DOI 10.1007/s11228-016-0374-7

**Monotonicity and Circular Cone Monotonicity** **Associated with Circular Cones**

**Jinchuan Zhou**^{1}**· Jein-Shan Chen**^{2}

Received: 27 April 2015 / Accepted: 2 May 2016 / Published online: 14 May 2016

© Springer Science+Business Media Dordrecht 2016

**Abstract The circular cone** *L**θ* is not self-dual under the standard inner product and
includes second-order cone as a special case. In this paper, we focus on the monotonicity
*of f*^{L}^{θ}*and circular cone monotonicity of f . Their relationship is discussed as well. Our*
*results show that the angle θ plays a different role in these two concepts.*

**Keywords Circular cone**· Monotonicity · Circular cone monotonicity
**Mathematics Subject Classification (2010) 26A27**· 26B35 · 49J52 · 65K10

**1 Introduction**

The circular cone [11,32] 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, 90*^{◦}*). Then, the n-dimensional circular cone denoted byL**θ*

can be expressed as

*L**θ* *:= {x = (x*1*, x*2*)** ^{T}* ∈ IR × IR

^{n}^{−1}

*| cos θx ≤ x*1

*}.*

The author’s work is supported by National Natural Science Foundation of China (11101248, 11271233) and Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016).

The author’s work is supported by Ministry of Science and Technology, Taiwan.

Jein-Shan Chen jschen@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

Note that*L*45^{◦} corresponds to the well-known second-order cone *K** ^{n}* (SOC, for short),
which is given by

*K*^{n}*:= {x = (x*1*, x*2*)** ^{T}* ∈ IR × IR

^{n−1}*| x*2

*≤ x*1

*}.*

There has been much study on SOC, see [5,6,8] and references therein; to the contrast, not much attention has been paid to circular cone at present. For optimization problems involved SOC, for example, second-order cone programming (SOCP) [1,2,17,19,21] and second-order cone complementarity problems (SOCCP) [3,9,14,16,28], the so-called SOC-functions (see [5–7])

*f*^{soc}*(x)= f (λ*1*(x))u*^{(1)}_{x}*+ f (λ*2*(x))u*^{(2)}_{x}*∀x = (x*1*, x*_{2}*)** ^{T}* ∈ IR × IR

*(1) play an essential role on both theory and algorithm aspects. In expression (1), f*

^{n−1}*: J → IR*

*with J*

*⊆ IR is a real-valued function and x is decomposed 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 toK** ^{n}*, given by

*λ*_{i}*(x)= x*1*+ (−1)*^{i}*x*2* and u*^{(i)}*x* = 1
2

1

*(−1)*^{i}*¯x*2

(3)
*for i* *= 1, 2 with ¯x*2 *:= x*2*/x*2* if x*2 * = 0, and ¯x*2 being any vector in IR* ^{n−1}*satisfying

* ¯x*2* = 1 if x*2 = 0. The decomposition (2) is called the spectral factorization associated
*with second-order cone for x. Likewise, there is a similar decomposition for x associated*
with circular cone case. More specifically, from [31, Theorem 3.1], the spectral factorization
associated with*L**θ**for x is in form of*

*x= λ*1*(x)· u*^{(1)}*x* *+ λ*2*(x)· u*^{(2)}*x* (4)

where

*λ*1*(x)* *:= x*1*− x*2*ctanθ*

*λ*_{2}*(x)* *:= x*1*+ x*2* tan θ* (5)

and ⎧

⎪⎪

⎨

⎪⎪

⎩

*u*^{(1)}*x* := 1
1+ ctan^{2}*θ*

1 0

*0 ctanθ· I*

1

*− ¯x*2

=

sin^{2}*θ*

*−(sin θ cos θ) ¯x*2

*u*^{(2)}* _{x}* := 1
1+ tan

^{2}

*θ*

1 0

*0 tan θ· I*

1

*¯x*2

=

cos^{2}*θ*
*(sin θ cos θ )¯x*2

(6)

*Analogously, for any given f* : IR → IR, we can define the following vector-valued function
for the setting of circular cone:

*f*^{L}^{θ}*(x):= f (λ*1*(x)) u*^{(1)}_{x}*+ f (λ*2*(x)) u*^{(2)}_{x}*.* (7)
For convenience, we sometime write out the explicit expression for (7) by plugging in λ_{i}*(x)*
*and u*^{(i)}* _{x}* :

*f*^{L}^{θ}*(x)*=

⎡

⎢⎣

*f (x*_{1}*− x*2*ctanθ)*

1+ ctan^{2}*θ* +*f (x*_{1}*+ x*2* tan θ)*
1+ tan^{2}*θ*

−*f (x*_{1}*− x*2*ctanθ)ctanθ*

1+ ctan^{2}*θ* +*f (x*_{1}*+ x*2* tan θ) tan θ*
1+ tan^{2}*θ*

*¯x*2

⎤

⎥*⎦ .* (8)

*Clearly, as θ* = 45^{◦}, the decomposition (4)–(8) reduces to (1)–(3). Since our main target is
*on circular cone, in the subsequent contexts of the whole paper, λ*_{i}*and u*^{(i)}* _{x}* stands for (5)
and (6), respectively.

