1Introduction Monotonicityandcircularconemonotonicityassociatedwithcircularcones

Monotonicity and circular cone monotonicity associated with circular cones

Jinchuan Zhou ¹ Department of Mathematics

School of Science

Shandong University of Technology Zibo 255049, P.R. China E-mail: jinchuanzhou@163.com

Jein-Shan Chen ² Department of Mathematics National Taiwan Normal University

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

April 26, 2015

(1st revised on March 24, 2016) (2nd revised on April 19, 2016)

(3rd revised on April 29, 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^Lθ 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.

AMS subject classifications. 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

1The author’s work is supported by National Natural Science Foundation of China (11101248, 11271233) and Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016).

2Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan.

half-aperture angle be θ with θ ∈ (0,^π₂). Then, the n-dimensional circular cone denoted by Lθ can be expressed as

L_θ := {x = (x₁, x₂)^T ∈ IR × IRⁿ⁻¹| cos θkxk ≤ x₁}.

Note that L₄₅^◦ corresponds the well-known second-order cone Kⁿ(SOC, for short), which is given by

Kⁿ:= {x = (x₁, x₂)^T ∈ IR × IRⁿ⁻¹| kx₂k ≤ x₁}.

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, 6, 7])

f^soc(x) = f (λ₁(x))u⁽¹⁾_x + f (λ₂(x))u⁽²⁾_x ∀x = (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ (1) play an essential role on both theory and algorithm aspects. In expression (1), f : J → IR with J ⊆ IR is a real-valued function and x is decomposed 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

λ_i(x) = x₁+ (−1)ⁱkx₂k and u⁽ⁱ⁾_x = 1 2

 1

(−1)ⁱx¯₂



(3)

for i = 1, 2 with ¯x2 := x2/kx2k if x2 6= 0, and ¯x2 being any vector in IRⁿ⁻¹ satisfying k¯x₂k = 1 if x₂ = 0. The decomposition (2) is called the spectral factorization associated with second-order cone for x. Likewise, there is a similar decomposition for x associ- ated with circular cone case. More specifically, from [31, Theorem 3.1], the spectral factorization associated with Lθ for x is in form of

x = λ₁(x) · u⁽¹⁾_x + λ₂(x) · u⁽²⁾_x (4) where

 λ₁(x) := x1− kx2kctanθ

λ₂(x) := x₁+ kx₂k tan θ (5)

and











u⁽¹⁾x := 1 1 + ctan²θ

1 0

0 ctanθ · I

  1

−¯x₂



=

 sin²θ

−(sin θ cos θ)¯x₂



u⁽²⁾x := 1 1 + tan²θ

1 0

0 tan θ · I

  1

¯ x₂



=

 cos²θ (sin θ cos θ)¯x₂

 (6)

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

f^Lθ(x) := f (λ₁(x)) u⁽¹⁾_x + f (λ₂(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^Lθ(x) =







f (x₁− kx₂kctanθ)

1 + ctan²θ +f (x₁ + kx₂k tan θ) 1 + tan²θ



−f (x1− kx2kctanθ)ctanθ

1 + ctan²θ + f (x1+ kx2k tan θ) tan θ 1 + tan²θ



¯ x₂





. (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, t⁰) with t, t⁰ ∈ IR ∪ {±∞}. Denote S the set of all x ∈ IRⁿ whose spectral values λ_i(x) for i = 1, 2 belong to J , i.e.,

S := {x ∈ IRⁿ| 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{λ1(x), λ1(y)} ≤ λ1(βx + (1 − β)y) ≤ λ2(βx + (1 − β)y) ≤ max{λ2(x), λ2(y)}.

We point out that there is a close relation between L_θ and Kⁿ (see [31]) as below Kⁿ = ALθ where A :=tan θ 0

0 I

 .

It is well-known that Kⁿ is a self-dual cone in the standard inner product hx, yi = Pn

i=1x_iy_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 Lθ is self-dual. More precisely, we define an inner product associated with A as hx, yi_A:= hAx, Ayi. Then

L^∗_θ = {x | hx, yi_A ≥ 0, ∀y ∈ L_θ} = {x | hAx, Ayi ≥ 0, ∀y ∈ A⁻¹Kⁿ}

= {x | hAx, yi ≥ 0, ∀y ∈ Kⁿ} = {x | Ax ∈ Kⁿ}

= A⁻¹Kⁿ= L_θ.

However, under this new inner product the second-order cone is not self-dual, because (Kⁿ)^∗ = {x | hx, yi_A≥ 0, ∀y ∈ Kⁿ} = {x | hAx, Ayi ≥ 0, ∀y ∈ Kⁿ}

= {x | hA²x, yi ≥ 0, ∀y ∈ Kⁿ} = {x | A²x ∈ Kⁿ} = A⁻²Kⁿ.

