1Introduction Propertiesofcircularconeandspectralfactorizationassociatedwithcircularcone

to appear in Journal of Nonlinear and Convex Analysis, 2013

Properties of circular cone and spectral factorization associated with circular cone

Jinchuan Zhou ¹ Department of Mathematics

School of Science

Shandong University of Technology Zibo 255049, P.R. China E-mail: [email protected]

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

Taipei 11677, Taiwan E-mail: [email protected]

October 11, 2012

Abstract. Circular cone includes second-order cone as a special case when the rotation angle is 45 degree. This paper gives an insight on circular cone, in which we describe the tangent cone, normal cone, second order tangent cone, and second order regularity of circular cone. Moreover, we establish the spectral factorization associated with circular cone. These results are crucial to subsequent study regarding various analysis towards optimizations associated with circular cone.

Keywords. Circular cone, second order regular, spectral factorization.

1 Introduction

The circular cone [9] 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) and Shandong Province Natural Science Foundation (ZR2010AQ026).

2Corresponding author. Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is supported by National Science Council of Taiwan.

half-aperture angle be θ where θ ∈ (0,^π₂). Then, we denote the n-dimensional circular cone by L_θ which is expressed as

L_θ :=x = (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ | cos θkxk ≤ x₁ . (1) Circular cone includes second-order cone (SOC), given by

Kⁿ := {x = (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ | kx₂k ≤ x₁}, (2) as a special case when the rotation angle is 45 degree. This can be verified by

Kⁿ = (x1, x₂)^T ∈ IR × IRⁿ⁻¹ | kx₂k ≤ x₁

= (x1, x2)^T ∈ IR × IRⁿ⁻¹ | x1 ≥ 0, 2kx2k² ≤ 2x²₁

= (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ | x₁ ≥ 0, 2x²₁+ 2kx₂k² ≤ 4x²₁

=



(x₁, x₂)^T ∈ IR × IRⁿ⁻¹  

q

2x²₁ + 2kx₂k² ≤ 2x₁



= (

(x₁, x₂)^T ∈ IR × IRⁿ⁻¹    

√2 2

q

x²₁+ kx₂k² ≤ x₁ )

= (

(x₁, x₂)^T ∈ IR × IRⁿ⁻¹    

√2

2 kxk ≤ x₁ )

= n

(x₁, x₂)^T ∈ IR × IRⁿ⁻¹   cosπ

4kxk ≤ x₁o .

Considerable attentions have been devoted to the second-order cone Kⁿ [5, 6, 7], a special case of self-dual cone. However, the study on the circular cone L_θ, a non-self-dual (or non-symmetric cone) is rather limited. In this paper, we show that there exists a close relationship between Kⁿand L_θ by establishing an inequality regrading distance between Kⁿ and L_θ. This nice property plays an essential role in our subsequence analysis and give us more information and insight on L_θ. In particular, we develop the formulae of tangent cone, normal cone, and second-order tangent cone of L_θ in terms of Kⁿ (the formula of the latter has been given by different scholars). Furthermore, we show that L_θ, as a non-self-dual and non-polytechnic cone, is also second-order regular. Note that we know the second-order cone and positive semi-definitive cone are both second order regular, but there are all symmetric. Thus this is an interesting case which indicates the second order regularity of a non-symmetric cone. Finally, we develop the spectral factorization of z in terms of L_θ by studying the projection on L_θ which will be useful in dealing with optimization associated circular cone.

In fact, it is not hard to see that

L_θ = (x1, x₂)^T ∈ IR × IRⁿ⁻¹ | x₁ ≥ 0, kxk²cos²θ ≤ x²₁

= (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ | x₁ ≥ 0, (x²₁+ kx₂k²) cos²θ ≤ x²₁

= (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ | x₁ ≥ 0, kx₂k² ≤ x²₁tan²θ

= (x₁, x₂)^T ∈ IR × IRⁿ⁻¹ | kx₂k ≤ x₁tan θ , which yields

x₁ x₂



∈ L_θ ⇐⇒ tan θx₁ x₂



∈ Kⁿ ⇐⇒  tan θ 0

0 1

 x₁ x₂



∈ Kⁿ. (3) For simplicity, let us denote

A := tan θ 0

0 1

 . Then, the above expression (3) is equivalent to

x₁ x₂



∈ L_θ ⇐⇒ Ax₁ x₂



∈ Kⁿ. (4)

We point out that the matrix A is positive definite whose inverse matrix is A⁻¹ = ctanθ 0

0 1



where ctanθ := 1 tan θ.

