𝑤)) − 𝑓(𝜆_{2}(𝑤))^{2},

(28)

where we have used the facts that‖𝑢^{1}_{𝑤}‖ = 1/√1 + ctan^{2}𝜃,

‖𝑢^{2}_{𝑤}‖ = 1/√1 + tan^{2}𝜃, and ⟨𝑢^{1}_{𝑤}, 𝑢^{2}_{𝑤}⟩ = 0. Since {𝑓^{V}}^{∞}_{V=1}con-
verges uniformly to𝑓 on ̃𝐶, the above equations show that
{(𝑓^{V})^{L}^{𝜃}}^{∞}_{V=1}converges uniformly to𝑓^{L}^{𝜃}onB(𝑧, 𝛿). If 𝑤_{2}= 0,
then it follows fromLemma 2that∇(𝑓^{V})^{L}^{𝜃}(𝑤) = (𝑓^{V})^{}(𝑤_{1})𝐼.

Hence it follows from (27) that

∇(𝑓^{V})^{L}^{𝜃}(𝑤) =(𝑓^{V})^{}(𝑤_{1}) ≤ 𝑘, (29)
since in this case𝜆_{𝑖}(𝑤) = 𝑤_{1}∈ ̃𝐶. If 𝑤_{2} ̸= 0, then

∇(𝑓^{V})^{L}^{𝜃}(𝑤)

=[[[[ [

𝜉 𝑤^{𝑇}_{2}

𝑤^{2}

𝑤_{2}

𝑤^{2} 𝑎𝐼 + (𝜉 − (ctan𝜃 − tan𝜃) − 𝑎)𝑤_{2}𝑤^{𝑇}_{2}

𝑤^{2}^{2}
]]
]]
]

=[[[[ [

𝜉 𝑤^{𝑇}_{2}

𝑤^{2}

𝑤_{2}

𝑤^{2} 𝑎𝐼 + (𝜉 − 𝑎)𝑤_{2}𝑤^{𝑇}_{2}

𝑤^{2}^{2}
]]
]]
]

+ [[ [

0 0

0 [− (ctan 𝜃 − tan 𝜃)]𝑤_{2}𝑤^{𝑇}_{2}

𝑤^{2}^{2}
]]
]

=[[[[ [

𝜉 𝑤^{𝑇}_{2}

𝑤^{2}

𝑤_{2}

𝑤^{2} 𝜉𝐼
]]
]]
]

+ (𝑎 − 𝜉) [[ [

0 0

0 𝐼 −𝑤_{2}𝑤^{𝑇}_{2}

𝑤^{2}^{2}
]]
]

− (ctan 𝜃 − tan 𝜃) [[ [

0 0

0 𝑤_{2}𝑤_{2}^{𝑇}

𝑤^{2}^{2}
]]
]
,

(30)

where𝑎, 𝜉, are given as in (20) with𝜆_{𝑖}(𝑧) replaced by 𝜆_{𝑖}(𝑤)
for𝑖 = 1, 2 and 𝑓 replaced by 𝑓^{V}. For simplicity of notations,
let us denote

𝐴 :=[[[ [

𝜉 𝑤^{𝑇}_{2}

𝑤^{2}

𝑤_{2}

𝑤^{2} 𝜉𝐼
]]
]
]

+ (𝑎 − 𝜉) [[ [

0 0

0 𝐼 −𝑤_{2}𝑤^{𝑇}_{2}

𝑤^{2}^{2}
]]
]
,

𝐵 := − (ctan 𝜃 − tan 𝜃) [[ [

0 0

0 𝑤_{2}𝑤^{𝑇}_{2}

𝑤^{2}^{2}
]]
]
.

(31)

Note that

|𝑎| =

𝑓^{V}(𝜆_{2}(𝑤)) − 𝑓^{V}(𝜆_{1}(𝑤))
𝜆_{2}(𝑤) − 𝜆_{1}(𝑤)

≤ 𝑘, (32)
where the inequality comes from the fact that𝑓^{V}is continu-
ously differentiable on ̃𝐶 and (27). Besides, we also note that

𝜉 =

(𝑓^{V})^{}(𝜆_{1}(𝑤))

1 + ctan^{2}𝜃 +(𝑓^{V})^{}(𝜆_{2}(𝑤))
1 + tan^{2}𝜃

≤ 1

1 + ctan^{2}𝜃(𝑓^{V})^{}(𝜆_{1}(𝑤)) + 1

1 + tan^{2}𝜃(𝑓^{V})^{}(𝜆_{2}(𝑤))