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^Lθ. It should be emphasized that the relation Kⁿ = AL_θ does not guarantee that there exists a similar close relation between f^Lθ and f^soc. For example, take f (t) to be a simple function max{t, 0}, which corresponds to the projection operator Π. For x ∈ L_θ, we have Ax ∈ Kⁿ which implies

ΠL_θ(x) = x = A⁻¹(Ax) = A⁻¹ΠKⁿ(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

ΠL_θ(x) = 0 and A⁻¹ΠKⁿ(Ax) =

₃

23 8



which says ΠL_θ(x) 6= A⁻¹ΠKⁿ(Ax). This example undoubtedly indicates that we cannot study f^Lθ by simply resorting to f^soc. Hence, it is necessary to study f^Lθ 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 cru- cial: (i) spectral factorization associated with circular cones; (ii) smooth and nonsmooth analysis for f^Lθ given as in (7); (iii) the so-called L_θ-convexity; and (iv) L_θ-monotonicity.

The first three points have been studied in [31], [4, 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^Lθ only depends on f .

To end this section, we say a few words about the notations and present the definitions of monotonicity and Lθ-monotonicity. A matrix M ∈ IR^n×n is said to be Lθ-invariant if M h ∈ L_θ for all h ∈ L_θ. We write x L_θ y to mean x − y ∈ L_θ and denote L^◦_θ the polar cone of L_θ, i.e.,

L^◦_θ := {y ∈ IRⁿ| hx, yi ≤ 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 eigenvalues, the minimum, and the maximum of eigenvalues of M , respectively. Besides, Sⁿ means the space of all symmetric matrices in IR^n×n and Sⁿ+ is the cone of positive semidefinite matrices. For a mapping g : IRⁿ → IR^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 Lθ-monotone on J if for any x, y ∈ S, x L_θ y =⇒ f^Lθ(x) L_θ f^Lθ(y);

(b) f^Lθ is said to be monotone on S if

f^Lθ(x) − f^Lθ(y), x − y ≥ 0, ∀x, y ∈ S.

(c) f^Lθ is said to be strictly monotone on S if

f^Lθ(x) − f^Lθ(y), x − y > 0, ∀x, y ∈ S, x 6= y.

(d) f^Lθ is said to be strongly monotone on S with µ > 0 if

f^Lθ(x) − f^Lθ(y), x − y ≥ µkx − yk², ∀x, y ∈ S.

2 Circular cone monotonicity of f

This section is devoted to the study of Lθ-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^TAy > 0 for some y. Then, the implication [z^TAz ≥ 0 =⇒ z^TBz ≥ 0] is valid if and only if B _Sⁿ₊ λA for some λ ≥ 0.

Proof. This is the well known S-Lemma, see [7, Lemma 3.1] or [25]. 2

Lemma 2.2. Given ζ ∈ IR, u ∈ IRⁿ⁻¹, and a symmetric matrix Ξ ∈ IR^n×n. Denote B := {z ∈ IRⁿ⁻¹|kzk ≤ 1}. Then, the following statements hold.

(a) Ξ being L_θ-invariant is equivalent to Ξctanθ z



∈ L_θ for any z ∈ B.

(b) If Ξ =ζ u^T

u H



with H ∈ Sⁿ⁻¹, then Ξ is L_θ-invariant is equivalent to

ζ ≥ kuk tan θ and there exists λ ≥ 0 such that

 ζ²− ctan²θkuk²− λ (ζ tan θ)u^T − ctanθu^TH (ζ tan θ)u − ctanθHu tan²θuu^T − H²+ λI



_Sⁿ

+ O.

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

ΞL_θ ∈ L_θ ⇐⇒ ΞA⁻¹Kⁿ∈ A⁻¹Kⁿ ⇐⇒ AΞA⁻¹Kⁿ ∈ Kⁿ

⇐⇒ AΞA⁻¹1 z



∈ Kⁿ⇐⇒ ΞA⁻¹1 z



∈ A⁻¹Kⁿ (9)

⇐⇒ ΞA⁻¹1 z



∈ L_θ ⇐⇒ Ξctanθ z



∈ L_θ,

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

(b) From (9), we know that

A Ξ A⁻¹1 z



=

 ζ tan θu^T ctanθu H

 1 z



= ζ + tan θu^Tz ctanθu + Hz



∈ Kⁿ, which means

ζ + u^Tz tan θ ≥ 0, ∀z ∈ B, (10)

and

ζ + u^Tz tan θ ≥ kctanθu + Hzk, ∀z ∈ B. (11) Note that condition (10) is equivalent to

ζ ≥ tan θ max{−u^Tz|z ∈ B} = tan θkuk and condition (11) is equivalent to

ζ + tan θu^Tz
2

≥ kctanθu + Hzk², i.e.,

z^T(tan²θuu^T − H²)z + 2 ζ tan θu^T − ctanθu^TH
z + ζ²− ctan²θu^Tu ≥ 0 ∀z ∈ B,

which can be rewritten as

1 z^T Θ1 z



≥ 0 ∀z ∈ B, (12)

with

Θ :=

 ζ²− ctan²θu^Tu (ζ tan θ)u^T − ctanθu^TH (ζ tan θ)u − ctanθHu tan²θuu^T − H²

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

k v^T1 0 0 −I

 k v



≥ 0 =⇒ k v^T Θk v



≥ 0, ∀k v



∈ IRⁿ. (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 6= 0, dividing by k² in the left side of (13) yields h

1 (v

k)^Ti1 0 0 −I

" 1 v k

#

≥ 0,

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

1 (v k)^Ti

Θ" 1 v k

#

≥ 0.