*Throughout this paper, we always assume that J is an open interval (finite or infinite) in*
*IR, i.e., J* *:= (t, t*^{}*)with t, t*^{} *∈ IR ∪ {±∞}. Denote S the set of all x ∈ IR** ^{n}*whose spectral

*values λ*

*i*

*(x)for i= 1, 2 belong to J , i.e.,*

*S:= {x ∈ IR*^{n}*| λ**i**(x)∈ J, i = 1, 2}.*

According to [24], we know S is open if and only if J is open. In addition, as J is an
*interval, we know S is convex because*

min*{λ*1*(x), λ*1*(y)} ≤ λ*1*(βx+ (1 − β)y) ≤ λ*2*(βx+ (1 − β)y) ≤ max{λ*2*(x), λ*2*(y)}.*

We point out that there is a close relation between*L**θ*and*K** ^{n}*(see [31]) as below

*K*

^{n}*= AL*

*θ*where

*A*:=

*tan θ 0*
0 *I*

*.*

It is well-known that *K** ^{n}* is a self-dual cone in the standard inner product

*x, y =*

*n*

*i*=1*x**i**y**i*. Due to*L*^{∗}* _{θ}* =

*L*

^{π}_{2}

*−θ*by [31, Theorem 2.1],

*L*

*θ*is not a self-dual cone unless

*θ*= 45

^{◦}. In fact, we can construct a new inner product which ensures the circular cone

*L*

*θ*

*is self-dual. More precisely, we define an inner product associated with A as*

*x, y*

*A*

*:= Ax, Ay. Then*

*L*^{∗}*θ* *= {x |
x, y**A**≥ 0, ∀y ∈L**θ**} = {x |
Ax, Ay ≥ 0, ∀y ∈ A*^{−1}*K** ^{n}*}

*= {x |
Ax, y ≥ 0, ∀y ∈K*^{n}*} = {x | Ax ∈K** ^{n}*}

*= A*^{−1}*K** ^{n}*=

*L*

*θ*

*.*

However, under this new inner product the second-order cone is not self-dual, because
*(K*^{n}*)*^{∗} *= {x |
x, y**A**≥ 0, ∀y ∈K*^{n}*} = {x |
Ax, Ay ≥ 0, ∀y ∈K** ^{n}*}

*= {x |
A*^{2}*x, y ≥ 0, ∀y ∈K*^{n}*} = {x | A*^{2}*x*∈*K*^{n}*} = A*^{−2}*K*^{n}*.*

Since we cannot find an inner product such that the circular cone and second-order cone
are both self-dual simultaneously, we must choose an inner product from the standard inner
*product or the new inner product associated with A. In view of the well-known properties*
regarding second-order cone and second-order cone programming (in which many results
are based on the Jordan algebra and second-order cones are considered as self-dual cones),
we adopt the standard inner product in this paper.

*Our main attention in this paper is on the vector-valued function f*^{L}* ^{θ}*. It should be empha-
sized that the relation

*K*

^{n}*= AL*

*θ*does not guarantee that there exists a similar close relation

*between f*

^{L}

^{θ}*and f*

^{soc}

*. For example, take f (t) to be a simple function max{t, 0}, which*

*corresponds to the projection operator . For x*∈

*L*

*θ*

*, we have Ax*∈

*K*

*which implies*

^{n}_{L}_{θ}*(x)= x = A*^{−1}*(Ax)= A*^{−1}_{K}^{n}*(Ax).*

*Unfortunately, the above relation fails to hold when x /*∈*L**θ**. To see this, we let tan θ* =
*1/4 and x= (−1, 1)** ^{T}*. Then, it can be verified

_{L}_{θ}*(x)= 0 and A*^{−1}_{K}^{n}*(Ax)*=

_{3}

23 8

*which says *_{L}_{θ}*(x) = A*^{−1}_{K}^{n}*(Ax). This example undoubtedly indicates that we cannot*
*study f*^{L}^{θ}*by simply resorting to f*^{soc}*. Hence, it is necessary to study f*^{L}* ^{θ}* directly, and the
results in this paper are neither trivial nor being taken for granted.

Much attention has been paid to symmetric cone optimizations, see [22,23,27] and references therein. Non-symmetric cone optimization research is much more recent; for

*example, the works on p-order cone [30], homogeneous cone [15,*29], matrix cone [12];

etc. Unlike the symmetric cone case in which the Euclidean Jordan algebra can unify the analysis [13], so far no unifying algebra structure has been found for non-symmetric cones.

In other words, we need to study each non-symmetric cones according to their different properties involved. For circular cone, a special non-symmetric cone, and circular cone optimization, like when dealing with SOCP and SOCCP, the following studies are crucial:

(i) spectral factorization associated with circular cones; (ii) smooth and nonsmooth analysis
*for f*^{L}* ^{θ}* given as in (7); (iii) the so-called

*L*

*θ*-convexity; and (iv)

*L*

*θ*-monotonicity. The first three points have been studied in [4,31,32], and [33], respectively. Here, we focus

*on the fourth item, that is, monotonicity. The SOC-monotonicity of f have been discussed*thoroughly in [5,7,24]; and the monotonicity of the spectral operator of symmetric cone has been studied in [18]. The main aim of this paper is studying those monotonicity properties

*in the framework of circular cone. Our results reveal that the angle θ plays different role in*

*these two concepts. More precisely, the circular cone monotonicity of f depends on f and*

*θ, whereas the monotonicity of f*

^{L}

^{θ}*only depends on f .*

To end this section, we say a few words about the notations and present the definitions
of monotonicity and*L**θ**-monotonicity. A matrix M* ∈ IR* ^{n×n}* is said to be

*L*

*θ*-invariant if

*Mh*∈

*L*

*θ*

*for all h*∈

*L*

*θ*

*. We write x*

*L*

*θ*

*yto mean x− y ∈L*

*θ*and denote

*L*

^{◦}

*the polar cone of*

_{θ}*L*

*θ*, i.e.,

*L*^{◦}*θ**:= {y ∈ IR*^{n}*|
x, y ≤ 0, ∀x ∈L**θ**}.*

*Denote e:= (1, 0, . . . , 0)*^{T}*and use λ(M), λ*_{min}*(M), λ*_{max}*(M)*for the set of all eigenval-
*ues, the minimum, and the maximum of eigenvalues of M, respectively. Besides,*S* ^{n}*means
the space of all symmetric matrices in IR