≤ [ 1

1 + ctan^{2}𝜃 + 1

1 + tan^{2}𝜃] 𝑘 = 𝑘,

(33)

=− ctan𝜃

1 + ctan^{2}𝜃(𝑓^{V})^{}(𝜆_{1}(𝑤)) + tan𝜃

1 + tan^{2}𝜃(𝑓^{V})^{}(𝜆_{2}(𝑤))

≤ [− ctan𝜃

1 + ctan^{2}𝜃 + tan𝜃
1 + tan^{2}𝜃]𝑘

= [ ctan𝜃

1 + ctan^{2}𝜃+ tan𝜃
1 + tan^{2}𝜃] 𝑘

= 2 tan 𝜃
1 + tan^{2}𝜃𝑘 ≤ 𝑘.

(34)

(i) For = 0, then ∇(𝑓^{V})^{L}^{𝜃}(𝑤) takes the form of 𝜉𝐼+(𝑎−

𝜉)𝑀_{𝑤}_{2}whose eigenvalues are𝜉 and 𝑎 by [5, Lemma 1].

In other words, in this case, we get from (32) and (33) that

∇(𝑓^{V})^{L}^{𝜃}(𝑤) = max {|𝑎| ,𝜉} ≤ 𝑘. (35)
(ii) For ̸= 0, since 𝐵 = −(ctan 𝜃 − tan 𝜃)(0, 𝑤_{2}/‖𝑤_{2}‖)^{𝑇}
(0, 𝑤_{2}/‖𝑤_{2}‖), the eigenvalues of 𝐵 are −(ctan 𝜃 −
tan𝜃) and 0 with multiplicity 𝑛 − 1. Note that

(ctan𝜃 − tan𝜃)

=

1 − ctan^{2}𝜃

1 + ctan^{2}𝜃(𝑓^{V})^{}(𝜆_{1}(𝑤)) +1 − tan^{2}𝜃

1 + tan^{2}𝜃(𝑓^{V})^{}(𝜆_{2}(𝑤))

≤ [

1 − ctan^{2}𝜃
1 + ctan^{2}𝜃

+

1 − tan^{2}𝜃
1 + tan^{2}𝜃

] 𝑘

=

ctan^{2}𝜃 − 1

1 + ctan^{2}𝜃 +1 − tan^{2}𝜃
1 + tan^{2}𝜃

𝑘

= 2

1 − tan^{2}𝜃
1 + tan^{2}𝜃

𝑘 ≤ 2𝑘.

(36) Note that

𝐴 =

𝑤^{2}𝐿^{𝑤}^{̃}+ (𝑎 − 𝜉) 𝑀_{𝑤}_{2}

=

𝑤^{2} [𝐿^{𝑤}^{̃}+ (𝑎 − 𝜉)𝑤^{2}

𝑀_{𝑤}_{̃}_{2}] ,

(37)

where𝑤 = (𝜉‖𝑤̃ _{2}‖/, 𝑤_{2}) and
𝐿_{𝑤}_{̃}:= [̃𝑤_{1} 𝑤̃_{2}^{𝑇}

𝑤̃_{2} 𝑤̃_{1}𝐼] . (38)
In this case the matrix𝐴 has eigenvalues of 𝜉 ± and 𝑎 with
multiplicity𝑛 − 2. Hence,

∇(𝑓^{V})^{L}^{𝜃}(𝑤)

≤ max {𝜉 + ,𝜉 − ,|𝑎|} + (ctan𝜃 − tan𝜃)

≤ max {𝜉 + ,|𝑎|} + (ctan𝜃 − tan𝜃) ≤ 4𝑘, (39)

where the last step is due to (32), (33), (34), and (36).

Putting (29), (35), and (39) together, we know that

∇(𝑓^{V})^{L}^{𝜃}(𝑤) ≤ 4𝑘 ∀𝑤 ∈ B (𝑧, 𝛿) , ∀V. (40)
Fix any𝑥, 𝑦 ∈ B(𝑧, 𝛿) with 𝑥 ̸= 𝑦. Since {(𝑓^{V})^{L}^{𝜃}}^{∞}_{V=1}converges
uniformly to𝑓^{L}^{𝜃}onB(𝑧, 𝛿), then for any 𝜖 > 0 there exists
V_{0}such that