Hence, the right side of (13) holds.

For k = 0, the left side of (13) is kvk ≤ 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 Θ ensures the existence of λ ≥ 0 such that

 ζ²− ctan²θu^Tu ζ tan θu^T − ctanθu^TH ζ tan θu − ctanθHu tan²θuu^T − H²



− λ1 0 0 −I



_Sⁿ

+ O.

Thus, the proof is complete. 2

Lemma 2.3. For a matrix being in form of H := x₁ x^T₂ x₂ αI + β ¯x₂x¯^T₂



, where α, β ∈ IR, then

max{x₁+ kx₂k, x₁− β} + max{0, α − x₁+ β}

≥ λ_max(H) ≥ λ_min(H)

≥ min{x₁− kx₂k, x₁− β} + min{0, α − x₁+ β}.

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

x₁ x^T₂ x₂ αI + β ¯x₂x¯^T₂



=x₁ x^T₂ x₂ x₁I



− β0 0

0 I − ¯x₂x¯^T₂



+0 0

0 (α − x₁ + β)I



and let

Ω₁ :=x₁ x^T₂ x₂ x₁I



− β0 0

0 I − ¯x₂x¯^T₂



, Ω₂ :=0 0

0 (α − x₁+ β)I

 .

Then, λ(Ω₁) = {x₁−kx₂k, x₁+kx₂k, x₁−β} by [6, Lemma 1] and λ(Ω₂) = {0, α−x₁+β}.

Thus, the desired result follows from the following facts:

λmin(Ω1 + Ω2) ≥ λmin(Ω1) + λmin(Ω2) and λmax(Ω1+ Ω2) ≤ λmax(Ω1) + λmax(Ω2).

This completes the proof. 2

Next, we turn our attention to the vector-valued function f^Lθ defined as in (7). Recall from [4, 32] that f^Lθ is differentiable at x if and only if f is differentiable at λ_i(x) for i = 1, 2 and

∇f^Lθ(x) =







f⁰(x₁)I x₂ = 0;

 ξ %¯x^T₂

%¯x₂ τ I + (η − τ )¯x₂x¯^T₂



x₂ 6= 0, (14)

where

τ := f (λ₂(x)) − f (λ₁(x))

λ₂(x) − λ₁(x) , ξ := f⁰(λ₁(x))

1 + ctan²θ + f⁰(λ₂(x)) 1 + tan²θ,

% := − ctanθ

1 + ctan²θf⁰(λ₁(x)) + tan θ

1 + tan²θf⁰(λ₂(x)), η := ctan²θ

1 + ctan²θf⁰(λ₁(x)) + tan²θ

1 + tan²θf⁰(λ₂(x)).

The following result shows that if f^Lθ is differentiable, then we can characterize the L_θ-monotonicity of f via the gradient ∇f^Lθ.

Theorem 2.1. Suppose that f : J → IR is differentiable. Then, f is L_θ-monotone on J if and only if ∇f^Lθ(x) is L_θ-invariant for all x ∈ S.

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

f^Lθ(x + th) − f^Lθ(x)

t L_θ 0. (15)

Since L_θis closed, taking the limit as t → 0⁺ yields ∇f^Lθ(x)h L_θ 0, i.e., ∇f^Lθ(x)h ∈ L_θ.

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

For any ζ ∈ L^◦_θ, we have

ζ, f^Lθ(x) − f^Lθ(y) = Z 1

0

ζ, ∇f^Lθ x + t(x − y)
(x − y) dt ≤ 0, (16) where the last step comes from ∇f^Lθ(x + t(x − y))(x − y) ∈ L_θ because x + t(x − y) ∈ S (since S is convex) and ∇f^Lθ is Lθ-invariant over S by hypothesis. Since ζ ∈ L^◦_θ is arbitrary, (16) implies f^Lθ(x) − f^Lθ(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^Lθ(x) L_θ f^Lθ(y). 2

Note that f is Lipschitz continuous on J if and only if f^Lθ 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 is L_θ-monotone on J ;

(b) ∂_Bf^Lθ(x) is L_θ-invariant for all x ∈ S;

(c) ∂f^Lθ(x) is L_θ-invariant for all x ∈ S.