*andS*

^{n×n}

^{n}_{+}is the cone of positive semidefinite

*matrices. For a mapping g*: IR

*→ IR*

^{n}

^{m}*, denote by D*

*g*the set of all differentiable points of

*g. For convenience, we define 0/0:= 0. Given a real-valued function f : J → IR,*(a)

*f*is said to be

*L*

*θ*

*-monotone on J if for any x, y∈ S,*

*x**L**θ* *y* *=⇒ f*^{L}^{θ}*(x)**L**θ* *f*^{L}^{θ}*(y)*;
(b) *f*^{L}^{θ}*is said to be monotone on S if*

*f*^{L}^{θ}*(x)− f*^{L}^{θ}*(y), x− y*

*≥ 0, ∀x, y ∈ S.*

(c) *f*^{L}^{θ}*is said to be strictly monotone on S if*

*f*^{L}^{θ}*(x)− f*^{L}^{θ}*(y), x− y*

*>0,* *∀x, y ∈ S, x = y.*

(d) *f*^{L}^{θ}*is said to be strongly monotone on S with μ > 0 if*

*f*^{L}^{θ}*(x)− f*^{L}^{θ}*(y), x− y*

*≥ μx − y*^{2}*,* *∀x, y ∈ S.*

**2 Circular Cone Monotonicity of****f**

This section is devoted to the study of*L**θ*-monotonicity. The main purpose is to provide
characterizations of*L**θ*-monotone functions. To this end, we need a few technical lemmas.

**Lemma 2.1 Let A, B be symmetric matrices and y**^{T}*Ay >* *0 for some y. Then, the*
*implication[z*^{T}*Az≥ 0 =⇒ z*^{T}*Bz≥ 0] is valid if and only if B *S^{n}_{+} *λA for some λ≥ 0.*

*Proof This is the well known S-Lemma, see [7, Lemma 3.1] or [25].*

**Lemma 2.2 Given ζ***∈ IR, u ∈ IR*^{n−1}*, and a symmetric matrix *∈ IR^{n×n}*. Denote*B :=

*{z ∈ IR*^{n−1}*|z ≤ 1}. Then, the following statements hold.*

(a) * beingL**θ**-invariant is equivalent to *

*ctanθ*
*z*

∈*L**θ**for any z∈ B.*

(b) *If *=

*ζ u*^{T}*u H*

*with H* ∈ S^{n−1}*, then isL**θ**-invariant is equivalent to*
*ζ* *≥ u tan θ*

*and there exists λ≥ 0 such that*

*ζ*^{2}*− ctan*^{2}*θu*^{2}*− λ (ζ tan θ)u*^{T}*− ctanθu*^{T}*H*
*(ζtan θ )u− ctanθH u tan*^{2}*θ uu*^{T}*− H*^{2}*+ λI*

S^{n}_{+}*O.*

*Proof (a) The result follows from the following observation:*

*L**θ* ∈*L**θ* *⇐⇒ A*^{−1}*K*^{n}*∈ A*^{−1}*K*^{n}*⇐⇒ AA*^{−1}*K** ^{n}*∈

*K*

^{n}*⇐⇒ AA*^{−1}

1
*z*

∈*K*^{n}*⇐⇒ A*^{−1}

1
*z*

*∈ A*^{−1}*K*^{n}

*⇐⇒ A*^{−1}

1
*z*

∈*L**θ* *⇐⇒ *

*ctanθ*
*z*

∈*L**θ**,* (9)

where the third equivalence comes from [7, Lemma 3.2].

(b) From (9), we know that
*A A*^{−1}

1
*z*

=

*ζ* *tan θ u*^{T}*ctanθ u* *H*

1
*z*

=

*ζ+ tan θu*^{T}*z*
*ctanθ u+ Hz*

∈*K*^{n}*,*
which means

*ζ+ u*^{T}*ztan θ* *≥ 0, ∀z ∈ B,* (10)

and

*ζ+ u*^{T}*ztan θ≥ ctanθu + Hz, ∀z ∈ B.* (11)
Note that condition (10) is equivalent to

*ζ* *≥ tan θ max{−u*^{T}*z|z ∈ B} = tan θu*

and condition (11) is equivalent to

*ζ+ tan θu*^{T}*z*2

*≥ ctanθu + Hz*^{2}*,*
i.e.,

*z*^{T}*(tan*^{2}*θ uu*^{T}*− H*^{2}*)z*+ 2

*ζtan θ u*^{T}*− ctanθu*^{T}*H*

*z+ ζ*^{2}− ctan^{2}*θ u*^{T}*u≥ 0 ∀z ∈ B,*
which can be rewritten as

*1 z*^{T}

* *

1
*z*

*≥ 0 ∀z ∈ B,* (12)

with

* *:=

*ζ*^{2}− ctan^{2}*θ u*^{T}*u* *(ζtan θ )u*^{T}*− ctanθu*^{T}*H*
*(ζtan θ )u− ctanθH u* tan^{2}*θ uu*^{T}*− H*^{2}

*.*
We now claim that (12) is equivalent to the following implication:

*k v*^{T}

1 0
0 *−I*

*k*
*v*

≥ 0 =⇒

*k v*^{T}

* *

*k*
*v*

*≥ 0, ∀*

*k*
*v*

∈ IR^{n}*.* (13)

First, we see that (12) corresponds to the case of k= 1 in (13). Hence, it only needs to show how to obtain (13) from (12). We proceed the arguments by considering the following two cases.

*For k = 0, dividing by k*^{2}in the left side of (13) yields

*1 (v*

*k)*^{T}

1 0
0 *−I*

1
*v*
*k*

*≥ 0,*
*which implies v/k*∈ B. Then, it follows from (12) that

*1 (v*

*k)*^{T}* *

1
*v*
*k*

*≥ 0.*

Hence, the right side of (13) holds.