(𝑓^{V})^{L}^{𝜃}(𝑤) − 𝑓^{L}^{𝜃}(𝑤) ≤ 𝜖, ∀𝑤 ∈ B (𝑧, 𝛿) , ∀V ≥ V^{0}.
(41)
Since𝑓^{V}is continuously differentiable,(𝑓^{V})^{L}^{𝜃}is continuously
differentiable byLemma 2. Thus,

𝑓^{L}^{𝜃}(𝑥) − 𝑓^{L}^{𝜃}(𝑦)

=𝑓^{L}^{𝜃}(𝑥) − (𝑓^{V})^{L}^{𝜃}(𝑥) + (𝑓^{V})^{L}^{𝜃}(𝑥) − (𝑓^{V})^{L}^{𝜃}(𝑦)
+ (𝑓^{V})^{L}^{𝜃}(𝑦) − 𝑓^{L}^{𝜃}(𝑦)

≤𝑓^{L}^{𝜃}(𝑥) − (𝑓^{V})^{L}^{𝜃}(𝑥) +(𝑓^{V})^{L}^{𝜃}(𝑥) − (𝑓^{V})^{L}^{𝜃}(𝑦)

+(𝑓^{V})^{L}^{𝜃}(𝑦) − 𝑓^{L}^{𝜃}(𝑦)

≤ 2𝜖 +∫^{1}

0 ∇(𝑓^{V})^{L}^{𝜃}(𝑦 + 𝑡 (𝑥 − 𝑦)) (𝑥 − 𝑦) 𝑑𝑡

≤ 2𝜖 + 4𝑘 𝑥 − 𝑦.

(42)

Because𝜖 > 0 is arbitrary, this ensures that

𝑓^{L}^{𝜃}(𝑥) − 𝑓^{L}^{𝜃}(𝑦) ≤ 4𝑘𝑥 − 𝑦 ∀𝑥,𝑦 ∈ B(𝑧,𝛿),
(43)
which says𝑓^{L}^{𝜃}is strictly continuous at𝑧.

“⇒” Suppose that 𝑓^{L}^{𝜃} is strictly continuous at 𝑧, then
there exist𝑘 > 0 and 𝛿 > 0 such that

𝑓^{L}^{𝜃}(𝑥) − 𝑓^{L}^{𝜃}(𝑦) ≤ 𝑘𝑥 − 𝑦 ∀𝑥,𝑦 ∈ B(𝑧,𝛿).

(44)
*Case 1.*𝑧_{2} ̸= 0. Take 𝜃, 𝜇 ∈ [𝜆_{1}(𝑧) − 𝛿_{1}, 𝜆_{1}(𝑧) + 𝛿_{1}] with 𝛿_{1} :=

min{𝛿, 𝜆_{2}(𝑧) − 𝜆_{1}(𝑧)}. Let

𝑥 := 𝜃𝑢^{1}_{𝑧}+ 𝜆_{2}(𝑧) 𝑢^{2}_{𝑧}, 𝑦 := 𝜇𝑢^{1}_{𝑧}+ 𝜆_{2}(𝑧) 𝑢^{2}_{𝑧}. (45)
Then,‖𝑥 − 𝑧‖ ≤ 𝛿 and ‖𝑦 − 𝑧‖ ≤ 𝛿 and it follows from (44)
that

𝑓(𝜃) − 𝑓(𝜇) = 1𝑢^{1}^{𝑧}𝑓^{L}^{𝜃}(𝑥) − 𝑓^{L}^{𝜃}(𝑦) ≤ 𝑘

𝑢^{1}^{𝑧}𝑥 − 𝑦

= 𝑘

𝑢^{1}^{𝑧}𝜃 − 𝜇𝑢^{1}^{𝑧} = 𝑘𝜃 − 𝜇,

(46)
which says 𝑓 is strictly continuous at 𝜆_{1}(𝑧). The similar
argument shows the strict continuity of𝑓 at 𝜆_{2}(𝑧).

*Case 2.*𝑧_{2}= 0. For any 𝜃, 𝜇 ∈ [𝑧_{1}−𝛿, 𝑧_{1}+𝛿], we have ‖𝜃𝑒−𝑧‖ =

|𝜃 − 𝑧_{1}| ≤ 𝛿 and ‖𝜇𝑒 − 𝑧‖ ≤ 𝛿 as well; that is, 𝜃𝑒, 𝜇𝑒 ∈ B(𝑧, 𝛿).