Proof. “(a) ⇒ (b)” Take V ∈ ∂_Bf^Lθ(x), then by definition of B-subdifferential there exists {x_k} ⊂ D_fLθ such that x_k → x and ∇f^Lθ(x_i) → V . According to (15), we obtain

∇f^Lθ(x_i)h L_θ 0 for h ∈ L_θ. Taking the limit yields V h L_θ 0. Since V ∈ ∂_Bf^Lθ(x) is arbitrary, this says that ∂_Bf^Lθ is L_θ-invariant.

“(b) ⇒ (c)” Take V ∈ ∂f^Lθ(x), then by definition, there exists V_i ∈ ∂_Bf^Lθ(x) and β_i ∈ [0, 1] such that V = P

iβ_iV_i and P

iβ_i = 1. Thus, for any h ∈ L_θ, we have V h = P

iβiVih ∈ Lθ, since Vi is Lθ-invariant and Lθ is convex. Hence ∂f^Lθ(x) is Lθ- invariant.

“(c) ⇒ (a)” The proof follows from Theorem 2.1 by replacing (16) with ζ, f^Lθ(x) − f^Lθ(y) = hζ, V (x − y)i ≤ 0,

for some V ∈ ∂f^Lθ(z) with z ∈ [x, y] by the mean-value theorem of Lipschitz functions [10]. 2

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 t₁, t₂ ∈ J with t₁ ≤ t₂, (tan θ − ctanθ)

f⁰(t₁) − f⁰(t₂)

≥ 0, (17)

and







f⁰(t1) f (t₂) − f (t₁) t₂− t₁ f (t₂) − f (t₁)

t₂− t₁ f⁰(t₂)





_S²

+ O, (18)

then f is L_θ-monotone on J .

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

Case 1: For x₂ = 0, in this case it is clear that ∇f^Lθ(x) being L_θ-invariant, i.e.,

∇f^Lθ(x)h = f⁰(x1)h ∈ Lθ for all h ∈ Lθ, is equivalent to saying f⁰(x1) ≥ 0.

Case 2: For x₂ 6= 0, let

H := τ I + (η − τ )¯x2x¯^T₂.

Then, applying Lemma 2.2 to the formula of ∇f^Lθ(x) in (14), ∇f^Lθ(x) is L_θ-invariant if and only if

ξ ≥ k%k tan θ (19)

and there exists λ ≥ 0 such that Υ :=

 ξ²− ctan²θ%²− λ ξ tan θ%¯x^T₂ − ctanθ%¯x^T₂H ξ tan θ%¯x₂− ctanθ%H ¯x₂ tan²θ%²x¯₂x¯^T₂ − H²+ λI



_Sⁿ

+ 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 θf⁰(λ₁(x)) − ctanθf⁰(λ₂(x)) ≤ − tan θf⁰(λ₁(x)) + tan θf⁰(λ₂(x))

≤ tan θf⁰(λ₁(x)) + ctanθf⁰(λ₂(x))

⇐⇒ f⁰(λ₂(x)) ≥ 0 and f⁰(λ₁(x)) ≥ 1 − ctan²θ

2 f⁰(λ₂(x)). (21)

This is ensured by (17) and (18). In fact, if tan θ ≥ ctanθ, then we know from (17) that f⁰(λ₁(x)) ≥ f⁰(λ₂(x)) ≥ (1 − ctan²θ)/2
f⁰(λ₂(x)) where the second inequality is due to f⁰(λ2(x)) ≥ 0 by (18). If tan θ ≤ ctanθ, then 1 − ctan²θ ≤ 0, and hence f⁰(λ₁(x)) ≥ 0 ≥ (1 − ctan²θ)/2
f⁰(λ₂(x)) since f⁰(λ_i(x)) ≥ 0 for i = 1, 2 by (18).

Now let us look into the entries of Υ. In the Υ₁₁-entry, we calculate ξ²− ctan²θ%²

= 1

(tan θ + ctanθ)²



(tan²θ − ctan²θ)f⁰(λ₁(x))²+ 2(1 + ctan²θ)f⁰(λ₁(x))f⁰(λ₂(x))



= 1

(tan θ + ctanθ)²



(tan²θ − ctan²θ)f⁰(λ₁(x))²+ (ctan²θ − tan²θ)f⁰(λ₁(x))f⁰(λ₂(x))



+ 1

(tan θ + ctanθ)² h

2 + tan²θ + ctan²θ i

f⁰(λ1(x))f⁰(λ2(x))

= µ + f⁰(λ₁(x))f⁰(λ₂(x)), with

µ

:= 1

(tan θ + ctanθ)²



(tan²θ − ctan²θ)f⁰(λ1(x))²+ (ctan²θ − tan²θ)f⁰(λ1(x))f⁰(λ2(x))



= tan θ − ctanθ

tan θ + ctanθf⁰(λ₁(x))h

f⁰(λ₁(x)) − f⁰(λ₂(x))i

≥ 0,

where the last step is due to (17). In the Υ₁₂-entry and Υ₂₁-entry, we calculate (ξ tan θ)%¯x^T₂ − ctanθ%¯x^T₂H

= 1

(tan θ + ctanθ)²



− tan²θ + ctan²θ
f⁰(λ1(x))²+ tan²θ − ctan²θ
f⁰(λ1(x))f⁰(λ2(x))



¯ x^T₂

= −tan θ − ctanθ

tan θ + ctanθf⁰(λ₁(x))h

f⁰(λ₁(x)) − f⁰(λ₂(x))i

¯ x^T₂

= −µ¯x^T₂.