To close this section, we say a few words about the notations. For a convex cone K, its dual cone is defined by

(K)^∗ = {v | hv, xi ≥ 0, ∀x ∈ K} , while its polar cone is given by

(K)^◦ = {v | hv, xi ≤ 0, ∀x ∈ K} .

2 Insight on circular cone

In this section, we give an insight on circular cone in which we shall study some properties of Lθ, including characterizing its tangle cone, normal cone, second-order tangent cone, etc.. To this end, we first describe the relationship between Kⁿ and L_θ.

Theorem 2.1. Let L_θ and Kⁿ be defined as in (1) and (2), respectively. Then, we have

(a) L_θ = A⁻¹Kⁿ and Kⁿ = AL_θ. (b) AKⁿ= L^π

2−θ and L^π

2−θ = A²L_θ. (c) L^∗_θ = L^π

2−θ and (L^∗_θ)^∗ = L_θ.

Proof. (a) This follows from equivalence (4) because L_θ = {x | x ∈ L_θ}

= {x | Ax ∈ Kⁿ}

= x | x ∈ A⁻¹Kⁿ

= A⁻¹Kⁿ. (b) According to part(a), we have

L^π

2−θ =

"

ctan(π

2 − θ) 0

0 1

#

Kⁿ = tan θ 0

0 1



Kⁿ = AKⁿ = A(AL_θ) = A²L_θ

which is the desired result.

(c) It is known that Kⁿ is self-dual. Hence, we have

Kⁿ = (Kⁿ)^∗ = {v | hv, ki ≥ 0, ∀k ∈ Kⁿ}

= {v | hv, Azi ≥ 0, ∀z ∈ L_θ}

= {v | hAv, zi ≥ 0, ∀z ∈ L_θ}

= {v | Av ∈ L^∗_θ}

= A⁻¹L^∗_θ which implies L^∗_θ = AKⁿ = L^π

2−θ by part(b). The remaining part is true for all closed convex cone. 2

Theorem 2.2. For any x, z ∈ IRⁿ, we have

kAk⁻¹ dist(Az, Kⁿ) ≤ dist(z, L_θ) ≤ kA⁻¹k dist(Az, Kⁿ) (5) and

kA⁻¹k⁻¹ dist(A⁻¹x, L_θ) ≤ dist(x, Kⁿ) ≤ kAk dist(A⁻¹x, L_θ). (6) Proof. First, we observe the following:

dist(x, Kⁿ)

= min

k∈Kⁿkx − kk = min

k∈AL_θkx − kk = min

z∈Lθ

kx − Azk = min

z∈Lθ

kA(A⁻¹x) − Azk

= min

z∈Lθ

kA(A⁻¹x − z)k ≤ kAk min

z∈Lθ

kA⁻¹x − zk = kAk dist A⁻¹x, L_θ
, (7)

dist(z, L_θ)

= min

u∈Lθ

kz − uk = min

u∈A⁻¹Kⁿkz − uk = min

k∈Kⁿkz − A⁻¹kk = min

k∈KⁿkA⁻¹(Az) − A⁻¹kk

= min

k∈KⁿkA⁻¹(Az − k)k ≤ kA⁻¹k min

k∈KⁿkAz − kk = kA⁻¹k dist(Az, Kⁿ). (8) These prove the second inequality in (5) and (6), respectively. Next, plugging z = A⁻¹x and x = Az in (7) and (8), respectively, yields the first inequality in (5) and (6), respectively. Thus, the proof is complete. 2

Theorem 2.2 indicates that the distances of arbitrary points to Kⁿ and L_θ are equiv- alent. This is an essential property for analyzing the tangent cone and normal cone of Lθ. Before we move on, we recall the definitions of tangent cone and normal cone. Given a subset S ⊂ IRⁿ and x ∈ S, the contingent cone T_S(x) and inner tangent cone T_Sⁱ(x) of S at x are defined respectively as

T_S(x) := {d ∈ IRⁿ| ∃t_n ↓ 0, dist (x + t_nd, S) = o(t_n)}

and

T_Sⁱ(x) := {d ∈ IRⁿ| dist(x + th, S) = o(t), t ≥ 0)}.

In general, these two cones can be different. However, when S is convex, they are equal to each other and to the closure of the radial cone, see [4, page 45]. Hence for convex sets, we simply speak of tangent cone rather than contingent or inner tangent cones.