It then follows from (44) that

𝑓(𝜃) − 𝑓(𝜇) =

(𝑓 (𝜃) − 𝑓 (𝜇) 0 )

=𝑓^{L}^{𝜃}(𝜃𝑒) − 𝑓^{L}^{𝜃}(𝜇𝑒)

≤ 𝑘 𝜃𝑒 − 𝜇𝑒 = 𝑘𝜃 − 𝜇.

(47)
This means𝑓 is strictly continuous at 𝜆_{𝑖}(𝑧) = 𝑧_{1}for𝑖 = 1, 2.

*Remark 5. As mentioned in*Section 1, the strict continuity
between𝑓^{L}^{𝜃}and𝑓 has been given in [1, Theorem 3.5]. Here
we provide an alternative proof, since our analysis technique
is different from that in [1, Theorem 3.5]. In particular, we
achieve an estimate regarding‖∇(𝑓^{V})^{L}^{𝜃}‖ via its eigenvalues,
which may have other applications.

According toLemma 1andTheorem 4, we obtain the fol- lowing result immediately.

**Theorem 6. Let 𝑓 : R → R and 𝑓**^{L}^{𝜃} *be defined as in (4).*

*Then,*𝑓^{L}^{𝜃}*is𝐵-differentiable at 𝑧 if and only if 𝑓 is 𝐵-differen-*
*tiable at*𝜆_{𝑖}*(𝑧), for 𝑖 = 1, 2.*

Next, inspired by [8, 9], we further study the H¨older
continuity relation between𝑓 and 𝑓^{L}^{𝜃}.

**Theorem 7. Let 𝑓 : R → R and 𝑓**^{L}^{𝜃} *be defined as in (4).*

*Then,*𝑓^{L}^{𝜃} *is H¨older continuous with exponent𝛼 ∈ (0, 1] if*
*and only if𝑓 is H¨older continuous with exponent 𝛼 ∈ (0, 1].*

*Proof. “⇐” Suppose that 𝑓 is H¨older continuous with expo-*
nent𝛼 ∈ (0, 1]. To proceed the proof, we consider the follow-
ing two cases.

*Case 1.*𝑧_{2} ̸= 0 and 𝑦_{2} ̸= 0. We assume without loss of generality
that‖𝑧_{2}‖ ≥ ‖𝑦_{2}‖. Thus,

𝑓^{L}^{𝜃}(𝑧) − 𝑓^{L}^{𝜃}(𝑦)

= 𝑓(𝜆^{1}(𝑧)) 𝑢^{1}_{𝑧}+ 𝑓 (𝜆_{2}(𝑧)) 𝑢^{2}_{𝑧}− 𝑓 (𝜆_{1}(𝑦)) 𝑢_{𝑦}^{1}

−𝑓 (𝜆_{2}(𝑦)) 𝑢^{2}_{𝑦}

= 𝑓(𝜆^{1}(𝑧)) [𝑢^{1}_{𝑧}− 𝑢^{1}_{𝑦}] + 𝑓 (𝜆_{2}(𝑧)) [𝑢^{2}_{𝑧}− 𝑢^{2}_{𝑦}]
+ [𝑓 (𝜆_{1}(𝑧)) − 𝑓 (𝜆_{1}(𝑦))] 𝑢^{1}_{𝑦}

+ [𝑓 (𝜆_{2}(𝑧)) − 𝑓 (𝜆_{2}(𝑦))] 𝑢^{2}_{𝑦}

≤ 𝑓(𝜆^{1}(𝑧)) [𝑢^{1}_{𝑧}− 𝑢^{1}_{𝑦}] + 𝑓 (𝜆_{2}(𝑧)) [𝑢^{2}_{𝑧}− 𝑢^{2}_{𝑦}]

+ 𝑓 (𝜆^{1}(𝑧)) − 𝑓 (𝜆_{1}(𝑦)) ⋅𝑢^{1}^{𝑦}
+ 𝑓 (𝜆^{2}(𝑧)) − 𝑓 (𝜆_{2}(𝑦)) ⋅𝑢^{2}^{𝑦} .