In the Υ₂₂-entry, we calculate

tan²θ%²x¯₂x¯^T₂ − H² = −τ²I + τ² + 1

(tan θ + ctanθ)² h

(tan²θ − ctan²θ)f⁰(λ₁(x))²

−2(1 + tan²θ)f⁰(λ1(x))f⁰(λ2(x)) i

!

¯ x2x¯^T₂

= −τ²I +



τ²+ µ − f⁰(λ1(x))f⁰(λ2(x))



¯ x2x¯^T₂. Hence, Υ can be rewritten as

Υ = Υ₁₁ Υ₁₂ Υ₂₁ Υ₂₂



= "µ + f⁰(λ₁(x))f⁰(λ₂(x)) − λ −µ¯x^T₂

−µ¯x₂ (λ − τ²)I +

τ²+ µ − f⁰(λ₁(x))f⁰(λ₂(x))

¯ x₂x¯^T₂

# .

Now, applying Lemma 2.3 to Υ, we have

λ_min(Υ) ≥ min (

Υ₁₁− |µ|, Υ₁₁−

τ²+ µ − f⁰(λ₁(x))f⁰(λ₂(x)) )

+ min (

0, 

λ − τ²

− Υ₁₁+

τ²+ µ − f⁰(λ₁(x))f⁰(λ₂(x)) )

= minn

f⁰(λ₁(x))f⁰(λ₂(x)) − λ, 2f⁰(λ₁(x))f⁰(λ₂(x)) − τ²− λo +2 min

n

0, λ − f⁰(λ1(x))f⁰(λ2(x)) o

. (22)

Using λ₁(x) ≤ λ₂(x) and condition (18) ensures

f⁰(λ₁(x)) ≥ 0, f⁰(λ₂(x)) ≥ 0, and f⁰(λ₁(x))f⁰(λ₂(x)) ≥  f (λ₂(x)) − f (λ1(x)) λ₂(x) − λ₁(x)

2

, which together with (15) yields

f⁰(λ₁(x))f⁰(λ₂(x)) ≥ 0 and f⁰(λ₁(x))f⁰(λ₂(x)) − τ² ≥ 0.

Thus, we can plug λ := f⁰(λ₁(x))f⁰(λ₂(x)) ≥ 0 into (22), which gives λ_min(Υ) ≥ 0. Hence Υ is positive semi-definite. This completes the proof. 2

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 is L_θ- monotone on J if and only if f⁰(t) ≥ 0 for all t ∈ J and (tan θ−ctanθ)(f⁰(t₁)−f⁰(t₂)) ≥ 0 for all t₁, t₂ ∈ J with t₁ ≤ t₂.

Proof. In light of the proof of Theorem 2.3, we know that f is L_θ-monotone if and only if for any x ∈ S,

f⁰(λ₂(x)) ≥ 0, f⁰(λ₁(x)) ≥ 1 − ctan²θ

2 f⁰(λ₂(x)), (23)

and there exists λ ≥ 0 such that

Υ =µ + f⁰(λ1(x))f⁰(λ2(x)) − λ ±µ

±µ µ − f⁰(λ₁(x))f⁰(λ₂(x)) + λ



_S²

+ O, (24)

where the form of Υ comes from the fact that ¯x₂ = ±1 in this case. It follows from (24) that µ²− (f⁰(λ₁(x))f⁰(λ₂(x)) − λ)²− µ² ≥ 0, which implies λ = f⁰(λ₁(x))f⁰(λ₂(x)).

Substituting it into (24) yields

Υ = µ ±µ

±µ µ



= µ 1 ±1

±1 1



_S²

+ O,

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







f⁰(λ₂(x)) ≥ 0, f⁰(λ₁(x)) ≥ 1 − ctan²θ

2 f⁰(λ₂(x)), (tan θ − ctanθ)f⁰(λ₁(x))f⁰(λ₁(x)) − f⁰(λ₂(x)) ≥ 0.