*For k*= 0, the left side of (13) is*v ≤ 0, which says v = 0, i.e., (k, v)** ^{T}* = 0. Therefore,
the right side of (13) holds clearly.

*Now, applying Lemma 2.1 to ensures the existence of λ*≥ 0 such that

*ζ*^{2}− ctan^{2}*θ u*^{T}*u* *ζtan θ u*^{T}*− ctanθu*^{T}*H*
*ζtan θ u− ctanθH u* tan^{2}*θ uu*^{T}*− H*^{2}

*− λ*

1 0
0 *−I*

S^{n}_{+}*O.*

Thus, the proof is complete.

* Lemma 2.3 For a matrix being in form of H* :=

*x*1 *x*_{2}^{T}*x*2 *αI+ β ¯x*2*¯x*_{2}^{T}

*, where α, β* *∈ IR,*
*then*

max*{x*1*+ x*2*, x*1*− β} + max{0, α − x*1*+ β}*

*≥ λ*max*(H )≥ λ*min*(H )*

*≥ min{x*1*− x*2*, x*1*− β} + min{0, α − x*1*+ β}.*

*Proof First, we split H as sum of three special matrices, i.e.,*

*x*1 *x*_{2}^{T}*x*_{2} *αI+ β ¯x*2*¯x*2^{T}

=

*x*1 *x*^{T}_{2}
*x*2 *x*1*I*

*− β*

0 0

*0 I− ¯x*2*¯x*_{2}^{T}

+

0 0

*0 (α− x*1*+ β)I*

and let 1:=

*x*_{1} *x*_{2}^{T}*x*_{2} *x*_{1}*I*

*− β*

0 0

*0 I− ¯x*2*¯x** ^{T}*2

*,* 2:=

0 0

*0 (α− x*1*+ β)I*

*.*
*Then, λ(*_{1}*)= {x*1*− x*2*, x*1*+ x*2*, x*1*− β} by [*6, Lemma 1] and λ(_{2}*)= {0, α −*
*x*1*+ β}. Thus, the desired result follows from the following facts:*

*λ*min*(*1*+ *2*)≥ λ*min*(*1*)+ λ*min*(*2*)* *and λ*max*(*1*+ *2*)≤ λ*max*(*1*)+ λ*max*(*2*).*

This completes the proof.

*Next, we turn our attention to the vector-valued function f*^{L}* ^{θ}* defined as in (7). Recall
from [4,32] that f

^{L}

^{θ}*is differentiable at x if and only if f is differentiable at λ*

*i*

*(x)*for

*i= 1, 2 and*

*∇f*^{L}^{θ}*(x)*=

⎧⎨

⎩

*f*^{}*(x*1*)I* *x*2= 0;

*ξ* *
¯x*_{2}^{T}*
¯x*2 *τ I+ (η − τ) ¯x*2*¯x*2^{T}

*x*2* = 0,* (14)

where

*τ* := *f (λ*_{2}*(x))− f (λ*1*(x))*

*λ*_{2}*(x)− λ*1*(x)* *,* *ξ*:= *f*^{}*(λ*_{1}*(x))*

1+ ctan^{2}*θ* + *f*^{}*(λ*_{2}*(x))*
1+ tan^{2}*θ,*
*
* := − *ctanθ*

1+ ctan^{2}*θf*^{}*(λ*_{1}*(x))*+ *tan θ*

1+ tan^{2}*θf*^{}*(λ*_{2}*(x)),*
*η* := ctan^{2}*θ*

1+ ctan^{2}*θf*^{}*(λ*_{1}*(x))*+ tan^{2}*θ*

1+ tan^{2}*θf*^{}*(λ*_{2}*(x)).*

*The following result shows that if f*^{L}* ^{θ}* is differentiable, then we can characterize the

*L*

*θ*

*-monotonicity of f via the gradient∇f*

^{L}*.*

^{θ}**Theorem 2.1 Suppose that f***: J → IR is differentiable. Then, f isL**θ**-monotone on J if*
*and only if∇f*^{L}^{θ}*(x) isL**θ**-invariant for all x∈ S.*

*Proof “⇒” Suppose that f isL**θ**-monotone. Take x∈ S and h ∈L**θ*, what we want to prove
is*∇f*^{L}^{θ}*(x)h*∈*L**θ*. From the*L**θ**-monotonicity of f , we know f*^{L}^{θ}*(x+ th) **L**θ* *f*^{L}^{θ}*(x)*
*for all t > 0. Note thatL**θ* is a cone. Hence

*f*^{L}^{θ}*(x+ th) − f*^{L}^{θ}*(x)*

*t* *L**θ* *0.* (15)

Since *L**θ* *is closed, taking the limit as t* → 0^{+} yields *∇f*^{L}^{θ}*(x)h* *L**θ* 0, i.e.,

*∇f*^{L}^{θ}*(x)h*∈*L**θ*.

“*⇐” Suppose that ∇f*^{L}^{θ}*(x)*is*L**θ**-invariant for all x∈ S. Take x, y ∈ S with x **L**θ* *y*(i.e.,
*x−y ∈L**θ**). In order to show the desired result, we need to argue that f*^{L}^{θ}*(x)**L**θ* *f*^{L}^{θ}*(y).*

*For any ζ* ∈*L*^{◦}* _{θ}*, we have

*ζ, f*^{L}^{θ}*(x)− f*^{L}^{θ}*(y)*

=

1 0

*ζ,∇f*^{L}^{θ}*(x+ t(x − y)) (x − y)*

*dt≤ 0,* (16)
where the last step comes from*∇f*^{L}^{θ}*(x+ t(x − y))(x − y) ∈L**θ**because x+ t(x − y) ∈*
*S* *(since S is convex) and∇f*^{L}* ^{θ}* is

*L*

*θ*

*-invariant over S by hypothesis. Since ζ*∈

*L*

^{◦}

*is arbitrary, (16) implies f*

_{θ}

^{L}

^{θ}*(x)− f*

^{L}

^{θ}*(y)∈ (L*

^{◦}

_{θ}*)*

^{◦}=

*L*

*θ*, where the last step is due to the fact that

*L*

*θ*

*is a closed convex cone. This means f*

^{L}

^{θ}*(x)*

*L*

*θ*

*f*

^{L}

^{θ}*(y).*

*Note that f is Lipschitz continuous on J if and only if f*^{L}* ^{θ}* is Lipschitz continuous on

*S, see [4,*32]. The nonsmooth version of Theorem 2.1 is given below.