Moreover, the Fr´echet/regular normal cone (also known as the prenormal cone), written as bN_S(x), is defined as

Nb_S(x) := {v ∈ IRⁿ| hv, z − xi ≤ o(kz − xk), for z ∈ S},

and the Mordukhovich/limiting normal cone (or simply normal cone) is defined as N_S(x) := lim sup

z^−→_Sx

Nb_S(z).

When S is convex, N_S(x) = bN_S(x) and is the polar cone of T_S(x), i.e., N_S(x) := {v ∈ IRⁿ | hv, di ≤ 0, ∀d ∈ T_S(x)} .

Theorem 2.3. For any z ∈ L_θ, we have (a) TL_θ(z) = A⁻¹TKⁿ(Az),

(b) NL_θ(z) = ANKⁿ(Az).

Proof. (a) Let us first show that T_Lθ(z) ⊆ A⁻¹T_Kⁿ(Az). Choose d ∈ T_Lθ(z). Then, by definition of tangent cone, we have

dist(z + td, L_θ) = o(t). (9)

Plugging x = A(z + td) into (6) yields

kA⁻¹k⁻¹ dist(z + td, L_θ) ≤ dist (A(z + td), Kⁿ) ≤ kAk dist(z + td, L_θ).

This together with (9) implies dist(Az + tAd, Kⁿ) = o(t). Thus, Ad ∈ T_Kⁿ(Az), which says d ∈ A⁻¹TKⁿ(Az).

Conversely, let d ∈ A⁻¹TKⁿ(Az). Since Ad ∈ TKⁿ(Az), from definition of tangent cone, we know

dist(Az + tAd, Kⁿ) = o(t). (10)

Replacing z in (5) by z + td gives

kAk⁻¹ dist(Az + tAd, Kⁿ) ≤ dist(z + td, Lθ) ≤ kA⁻¹k dist(Az + tAd, Kⁿ).

This together with (10) implies dist(z + td, L_θ) = o(t), which says d ∈ TL_θ(z).

(b) The desired result follows from

NL_θ(z) = {v ∈ IRⁿ | hv, di ≤ 0, ∀d ∈ TL_θ(z)}

= v ∈ IRⁿ | hv, A⁻¹wi ≤ 0, ∀w ∈ TKⁿ(Az)

= v ∈ IRⁿ | hA⁻¹v, wi ≤ 0, ∀w ∈ T_Kⁿ(Az)

= v ∈ IRⁿ | A⁻¹v ∈ NKⁿ(Az)

= ANKⁿ(Az).

2

Theorem 2.3 tells us that the explicit formula of tangent cone TL_θ(z) can be estab- lished by TKⁿ(Az), which has been given in [3].

It is well known that in the study of second order analysis for optimization problems, we need the following inner and outer second order tangent sets to describe the possible curvature of the feasible region. Below, we state their official definitions.

Definition 2.1. [4, Definition 3.28] The set limits T_S^i,2(x, d) :=



w ∈ IRⁿ    

dist



x + td +1 2t²w, S



= o(t²), t ≥ 0



and

T_S²(x, d) =



w ∈ IRⁿ    

∃ tn↓ 0 such that dist



x + tnd +1 2t²_nw, S



= o(t²_n)



are called the inner and outer second order tangent sets, respectively, to the set S at x in the direction d.

Definition 2.2. [4, Definition 3.32] We say that the set S is second order directionally differentiable at a point x ∈ S in a direction d ∈ T_S(x), if T_Sⁱ(x) = T_S(x) and T_S^i,2(x, d) = T_S²(x, d). We simply say that S is second order directionally differentiable at a point x ∈ S if it is second order directionally differentiable in all directions d ∈ T_S(x).

Theorem 2.4. Let z ∈ L_θ and d ∈ T_Lθ(z). Then, T_L^i,2

θ(z, d) = T_L²

θ(z, d) = A⁻¹T_K²n(Az, Ad).

Proof. The first equality is due to the second order directionally differentiable of Kⁿ as shown in [10, Proposition 3.1] and the second equality can be proved by the same arguments as in Theorem 2.3. 2

Definition 2.3. [4, Definition 3.85] We say that a subset S ⊂ IRⁿ is second order regular at x if it satisfies

(i) T_S²(x, d) = T_S^i,2(x, d) for all d ∈ T_S(x);

(ii) for any d ∈ T_S(x) and for any sequence x + t_nd +¹₂t²_nr_n∈ S such that t_nr_n→ 0, the following condition holds:

n→∞lim dist r_n, T_S²(x, d)
= 0.