(48)

Let us analyze each term in the above inequality. First, we look into the first term:

𝑓 (𝜆^{1}(𝑧)) [𝑢^{1}_{𝑧}− 𝑢^{1}_{𝑦}] + 𝑓 (𝜆_{2}(𝑧)) [𝑢^{2}_{𝑧}− 𝑢^{2}_{𝑦}]

= tan𝜃

1 + tan^{2}𝜃𝑓(𝜆^{1}(𝑧)) − 𝑓 (𝜆_{2}(𝑧)) ⋅
𝑧_{2}

𝑧^{2} − 𝑦_{2}

𝑦^{2}

≤ tan𝜃

1 + tan^{2}𝜃[𝑓]_{𝛼}𝜆^{1}(𝑧) − 𝜆_{2}(𝑧)^{𝛼}

𝑧_{2}

𝑧^{2} − 𝑦_{2}

𝑦^{2}

= tan𝜃

1 + tan^{2}𝜃[𝑓]_{𝛼}(tan 𝜃 + ctan 𝜃)^{𝛼}𝑧^{2}^{𝛼}⋅

𝑧_{2}

𝑧^{2} − 𝑦_{2}

𝑦^{2}

≤ tan𝜃

1 + tan^{2}𝜃[𝑓]_{𝛼}(tan 𝜃 + ctan 𝜃)^{𝛼}𝑧^{2}^{𝛼} 2

𝑧^{2}𝑧^{2}− 𝑦_{2}

= 2 tan𝜃

1 + tan^{2}𝜃[𝑓]_{𝛼}(tan 𝜃 + ctan 𝜃)^{𝛼}

𝑧_{2}− 𝑦_{2}

𝑧^{2}

1−𝛼𝑧^{2}− 𝑦_{2}^{𝛼}

≤ tan𝜃

1 + tan^{2}𝜃[𝑓]_{𝛼}(tan 𝜃 + ctan 𝜃)^{𝛼}2^{2−𝛼}𝑧^{2}− 𝑦_{2}^{𝛼}

≤ tan𝜃

1 + tan^{2}𝜃[𝑓]_{𝛼}(tan 𝜃 + ctan 𝜃)^{𝛼}2^{2−𝛼}𝑧 − 𝑦^{𝛼},

(49) where the first inequality is due to the H¨older continu- ity of 𝑓, the second inequality comes from the fact that

‖(𝑧_{2}/‖𝑧_{2}‖) − (𝑦_{2}/‖𝑦_{2}‖)‖ ≤ (2/‖𝑧_{2}‖)‖𝑧_{2}− 𝑦_{2}‖ (cf. [8, Lemma
2.2]), and the third inequality follows from the fact that

‖𝑧_{2}− 𝑦_{2}‖ ≤ ‖𝑧_{2}‖ + ‖𝑦_{2}‖ ≤ 2‖𝑧_{2}‖ (since ‖𝑦_{2}‖ ≤ ‖𝑧_{2}‖). Next, we
look into the second term:

𝑓(𝜆^{1}(𝑧)) − 𝑓 (𝜆_{1}(𝑦))𝑢^{1}^{𝑦}

≤ [𝑓]_{𝛼}𝜆^{1}(𝑧) − 𝜆_{1}(𝑦)^{𝛼} 1

√1 + ctan^{2}𝜃

≤ [𝑓]_{𝛼}(√2 max {tan 𝜃, ctan 𝜃})^{𝛼}𝑧 − 𝑦^{𝛼}.

(50)

Similarly, the third term also satisfies

𝑓(𝜆^{2}(𝑧)) − 𝑓 (𝜆_{2}(𝑦))𝑢^{2}^{𝑦}

≤ [𝑓]_{𝛼}𝜆^{2}(𝑧) − 𝜆_{2}(𝑦)^{𝛼} 1

√1 + tan^{2}𝜃

≤ [𝑓]_{𝛼}(√2 max {tan 𝜃, ctan 𝜃})^{𝛼}𝑧 − 𝑦^{𝛼}.
(51)

Combining (49)–(51) proves that𝑓^{L}^{𝜃}is H¨older continuous
with exponent𝛼 ∈ (0, 1].

*Case 2. Either*𝑧_{2} = 0 or 𝑦_{2}= 0. In this case, we take 𝑢^{𝑖}_{𝑧} = 𝑢^{𝑖}_{𝑦},
for𝑖 = 1, 2 according to the spectral decomposition. There-
fore, we obtain

𝑓^{L}^{𝜃}(𝑧) − 𝑓^{L}^{𝜃}(𝑦)

= 𝑓(𝜆^{1}(𝑧)) 𝑢^{1}_{𝑧}+ 𝑓 (𝜆_{2}(𝑧)) 𝑢^{2}_{𝑧}− 𝑓 (𝜆_{1}(𝑦)) 𝑢^{1}_{𝑦}

−𝑓 (𝜆_{2}(𝑦)) 𝑢^{2}_{𝑦}

= [𝑓(𝜆^{1}(𝑧)) − 𝑓 (𝜆_{1}(𝑦))] 𝑢^{1}_{𝑧}
+ [𝑓 (𝜆_{2}(𝑧)) − 𝑓 (𝜆_{2}(𝑦))] 𝑢^{2}_{𝑧}