**Theorem 2.2 Suppose that f***: J → IR is Lipschitz continuous on J . Then the following*
*statements are equivalent.*

(a) *f isL**θ**-monotone on J ;*

(b) *∂*_{B}*f*^{L}^{θ}*(x) isL**θ**-invariant for all x∈ S;*

(c) *∂f*^{L}^{θ}*(x) isL**θ**-invariant for all x∈ S.*

*Proof “(a)⇒ (b)” Take V ∈ ∂**B**f*^{L}^{θ}*(x), then by definition of B-subdifferential there exists*
*{x**k**} ⊂ D*_{f}*Lθ* *such that x*_{k}*→ x and ∇f*^{L}^{θ}*(x*_{i}*)* *→ V . According to (*15), we obtain

*∇f*^{L}^{θ}*(x*_{i}*)h**L**θ* *0 for h*∈*L**θ**. Taking the limit yields V h**L**θ* *0. Since V* *∈ ∂**B**f*^{L}^{θ}*(x)*is
*arbitrary, this says that ∂*_{B}*f*^{L}* ^{θ}*is

*L*

*θ*-invariant.

“(b)*⇒ (c)” Take V ∈ ∂f*^{L}^{θ}*(x), then by definition, there exists V*_{i}*∈ ∂**B**f*^{L}^{θ}*(x)and β** _{i}* ∈

*[0, 1] such that V =*

*i**β**i**V**i* and

*i**β**i* *= 1. Thus, for any h ∈* *L**θ**, we have V h* =

*i**β**i**V**i**h*∈*L**θ**, since V**i*is*L**θ*-invariant and*L**θ**is convex. Hence ∂f*^{L}^{θ}*(x)*is*L**θ*-invariant.

“(c)⇒ (a)” The proof follows from Theorem 2.1 by replacing (16) with

*ζ, f*^{L}^{θ}*(x)− f*^{L}^{θ}*(y)*

*=
ζ, V (x − y) ≤ 0,*

*for some V* *∈ ∂f*^{L}^{θ}*(z)with z∈ [x, y] by the mean-value theorem of Lipschitz functions*
[10].

With these preparations, we provide a sufficient condition for the*L**θ*-monotonicity.

**Theorem 2.3 Suppose that f***: J → IR is differentiable. If for all t*1*, t*2*∈ J with t*1*≤ t*2*,*
*(tan θ− ctanθ)*

*f*^{}*(t*_{1}*)− f*^{}*(t*_{2}*)*

*≥ 0,* (17)

*and* ⎡

⎢⎣ *f*^{}*(t*1*)* *f (t*2*)− f (t*1*)*
*t*2*− t*1

*f (t*2*)− f (t*1*)*
*t*_{2}*− t*1

*f*^{}*(t*2*)*

⎤

⎥⎦ _{S}^{2}_{+}*O,* (18)

*then f isL**θ**-monotone on J .*

*Proof According to Theorem 2.1, it suffices to show that∇f*^{L}^{θ}*(x)*is*L**θ*-invariant for all
*x∈ S. We proceed by discussing the following two cases.*

**Case 1 For x**_{2} *= 0, in this case it is clear that ∇f*^{L}^{θ}*(x)* being *L**θ*-invariant, i.e.,

*∇f*^{L}^{θ}*(x)h= f*^{}*(x*1*)h*∈*L**θ**for all h*∈*L**θ**, is equivalent to saying f*^{}*(x*1*)*≥ 0.

**Case 2 For x**_{2} = 0, let

*H* *:= τI + (η − τ) ¯x*2*¯x*2^{T}*.*

Then, applying Lemma 2.2 to the formula of*∇f*^{L}^{θ}*(x)*in (14),*∇f*^{L}^{θ}*(x)*is*L**θ*-invariant if
and only if

*ξ* *≥
tan θ* (19)

*and there exists λ*≥ 0 such that
*ϒ*:=

*ξ*^{2}− ctan^{2}*θ
*^{2}*− λ* *ξtan θ
¯x*_{2}^{T}*− ctanθ
¯x*_{2}^{T}*H*
*ξtan θ
¯x*2*− ctanθ
H ¯x*2 tan^{2}*θ
*^{2}*¯x*2*¯x*_{2}^{T}*− H*^{2}*+ λI*

S^{n}_{+}*O.* (20)
Hence, to achieve the desired result, it is equivalent to showing that the conditions (17)
and (18) can guarantee the validity of the conditions (19) and (20). To check this, we first
note that (19) is equivalent to

*− tan θf*^{}*(λ*1*(x))− ctanθf*^{}*(λ*2*(x))≤ − tan θf*^{}*(λ*1*(x))+ tan θf*^{}*(λ*2*(x))*

*≤ tan θf*^{}*(λ*_{1}*(x))+ ctanθf*^{}*(λ*_{2}*(x))*

*⇐⇒ f*^{}*(λ*2*(x))≥ 0 and f*^{}*(λ*1*(x))*≥1− ctan^{2}*θ*

2 *f*^{}*(λ*2*(x)).* (21)

This is ensured by (17) and (18). In fact, if tan θ *≥ ctanθ, then we know from (*17) that
*f*^{}*(λ*_{1}*(x))≥ f*^{}*(λ*_{2}*(x))*≥

*(1*− ctan^{2}*θ )/2*

*f*^{}*(λ*_{2}*(x))*where the second inequality is due

*to f*^{}*(λ*_{2}*(x))*≥ 0 by (18). If tan θ *≤ ctanθ, then 1 − ctan*^{2}*θ* *≤ 0, and hence f*^{}*(λ*_{1}*(x))*≥
0≥

*(1*− ctan^{2}*θ )/2*

*f*^{}*(λ*2*(x))since f*^{}*(λ**i**(x))≥ 0 for i = 1, 2 by (*18).