Due to the arbitrariness of λ_i(x) ∈ J and applying similar arguments following (21), the above conditions give f⁰(t) ≥ 0 for all t ∈ J and (tan θ − ctanθ)(f⁰(t1) − f⁰(t2)) ≥ 0 for all t₁, t₂ ∈ J with t₁ ≤ t₂. Thus, the proof is complete. 2

3 Monotonicity of f

In Section 2, we have shown that the circular cone monotonicity f^Lθ depends on both the monotonicity of f and the range of the angle θ. Now the following questions arise: how about on the relationship of monotonicity of f^Lθ and the L_θ-monotonicity of f ? Whether the monotonicity of f^Lθ also depends on θ? This is the main motivation of this section.

First, for a mapping H : IRⁿ → IRⁿ, let us denote ∂H(x) _Sⁿ₊ O (or ∂H(x) _Sⁿ₊ O) to mean that each elements in ∂H(x) is positive semi-definite (or positive definite), i.e.,

∂H(x) _Sⁿ

+ O (or _Sⁿ

+ O) ⇐⇒ A _Sⁿ

+ O (or _Sⁿ

+ 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^Lθ is monotone on S if and only if ∂f^Lθ(x) _Sⁿ

+ O for all x ∈ S;

(b) If ∂f^Lθ(x) _Sⁿ

+ O for all x ∈ S, then f^Lθ is strictly monotone on S;

(c) f^Lθ is strongly monotone on S if and only if there exists µ > 0 such that ∂f^Lθ(x) _Sⁿ

+

µI for all x ∈ S.

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

∂_Bf^Lθ(x) _Sⁿ

+ O ⇐⇒ ∂f^Lθ(x) _Sⁿ

+ O and ∂_Bf^Lθ(x) _Sⁿ

+ O ⇐⇒ ∂f^Lθ(x) _Sⁿ

+ O.

Proof. The result follows immediately from the fact ∂f^Lθ(x) = conv∂_Bf^Lθ(x). 2 Lemma 3.3. The following statements hold.

(a) For any h ∈ IRⁿ t₁

h₁− ctanθ¯x^T₂h₂2

+ t₂

h₁+ tan θ¯x^T₂h₂2

+ t₃

kh₂k² − (¯x^T₂h₂)²

≥ 0 (25) if and only if t_i ≥ 0 for i = 1, 2, 3.

(b) For any h ∈ IRⁿ/{0}

t₁

h₁− ctanθ¯x^T₂h₂2

+ t₂

h₁+ tan θ¯x^T₂h₂2

+ t₃

kh₂k²− (¯x^T₂h₂)²

> 0 (26) if and only if t_i > 0 for i = 1, 2, 3.

Proof. (a) The sufficiency is clear, since |¯x^T₂h₂| ≤ k¯x₂kkh₂k = kh₂k by Cauchy-Schwartz inequality. Now let us show the necessity. Taking h = (1, −ctanθ ¯x₂)^T, then (25) equal to t1(1 + ctan²θ)² ≥ 0, which implies t1 ≥ 0. Similarly, let h = (1, tan θ ¯x2)^T, then (25) yields t₂(1 + tan²θ)² ≥ 0, implying t₂ ≥ 0. Finally, let h = (0, u)^T with u satisfying hu, ¯x₂i = 0 and kuk = 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. If kh₂k−¯x^T₂h₂ > 0, then the result holds since t₃ > 0. If kh₂k−¯x^T₂h₂ = 0, then h2 = β ¯x2. So the left side of (26) takes t1(h1 − βctanθ)² + t2(h1 + β tan θ)² > 0, because h₁ − βctanθ = h₁+ β tan θ = 0 only happened when β = 0 and h₁ = 0. This means h₂ = 0 since h₂ = β ¯x₂, so h = 0. 2

If f is differentiable, it is known that f^Lθ is differentiable [4, 32]. It then follows from Lemma 3.1 that f^Lθ is monotone on S if and only if ∇f^Lθ(x) _Sⁿ

+ O for all x ∈ S. Hence, to characterize the monotonicity of f^Lθ, the first thing is to estimate ∇f^Lθ(x) _Sⁿ₊ O via f .

Theorem 3.1. Given x ∈ IRⁿ and suppose that f is differentiable at λ_i(x) for i = 1, 2.

Then ∇f^Lθ(x) _Sⁿ

+ O if and only if f⁰(λ_i(x)) ≥ 0 for i = 1, 2 and f (λ₂(x)) ≥ f (λ₁(x)).

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

Case 1. For x₂ = 0, using ∇f^Lθ(x) = f⁰(x₁)e, it is clear that ∇f^Lθ(x) _Sⁿ

+ O is equivalent to saying f⁰(x₁) ≥ 0. Then, the desired result follows.

Case 2. For x2 6= 0, denote

b₁ := f⁰(λ₁(x))

1 + ctan²θ and b₂ = f⁰(λ₂(x)) 1 + tan²θ.