≤ 𝑓 (𝜆^{1}(𝑧)) − 𝑓 (𝜆_{1}(𝑦)) ⋅𝑢^{𝑧}^{1}
+ 𝑓 (𝜆^{2}(𝑧)) − 𝑓 (𝜆_{2}(𝑦)) ⋅𝑢^{2}^{𝑧}

≤ [𝑓]_{𝛼}𝜆^{1}(𝑧) − 𝜆_{1}(𝑦)^{𝛼} 1

√1 + ctan^{2}𝜃
+ [𝑓]_{𝛼}𝜆^{2}(𝑦) − 𝜆_{2}(𝑧)^{𝛼} 1

√1 + tan^{2}𝜃

≤ 2[𝑓]_{𝛼}(√2 max {tan 𝜃, ctan 𝜃})^{𝛼}𝑧 − 𝑦^{𝛼},

(52)

which says𝑓^{L}^{𝜃}is H¨older continuous.

“⇒” Recall that 𝑓^{L}^{𝜃}(𝜏𝑒) = (𝑓(𝜏), 0)^{𝑇}. Hence, for any
𝜏, 𝜁 ∈ R,

𝑓(𝜏) − 𝑓(𝜁) = 𝑓^{L}^{𝜃}(𝜏𝑒) − 𝑓^{L}^{𝜃}(𝜁𝑒)

≤ [𝑓^{L}^{𝜃}]_{𝛼}⋅ 𝜏𝑒 − 𝜁𝑒^{𝛼}= [𝑓^{L}^{𝜃}]_{𝛼}⋅ 𝜏 − 𝜁^{𝛼},
(53)
which says𝑓 is H¨older continuous.

**3.** **𝜌-Order Semismoothness** **and** **𝐵-Subdifferential Formula**

The property of semismoothness plays an important role in
nonsmooth Newton methods [10,11]. For more information
on semismooth functions, see [12–15]. The relationship of
semismooth between 𝑓^{L}^{𝜃} and 𝑓 has been given in [1,
Theorem 4.1]. But the exact formula of the𝐵-subdifferential

𝜕_{𝐵}(𝑓^{L}^{𝜃}) is not presented. Hence the main aim of this
section is twofold: one is establishing the exact formula of𝐵-
subdifferential; another is studing the𝜌-order semismooth-
ness for𝜌 > 0.

**Lemma 8. Define 𝜓(𝑧) = ‖𝑧‖ and Φ(𝑧) = 𝑧/‖𝑧‖ for 𝑧 ̸= 0.**

*Then,𝜓 and Φ are strongly semismooth at 𝑧 ̸= 0.*

*Proof. Since*𝑧 ̸= 0, it is clear that 𝜓 and Φ are twice continu-
ously differentiable and hence the gradient is Lipschitz con-
tinuous near𝑧. Therefore, 𝜓 and Φ are strongly semismooth
at𝑧, see [16, Proposition 7.4.5].

The relationship of 𝜌-order semismoothness between
𝑓^{L}^{𝜃}and𝑓 is given below. Recall from [7] that in the definition
of𝜌-order semismooth, we can restrict 𝑥+ℎ in (11) belonging
to differentiable points.

**Theorem 9. Let 𝑓 : R → R and 𝑓**^{L}^{𝜃} *be defined as in (4).*

*Given𝜌 > 0, then the following statements hold.*

*(a) If𝑓 is 𝜌-order semismooth at 𝜆*_{𝑖}*(𝑧) for 𝑖 = 1, 2, then*
𝑓^{L}^{𝜃}*is min{1, 𝜌}-order semismooth at 𝑧.*

*(b) If*𝑓^{L}^{𝜃} *is𝜌-order semismooth at 𝑧, then 𝑓 is 𝜌-semi-*
*smooth at*𝜆_{𝑖}*(𝑧) for 𝑖 = 1, 2.*

*(c) For*𝑧_{2}*= 0, 𝑓*^{L}^{𝜃}*is𝜌-semismooth at 𝑧 if and only if 𝑓 is*
*𝜌-order semismooth at 𝜆*_{𝑖}(𝑧) = 𝑧_{1}*for𝑖 = 1, 2.*

*Proof. (a) Take*ℎ ∈ R^{𝑛}satisfying𝑧 + ℎ ∈ 𝐷_{𝑓}L𝜃. We consider
the following two cases to complete the proof.