*Now let us look into the entries of ϒ. In the ϒ*11-entry, we calculate
*ξ*^{2}− ctan^{2}*θ
*^{2}

= 1

*(tan θ+ ctanθ)*^{2}

*(tan*^{2}*θ*− ctan^{2}*θ )f*^{}*(λ*1*(x))*^{2}*+ 2(1 + ctan*^{2}*θ )f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))*

= 1

*(tan θ+ ctanθ)*^{2}

*(tan*^{2}*θ*− ctan^{2}*θ )f*^{}*(λ*_{1}*(x))*^{2}*+ (ctan*^{2}*θ*− tan^{2}*θ )f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*

+ 1

*(tan θ+ ctanθ)*^{2}

2+ tan^{2}*θ*+ ctan^{2}*θ*

*f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*

*= μ + f*^{}*(λ*1*(x))f*^{}*(λ*2*(x)),*
with

*μ* := 1

*(tan θ+ ctanθ)*^{2}

*(tan*^{2}*θ*− ctan^{2}*θ )f*^{}*(λ*_{1}*(x))*^{2}*+ (ctan*^{2}*θ*− tan^{2}*θ )f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*

= *tan θ− ctanθ*

*tan θ+ ctanθf*^{}*(λ*_{1}*(x))*

*f*^{}*(λ*_{1}*(x))− f*^{}*(λ*_{2}*(x))*

*≥ 0,*

where the last step is due to (17). In the ϒ_{12}*-entry and ϒ*_{21}-entry, we calculate
*(ξ**tan θ )
**¯x** ^{T}*2

*− ctanθ ¯x*

*2*

^{T}*H*

= 1

*(tan θ**+ ctanθ)*^{2}

− tan^{2}*θ*+ ctan^{2}*θ*

*f*^{}*(λ*1*(x))*^{2}+

tan^{2}*θ*− ctan^{2}*θ*

*f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))*

*¯x*2^{T}

= −*tan θ**− ctanθ*

*tan θ**+ ctanθ**f*^{}*(λ*1*(x))*

*f*^{}*(λ*1*(x))**− f*^{}*(λ*2*(x))*

*¯x*2^{T}

*= −μ ¯x*2^{T}*.*

*In the ϒ*_{22}-entry, we calculate
tan^{2}*θ
*^{2}*¯x*2*¯x*2^{T}*− H*^{2} *= −τ*^{2}*I*+

*τ*^{2}+ 1

*(tan θ+ ctanθ)*^{2}

*(tan*^{2}*θ*− ctan^{2}*θ )f*^{}*(λ*_{1}*(x))*^{2}

*−2(1 + tan*^{2}*θ )f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*

*¯x*2*¯x*^{T}_{2}

*= −τ*^{2}*I*+

*τ*^{2}*+ μ − f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))*

*¯x*2*¯x*2^{T}*.*

*Hence, ϒ can be rewritten as*
*ϒ* =

*ϒ*_{11} *ϒ*_{12}
*ϒ*21 *ϒ*22

=

*μ+ f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))− λ* *−μ ¯x*2^{T}

*−μ ¯x*2 *(λ− τ*^{2}*)I*+

*τ*^{2}*+ μ − f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))*

*¯x*2*¯x*_{2}^{T}

*.*
*Now, applying Lemma 2.3 to ϒ, we have*

*λ*min*(ϒ )* ≥ min

*ϒ*11*− |μ|, ϒ*11−

*τ*^{2}*+ μ − f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))*

+ min
*0,*

*λ− τ*^{2}

*− ϒ*11+

*τ*^{2}*+ μ − f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*

= min

*f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))− λ, 2f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))− τ*^{2}*− λ*
+2 min!

*0, λ− f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*"

*.* (22)

*Using λ*_{1}*(x)≤ λ*2*(x)*and condition (18) ensures

*f*^{}*(λ*1*(x))≥ 0, f*^{}*(λ*2*(x))≥ 0, and f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))*≥

*f (λ*2*(x))− f (λ*1*(x))*
*λ*2*(x)− λ*1*(x)*

2

*,*
which together with (15) yields

*f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))≥ 0 and f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))− τ*^{2}*≥ 0.*

*Thus, we can plug λ:= f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))*≥ 0 into (22), which gives λ_{min}*(ϒ )* ≥ 0.

*Hence ϒ is positive semi-definite. This completes the proof.*

*Remark 2.1 The condition (17) holds automatically when θ* = 45^{◦}. In other case, some
*addition requirement needs to be imposed on f . For instance, f is required to be convex as*
*θ* *∈ (0, 45*^{◦}*)while f is required to be concave as θ* *∈ (45*^{◦}*,*90^{◦}*). This indicates that the*
*angle plays an essential role in the framework of circular cone, i.e., the assumption on f is*
dependent on the range of the angle.

Based on the above, we can achieve a necessary and sufficient condition for *L**θ*-
*monotonicity in the special case of n*= 2.