Then, ξ = b₁+ b₂, % = −b₁ctanθ + b₂tan θ, and η = b₁ctan²θ + b₂tan²θ. Hence, for all h = (h₁, h₂)^T ∈ IR × IRⁿ⁻¹, we have

hh, ∇f^Lθ(x)hi

= (b1+ b2)h²₁+ 2%h1x¯^T₂h2+ τ kh2k²+ (b1ctan²θ + b2tan²θ − τ )(¯x^T₂h2)²

= (b₁+ b₂)h²₁+ 2(−b₁ctanθ + b₂tan θ)h₁x¯^T₂h₂+ (b₁ctan²θ + b₂tan²θ)(¯x^T₂h₂)² +τ

kh₂k²− (¯x^T₂h₂)²

= b₁

h²₁− 2ctanθh₁x¯^T₂h₂+ ctan²θ(¯x^T₂h₂)² + b₂

h²₁+ 2 tan θh₁x¯^T₂h₂+ tan²θ(¯x^T₂h₂)²

+τ

kh₂k²− (¯x^T₂h₂)²

= b₁

h₁− ctanθ¯x^T₂h₂2

+ b₂

h₁+ tan θ¯x^T₂h₂2

+ τ

kh₂k² − (¯x^T₂h₂)² . In light of Lemma 3.3, the desired result is equivalent to

b₁ ≥ 0, b₂ ≥ 0, τ = f (λ₂(x)) − f (λ₁(x)) λ₂(x) − λ₁(x) ≥ 0,

i.e., f⁰(λ₁(x)) ≥ 0, f⁰(λ₂(x)) ≥ 0, and f (λ₂(x)) ≥ f (λ₁(x)) due to λ₂(x) > λ₁(x) in this case. 2

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

Corollary 3.1. Given x ∈ IRⁿ and suppose that f is differentiable at λ_i(x) for i = 1, 2.

Then for x₂ 6= 0, ∇f^Lθ(x) _Sⁿ

+ O if and only if f⁰(λ_i(x)) > 0 for i = 1, 2 and f (λ₂(x)) >

f (λ₁(x)); for x₂ = 0, ∇f^Lθ(x) _Sⁿ₊ O if and only if f⁰(x₁) > 0.

When f is non-differentiable, we resort to the subdifferential ∂B(f^Lθ), whose estimate is given in [32].

Lemma 3.4. Let f : IR → IR be strictly continuous. Then, for any x ∈ IRⁿ, the B- differential ∂_B(f^Lθ)(x) is well defined and nonempty. Moreover,

(i) if x₂ 6= 0, then

∂B(f^Lθ)(x) =















 ξ %x^T₂/kx₂k

%x₂/kx₂k τ I + (η − τ )x₂x^T₂/kx₂k²

          

τ = f λ2(x)
− f λ1(x)
λ₂(x) − λ₁(x) ξ − %ctanθ ∈ ∂Bf (λ1(x)) ξ + % tan θ ∈ ∂_Bf (λ₂(x)) η = ξ − %(ctanθ − tan θ)















;

(27) (ii) if x₂ = 0, then

∂_B(f^Lθ)(x) ⊂











 ξ %w^T

%w τ I + (η − τ )ww^T

        

τ ∈ ∂f (λ₁(x)), kwk = 1 ξ − %ctanθ ∈ ∂_Bf (λ₁(x)) ξ + % tan θ ∈ ∂_Bf (λ₁(x)) η = ξ − %(ctanθ − tan θ)











. (28)

Lemma 3.4 presents an upper estimation on ∂_Bf^Lθ(x) when x₂ = 0. Here we give an lower estimation.

Lemma 3.5. Suppose that f is locally Lipschitz at x with x₂ = 0. Then,

∂_Bf (x₁)I ⊆ ∂_Bf^Lθ(x).

Proof. First, take u ∈ ∂_Bf (x₁). Since f is locally Lipschitz at x₁, there exists t ∈ D_f satisfying t → x₁. Note that λ_i(te) = t for i = 1, 2. Then, f^Lθ is differentiable at te, i.e., te ∈ D_fLθ and ∇f^Lθ(te) = f⁰(t)I. Hence,

uI = lim

t→x1

f⁰(t)I = lim

te→x∇f^Lθ(te) ∈ ∂_Bf^Lθ(x).

Since u is an arbitrary element in ∂Bf (x1), the desired result follows. 2

Using Lemmas 3.4 and 3.5, the nonsmooth version of Theorem 3.1 is given below.

Theorem 3.2. Given x ∈ IRⁿ, then ∂_Bf^Lθ(x) _Sⁿ

+ O if and only if ∂_Bf (λ_i(x)) ≥ 0 for i = 1, 2, and f (λ₂(x)) ≥ f (λ₁(x)).

Proof. Consider the following two cases.