*Case 1. For*𝑧_{2} ̸= 0, 𝑧_{2}+ ℎ_{2} ̸= 0 as ℎ is sufficiently close to 0.

Since𝑧+ℎ ∈ 𝐷_{𝑓}L𝜃, we know that𝜆_{𝑖}(𝑧+ℎ) ∈ 𝐷_{𝑓}for𝑖 = 1, 2 by
Lemma 2. Then, according toLemma 1, the first component
of

𝑓^{L}^{𝜃}(𝑧 + ℎ) − 𝑓^{L}^{𝜃}(𝑧) − (𝑓^{L}^{𝜃})^{}(𝑧 + ℎ; ℎ) (54)
is expressed as

𝑓 (𝜆_{1}(𝑧 + ℎ))

1 + ctan^{2}𝜃 − 𝑓 (𝜆_{1}(𝑧))

1 + ctan^{2}𝜃 − 1
1 + ctan^{2}𝜃

× 𝑓^{}(𝜆_{1}(𝑧 + ℎ) ; ℎ_{1}−(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} ctan𝜃)
+𝑓 (𝜆_{2}(𝑧 + ℎ))

1 + tan^{2}𝜃 −𝑓 (𝜆_{2}(𝑧))
1 + tan^{2}𝜃 − 1

1 + tan^{2}𝜃

× 𝑓^{}(𝜆_{2}(𝑧 + ℎ) ; ℎ_{1}+(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} tan𝜃).

(55)

Because‖ ⋅ ‖ is continuously differentiable over 𝑧_{2} ̸= 0, it is
strongly semismooth at𝑧_{2}byLemma 8. Therefore,

𝑧^{2}+ ℎ_{2} = 𝑧^{2} + (𝑧^{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} + 𝑂(ℎ^{2}^{2})

= 𝑧^{2} + (𝑧^{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} + 𝑂(‖ℎ‖^{2}) .

(56)

Combining this and the𝜌-semismoothness of 𝑓 at 𝜆_{1}(𝑧), we
have

𝑓 (𝜆_{1}(𝑧 + ℎ))

= 𝑓 (𝜆_{1}(𝑧)) + 𝑓^{}(𝜆_{1}(𝑧 + ℎ)) (𝜆_{1}(𝑧 + ℎ) − 𝜆_{1}(𝑧))
+ 𝑂 (𝜆^{1}(𝑧 + ℎ) − 𝜆_{1}(𝑧)^{1+𝜌})

= 𝑓 (𝜆_{1}(𝑧)) + 𝑓^{}(𝜆_{1}(𝑧 + ℎ)) (𝜆_{1}(𝑧 + ℎ) − 𝜆_{1}(𝑧))
+ 𝑂 (‖ℎ‖^{1+𝜌})

= 𝑓 (𝜆_{1}(𝑧)) + 𝑓^{}(𝜆_{1}(𝑧 + ℎ))

× (ℎ_{1}− (𝑧^{2}+ ℎ_{2} − 𝑧^{2})ctan𝜃) + 𝑂(‖ℎ‖^{1+𝜌})

= 𝑓 (𝜆_{1}(𝑧)) + 𝑓^{}(𝜆_{1}(𝑧 + ℎ)) (ℎ_{1}−(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} ctan𝜃)
+ 𝑂 (‖ℎ‖^{2}) + 𝑂 (‖ℎ‖^{1+𝜌})

= 𝑓 (𝜆_{1}(𝑧)) + 𝑓^{}(𝜆_{1}(𝑧 + ℎ)) (ℎ_{1}−(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} ctan𝜃)
+ 𝑂 (‖ℎ‖^{1+min{1,𝜌}}) ,

(57)
where the second equation is due toLemma 3and the last
equality comes from the boundedness of𝑓^{}, since𝑓 is strictly
continuous at𝜆_{1}(𝑧). Similar argument holds for 𝑓(𝜆_{2}(𝑧+ℎ)).

Hence the first component of (54) is𝑂(‖ℎ‖^{1+min{1,𝜌}}).