**Theorem 2.4 Suppose that f***: J → IR is differentiable on J and n = 2. Then, f isL**θ**-*
*monotone on J if and only if f*^{}*(t)≥ 0 for all t ∈ J and (tan θ −ctanθ)(f*^{}*(t*1*)−f*^{}*(t*2*))*≥ 0
*for all t*1*, t*2*∈ J with t*1*≤ t*2*.*

*Proof In light of the proof of Theorem 2.3, we know that f isL**θ*-monotone if and only if
*for any x∈ S,*

*f*^{}*(λ*2*(x))≥ 0, f*^{}*(λ*1*(x))*≥1− ctan^{2}*θ*

2 *f*^{}*(λ*2*(x)),* (23)

*and there exists λ*≥ 0 such that
*ϒ*=

*μ+ f*^{}*(λ*_{1}*(x))f*^{}*(λ*_{2}*(x))− λ* *±μ*

*±μ* *μ− f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))+ λ*

_{S}^{2}_{+}*O,* (24)
*where the form of ϒ comes from the fact that* *¯x*2 = ±1 in this case. It follows from (24)
*that μ*^{2}*− (f*^{}*(λ*1*(x))f*^{}*(λ*2*(x))− λ)*^{2}*− μ*^{2} *≥ 0, which implies λ = f*^{}*(λ*1*(x))f*^{}*(λ*2*(x)).*

Substituting it into (24) yields
*ϒ*=

*μ* *±μ*

*±μ μ*

*= μ*

1 ±1

±1 1

_{S}^{2}_{+}*O,*

*which in turn implies μ*≥ 0. Hence, the conditions (23) and (24) are equivalent to

⎧⎨

⎩*f*^{}*(λ*2*(x))≥ 0, f*^{}*(λ*1*(x))*≥1− ctan^{2}*θ*

2 *f*^{}*(λ*2*(x)),*
*(tan θ− ctanθ)f*^{}*(λ*1*(x))*

*f*^{}*(λ*1*(x))− f*^{}*(λ*2*(x))*

*≥ 0.*

*Due to the arbitrariness of λ**i**(x)∈ J and applying similar arguments following (*21), the
*above conditions give f*^{}*(t)≥ 0 for all t ∈ J and (tan θ − ctanθ)(f*^{}*(t*1*)− f*^{}*(t*2*))*≥ 0 for
*all t*_{1}*, t*_{2}*∈ J with t*1*≤ t*2. Thus, the proof is complete.

**3 Monotonicity of****f**^{L}^{θ}

In Section 2, we have shown that the circular cone monotonicity of f depends on both
*the monotonicity of f and the range of the angle θ . Now the following questions arise:*

*how about on the relationship between the monotonicity of f*^{L}* ^{θ}* and the

*L*

*θ*-monotonicity

*of f ? Whether the monotonicity of f*

^{L}

^{θ}*also depends on θ ? This is the main motivation*

*of this section. First, for a mapping H*: IR

*→ IR*

^{n}

^{n}*, let us denote ∂H (x)*S

^{n}_{+}

*O*(or

*∂H (x)*S^{n}_{+} *O) to mean that each elements in ∂H (x) is positive semi-definite (or positive*
definite), i.e.,

*∂H (x)*S^{n}_{+}*O (or*S^{n}_{+}*O)* *⇐⇒ A *S^{n}_{+}*O (or*S^{n}_{+}*O),* *∀A ∈ ∂H(x).*

Taking into account of the result in [26], we readily have

**Lemma 3.1 Let f be Lipschitz continuous on J . The following statements hold:**

(a) *f*^{L}^{θ}*is monotone on S if and only if ∂f*^{L}^{θ}*(x)*S^{n}_{+} *O for all x∈ S;*

(b) *If ∂f*^{L}^{θ}*(x)*S^{n}_{+}*O for all x∈ S, then f*^{L}^{θ}*is strictly monotone on S*;

(c) *f*^{L}^{θ}*is strongly monotone on S if and only if there exists μ > 0 such that ∂f*^{L}^{θ}*(x)*S^{n}_{+}

*μI for all x∈ S.*

**Lemma 3.2 Suppose that f is Lipschitz continuous. Then,**

*∂*_{B}*f*^{L}^{θ}*(x)*S^{n}_{+}*O⇐⇒ ∂f*^{L}^{θ}*(x)*S^{n}_{+}*O and ∂**B**f*^{L}^{θ}*(x)*S^{n}_{+}*O⇐⇒ ∂f*^{L}^{θ}*(x)*S^{n}_{+}*O.*

*Proof The result follows immediately from the fact ∂f*^{L}^{θ}*(x)= conv∂**B**f*^{L}^{θ}*(x).*

**Lemma 3.3 The following statements hold.**

(a) *For any h*∈ IR^{n}*t*1

*h*1*− ctanθ ¯x*2^{T}*h*2

2

*+ t*2

*h*1*+ tan θ ¯x** ^{T}*2

*h*2

2

*+ t*3

*h*2^{2}*− ( ¯x*2^{T}*h*2*)*^{2}

≥ 0 (25)

*if and only if t**i* *≥ 0 for i = 1, 2, 3.*

(b) *For any h*∈ IR* ^{n}*\{0}

*t*1

*h*1*− ctanθ ¯x*2^{T}*h*2

2

*+ t*2

*h*1*+ tan θ ¯x*2^{T}*h*2

2

*+ t*3

*h*2^{2}*− ( ¯x*2^{T}*h*2*)*^{2}

*>*0 (26)
*if and only if t*_{i}*>0 for i= 1, 2, 3.*

*Proof (a) The sufficiency is clear, since| ¯x*^{T}_{2}*h*2*| ≤ ¯x*2*h*2* = h*2 by Cauchy-Schwartz
*inequality. Now let us show the necessity. Taking h= (1, −ctanθ ¯x*2*)** ^{T}*, then (25) equal to

*t*1

*(1*+ctan

^{2}

*θ )*

^{2}

*≥ 0, which implies t*1

*≥ 0. Similarly, let h = (1, tan θ ¯x*2

*)*

*, then (25) yields*

^{T}*t*2

*(1*+ tan

^{2}

*θ )*

^{2}

*≥ 0, implying t*2

*≥ 0. Finally, let h = (0, u)*

^{T}*with u satisfying u, ¯x*2 = 0 and

*u = 1, then it follows from (*25) that t3≥ 0.