Case 1. For x₂ 6= 0, according to (27), we know for V ∈ ∂_Bf^Lθ(x), there exists v_i ∈

∂_Bf (λ_i(x)) for i = 1, 2 such that ξ − ρctanθ = v₁, ξ + ρ tan θ = v₂, and V =

 ξ %¯x^T₂

%¯x2 τ I + (η − τ )¯x2x¯^T₂

 . Hence,

ξ = v₁tan θ + v₂ctanθ

tan θ + ctanθ , ρ = v₂− v₁

tan θ + ctanθ and η = v₁ctanθ + v₂tan θ tan θ + ctanθ and

hh, V hi

= v₁tan θ + v₂ctanθ

tan θ + ctanθ h²₁ + 2 v₂− v₁

tan θ + ctanθx¯^T₂h₂h₁+ v₁ctanθ + v₂tan θ

tan θ + ctanθ (¯x^T₂h₂)² +τh

kh₂k²− (¯x^T₂h₂)²i

= tan θ

tan θ + ctanθv₁h

h₁− (¯x^T₂h₂)ctanθi2

+ ctanθ

tan θ + ctanθv₂h

h₁+ (¯x^T₂h₂) tan θi2

+τh

kh₂k²− (¯x^T₂h₂)²i

. (29)

Applying Lemma 3.3, we have hh, V hi ≥ 0 for all h ∈ IRⁿ if and only if v1, v2 ≥ 0, and τ ≥ 0. In other words, ∂_Bf^Lθ(x) _Sⁿ

+ O if and only if ∂_Bf (λ_i(x)) ≥ 0 for i = 1, 2 and f (λ₂(x)) ≥ f (λ₁(x)).

Case 2. For x2 = 0, the sufficiency follows by along the same arguments as above. In fact, the every element of the left set (28) is positive semidefinite. Then, the necessity follows from Lemma 3.5. 2

Likewise, we have a nonsmooth version of Corollary 3.1 which is given below.

Corollary 3.2. Given x ∈ IRⁿ, then for x₂ 6= 0, ∂_Bf^Lθ(x)  0 if and only if ∂_Bf (λ_i(x)) >

0 for i = 1, 2, and f (λ2(x)) > f (λ1(x)); for x2 = 0, ∂Bf^Lθ(x)  0 if and only if

∂_Bf (x₁) > 0.

One point needs to be mentioned here. Because we cannot characterize the strict monotonicity by subgradients (see Lemma 3.1), we resort to the definition of monotonic- ity. Indeed, inspired by [18], we obtain the following result.

Theorem 3.3. Suppose that f is locally Lipschitz continuous on J . Then, (a) f^Lθ is monotone on S if and only if f is nondecreasing on J ;

(b) f^Lθ is strictly monotone on S if and only if f is strictly increasing on J ;

(c) f^Lθ is strongly monotone on S with µ > 0 if and only if f is strongly increasing on S with µ > 0.

Proof. (a) =⇒. Take t₁, t₂ ∈ J with t₁ ≤ t₂. Since f^Lθ is monotone, we have 0 ≤ h(t1− t2)e, f^Lθ(t1e) − f^Lθ(t2e)i = (t1 − t2)(f (t1) − f (t2)), which implies f (t₁) ≤ f (t₂).

⇐=. Take x ∈ S. Since f is monotone on J, we know ∂_Bf (t) ≥ 0 for all t ∈ J , and hence ∂f (t) ≥ 0 for t ∈ J . Because λ_i(x) ∈ J for i = 1, 2, it implies ∂_Bf (λ_i(x)) ≥ 0 for i = 1, 2. In addition, f (λ₂(x)) − f (λ₁(x)) = γ(λ₂(x) − λ₁(x)) for some γ ∈ ∂f (t₀) and t₀ ∈ (λ₁(x), λ₂(x)) ⊆ J . Hence, f (λ₂(x)) ≥ f (λ₁(x)) due to γ ∈ ∂f (t₀) ≥ 0. Applying Theorem 3.2 yields ∂_Bf^Lθ(x) _Sⁿ

+ O, which means f^Lθ is monotone on S by Lemma 3.1.

(b) =⇒. Given t₁, t₂ ∈ J with t₁ > t₂. Since f^Lθ is strictly monotone on S, we have



f (t1) − f (t2)



(t1− t2) = hf^Lθ(t1e) − f^Lθ(t2e), (t1− t2)ei > 0,

which together with the fact t₁ > t₂ yields f (t₁) > f (t₂). This says f is strictly increasing on J .

⇐=. Let x, y ∈ S with x 6= y. We shall show the strict monotonicity of f^Lθ by checking definition. To proceed, we consider the following four cases.

Case 1. For x₂ = y₂ = 0, it is clear that hf^Lθ(x) − f^Lθ(y), x − yi =

f (x₁) − f (y₁)

(x₁− y₁) > 0,

where the last step is due to the fact that f is strictly monotone and x1 6= y1, since (x₁, 0) = x 6= y = (y₁, 0) under this case.

Case 2. For x₂ = 0 and y₂ 6= 0, we have hf^Lθ(y) − f^Lθ(x), y − xi