Next, let us look into the second component of (54),
which involved𝜆_{1}(𝑧). ByLemma 1again, it can be expressed
as

− ctan𝜃

1 + ctan^{2}𝜃𝑓 (𝜆_{1}(𝑧 + ℎ)) 𝑧_{2}+ ℎ_{2}

𝑧^{2}+ ℎ_{2}

+ ctan𝜃

1 + ctan^{2}𝜃𝑓^{}(𝜆_{1}(𝑧 + ℎ) ; ℎ_{1}−(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} ctan𝜃)

× 𝑧_{2}+ ℎ_{2}

𝑧^{2}+ ℎ_{2}

+ ctan𝜃

1 + ctan^{2}𝜃𝑓 (𝜆_{1}(𝑧)) 𝑧_{2}

𝑧^{2}

+ ctan𝜃
1 + ctan^{2}𝜃

𝑓 (𝜆_{1}(𝑧 + ℎ))

𝑧^{2}+ ℎ_{2} 𝑀^{(𝑧}^{2}^{+ℎ}^{2}^{)}ℎ.

(58)

Note thatΦ is continuous differentiable (and hence is semi-
smooth) with ∇Φ(𝑧_{2}) = (1/‖𝑧_{2}‖)(𝐼 − (𝑧_{2}𝑧^{𝑇}_{2}/‖𝑧_{2}‖^{2})) and
𝑀_{(𝑧}_{2}_{+ℎ}_{2}_{)}ℎ = ‖𝑧_{2} + ℎ_{2}‖∇Φ(𝑧_{2}+ ℎ_{2})ℎ_{2}. Thus, expression (58)
can be rewritten as

− ctan𝜃

1 + ctan^{2}𝜃𝑓 (𝜆_{1}(𝑧 + ℎ)) Φ (𝑧_{2}+ ℎ_{2})
+ ctan𝜃

1 + ctan^{2}𝜃𝑓^{}(𝜆_{1}(𝑧 + ℎ) ; ℎ_{1}−(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} ctan𝜃)

× Φ (𝑧_{2}+ ℎ_{2})
+ ctan𝜃

1 + ctan^{2}𝜃𝑓 (𝜆_{1}(𝑧)) Φ (𝑧_{2})
+ ctan𝜃

1 + ctan^{2}𝜃𝑓 (𝜆_{1}(𝑧 + ℎ)) ∇Φ (𝑧_{2}+ ℎ_{2}) ℎ_{2}

= ctan𝜃
1 + ctan^{2}𝜃

× [ − 𝑓 (𝜆_{1}(𝑧 + ℎ)) + 𝑓 (𝜆_{1}(𝑧))

+𝑓^{}(𝜆_{1}(𝑧 + ℎ) ; ℎ_{1}−(𝑧_{2}+ ℎ_{2})^{𝑇}ℎ_{2}

𝑧^{2}+ ℎ_{2} ctan𝜃)]

× Φ (𝑧_{2}+ ℎ_{2})
+𝑓 (𝜆_{1}(𝑧)) ctan 𝜃

1 + ctan^{2}𝜃

× [−Φ (𝑧_{2}+ ℎ_{2}) + Φ (𝑧_{2}) + ∇Φ (𝑧_{2}+ ℎ_{2}) ℎ_{2}]
+ ctan𝜃

1 + ctan^{2}𝜃∇Φ (𝑧_{2}+ ℎ_{2}) ℎ_{2}

× [𝑓 (𝜆_{1}(𝑧 + ℎ)) − 𝑓 (𝜆_{1}(𝑧))]

= 𝑂 (‖ℎ‖^{1+min{1,𝜌}}) + 𝑂 (‖ℎ‖^{2}) + 𝑂 (‖ℎ‖^{2})

= 𝑂 (‖ℎ‖^{1+min{1,𝜌}}) .

(59)
The second equation comes from (57), strongly semismooth-
ness ofΦ at 𝑧_{2}, and

∇Φ(𝑧^{2}+ ℎ_{2}) ℎ_{2}[𝑓 (𝜆_{1}(𝑧 + ℎ)) − 𝑓 (𝜆_{1}(𝑧))] = 𝑂 (‖ℎ‖^{2}) ,
(60)
since𝑓 is Lipschitz at 𝜆_{1}(𝑧) (which is ensured by the 𝜌-order
semismoothness of𝑓). Analogous arguments apply for the
second component of (54) involving 𝜆_{2}(𝑧). From all the
above, we may conclude that

𝑓^{L}^{𝜃}(𝑧 + ℎ) − 𝑓^{L}^{𝜃}(𝑧) − (𝑓^{L}^{𝜃})^{}(𝑧 + ℎ; ℎ)

= 𝑂 (‖ℎ‖^{1+min{1,𝜌}}) , (61)

which says𝑓^{L}^{𝜃}is min{1, 𝜌}-order semismooth at 𝑧 under this
case.