(b) The necessity is the same of the argument as given in part (a). For sufficiency, take a
*nonzero vector h. Ifh*2*− ¯x** ^{T}*2

*h*

_{2}

*>0, then the result holds since t*

_{3}

*>*0. If

*h*2

*− ¯x*2

^{T}*h*

_{2}= 0,

*then h*

_{2}

*= β ¯x*2. So the left side of (26) takes t

_{1}

*(h*

_{1}

*− βctanθ)*

^{2}

*+ t*2

*(h*

_{1}

*+ β tan θ)*

^{2}

*>*0,

*because h*

_{1}

*− βctanθ = h*1

*+ β tan θ = 0 only happened when β = 0 and h*1 = 0. This

*means h*

_{2}

*= 0 since h*2

*= β ¯x*2

*, so h*= 0.

*If f is differentiable, it is known that f*^{L}* ^{θ}* is differentiable [4,32]. It then follows from

*Lemma 3.1 that f*

^{L}

^{θ}*is monotone on S if and only if∇f*

^{L}

^{θ}*(x)*S

^{n}_{+}

*Ofor all x∈ S. Hence,*

*to characterize the monotonicity of f*

^{L}*, the first thing is to estimate*

^{θ}*∇f*

^{L}

^{θ}*(x)*S

^{n}_{+}

*O*via

*f*.

* Theorem 3.1 Given x* ∈ IR

^{n}*and suppose that f is differentiable at λ*

*i*

*(x) for i*

*= 1, 2.*

*Then∇f*^{L}^{θ}*(x)*S^{n}_{+}*O if and only if f*^{}*(λ**i**(x))≥ 0 for i = 1, 2 and f (λ*2*(x))≥ f (λ*1*(x)).*

*Proof The proof is divided into the following two cases.*

* Case 1 For x*2

*= 0, using ∇f*

^{L}

^{θ}*(x)*

*= f*

^{}

*(x*1

*)e, it is clear that∇f*

^{L}

^{θ}*(x)*S

^{n}_{+}

*O*is

*equivalent to saying f*

^{}

*(x*1

*)*≥ 0. Then, the desired result follows.

**Case 2 For x**_{2} = 0, denote

*b*1:= *f*^{}*(λ*1*(x))*

1+ ctan^{2}*θ* and *b*2= *f*^{}*(λ*2*(x))*
1+ tan^{2}*θ.*

*Then, ξ= b*1*+ b*2*,
= −b*1*ctanθ+ b*2*tan θ , and η= b*1ctan^{2}*θ+ b*2tan^{2}*θ. Hence, for*
*all h= (h*1*, h*2*)** ^{T}* ∈ IR × IR

*, we have*

^{n−1}*
h, ∇f*^{L}^{θ}*(x)h*

*= (b*1*+ b*2*)h*^{2}_{1}*+ 2
h*1*¯x*^{T}_{2}*h*_{2}*+ τh*2^{2}*+ (b*1ctan^{2}*θ+ b*2tan^{2}*θ− τ)( ¯x*_{2}^{T}*h*_{2}*)*^{2}

*= (b*1*+ b*2*)h*^{2}_{1}*+ 2(−b*1*ctanθ+ b*2*tan θ )h*_{1}*¯x*_{2}^{T}*h*_{2}*+ (b*1ctan^{2}*θ+ b*2tan^{2}*θ )(¯x*_{2}^{T}*h*_{2}*)*^{2}
*+τ*

*h*2^{2}*− ( ¯x*_{2}^{T}*h*_{2}*)*^{2}

*= b*1

*h*^{2}_{1}*− 2ctanθh*1*¯x*2^{T}*h*_{2}+ ctan^{2}*θ (¯x*2^{T}*h*_{2}*)*^{2}

*+ b*2

*h*^{2}_{1}*+ 2 tan θh*1*¯x*2^{T}*h*_{2}+ tan^{2}*θ (¯x*2^{T}*h*_{2}*)*^{2}

*+τ*

*h*2^{2}*− ( ¯x*2^{T}*h*_{2}*)*^{2}

*= b*1

*h*_{1}*− ctanθ ¯x*_{2}^{T}*h*_{2}

2

*+ b*2

*h*_{1}*+ tan θ ¯x*_{2}^{T}*h*_{2}

2

*+ τ*

*h*2^{2}*− ( ¯x*_{2}^{T}*h*_{2}*)*^{2}

*.*

In light of Lemma 3.3, the desired result is equivalent to
*b*1*≥ 0, b*2*≥ 0, τ =* *f (λ*2*(x))− f (λ*1*(x))*

*λ*2*(x)− λ*1*(x)* *≥ 0,*

*i.e., f*^{}*(λ*1*(x))≥ 0, f*^{}*(λ*2*(x))≥ 0, and f (λ*2*(x))≥ f (λ*1*(x))due to λ*2*(x) > λ*1*(x)*in this
case.

By following almost the same arguments as given in Theorem 3.1, we further obtain the following consequence.

* Corollary 3.1 Given x* ∈ IR

^{n}*and suppose that f is differentiable at λ*

*i*

*(x) for i*

*= 1, 2.*

*Then for x*_{2}* = 0, ∇f*^{L}^{θ}*(x)*S^{n}_{+}*O if and only if f*^{}*(λ*_{i}*(x)) >0 for i= 1, 2 and f (λ*2*(x)) >*

*f (λ*_{1}*(x)); for x*_{2}*= 0, ∇f*^{L}^{θ}*(x)*S^{n}_{+}*O if and only if f*^{}*(x*_{1}*) >0.*

*When f is non-differentiable, we resort to the subdifferential ∂*_{B}*(f*^{L}^{θ}*), whose estimate*
is given in [32].