Theorem 2.5. The circular cone Lθ is second order regular.

Proof. Let z ∈ L_θ and d ∈ T_Lθ(z). According to Theorem 2.4, it suffices to show that for any sequence z + t_nd + ¹₂t²_nr_n∈ L_θ with t_nr_n→ 0, there holds

n→∞lim dist rn, T_L²_θ(z, d)
= 0. (11) We will complete the proof by using the relationship between L_θ and Kⁿ. Since z + t_nd +

1

2t²_nr_n∈ L_θ, we know Az + t_nAd +¹₂t²_nAr_n ∈ K by Theorem 2.1(a). Note that t_nAr_n→ 0

because kt_nAr_nk ≤ kAk · kt_nr_nk. In addition, Kⁿ is second order regular (see [10] for detailed proof), from Definition 2.3, we have

n→∞lim dist Arn, T_K²n(Az, Ad)
= 0. (12) On the other hand, we observe that

dist r_n, T_L²

θ(z, d)

= dist r_n, A⁻¹T_K²n(Az, Ad)

= dist A⁻¹(Ar_n), A⁻¹T_K²n(Az, Ad)

≤ kA⁻¹k dist Ar_n, T_K²n(Az, Ad)
. This together with (12) implies the validity of (11). 2

3 Spectral factorization associated with circular cone

In this section, we will develop the spectral factorization associated with circular cone which is the basis of further investigations for optimization associated with circular cone.

To this end, we start with studying the projection on L_θ, i.e., ΠL_θ(z) := arg min

x∈Lθ

kz − xk = {x ∈ L_θ | kz − xk ≤ kz − uk, ∀u ∈ L_θ} .

It should be mentioned that the projection cannot be obtained by using the relation- ship between L_θ and Kⁿ because

kA⁻¹xk ≤ kA⁻¹yk ; kxk ≤ kyk whenever θ 6= π/4.

For example, let x = (8, 1), y = (4, 2), and θ = cot⁻¹(1/8). Then, kA⁻¹xk =√

2 < √

17/2 = kA⁻¹yk, but kxk =√

65 >√

20 = kyk.

Therefore, we seek another way to characterize the projection. First, we note that for any closed convex cone Ω

Π−Ω(x) = −Π_Ω(−x).

In fact, letting a = Π_−Ω(x) yields

k(−x) − (−a)k = kx − ak ≤ kx − (−y)k = k(−x) − yk ∀y ∈ Ω,

where the inequality comes from the fact that a = Π−Ω(x) by definition of projection.

This means that −a = Π_Ω(−x). Besides, it is well known that any vector z ∈ IRⁿ can be written as

z = Π_Ω(z) + Π_Ω^◦(z).

Hence,

z = ΠL_θ(z) + ΠL^◦_θ(z) = ΠL_θ(z) + Π−L^∗_θ(z)

= Π_Lθ(z) − Π_L^∗

θ(−z) = Π_Lθ(z) − Π_Lπ

2−θ(−z). (13)

Due to the special structure of L_θ, the explicit formula of projection is given below.

ΠL_θ(z) =







z, if z ∈ L_θ 0, if z ∈ −L^∗_θ u, otherwise,

(14)

where

u =







z₁ + kz₂k tan θ 1 + tan²θ

 z₁+ kz₂k tan θ 1 + tan²θ tan θ

 z₂ kz₂k





.

In fact, formula (14) can be found in several places, for example, [8], [1, page 508] or [2, Theorem 3.3.6]. For completeness we provide the detailed argument on (14), nonetheless, by a different approach from that in [2, Theorem 3.3.6], which leads us to establish the spectral factorization associated with L_θ.

The first two cases in (14) follow from (13) directly. Now, consider the third case.

Note that it corresponds to z₁tan θ < kz₂k and −z₁ctanθ < kz₂k. Hence we must have z₂ 6= 0, because otherwise, we would have z₁ < 0 and z₁ > 0, which is impossible. Let us calculate the projection in the third case by solving the Karush-Kuhn-Tucker conditions for the following convex programming problems

min 1

2kx − zk² s.t. x ∈ L_θ which is equivalent to

min 1

2kx − zk²

s.t. kx₂k − x₁tan θ ≤ 0.

The KKT point of the above convex programming is to find x ∈ L_θ and λ ≥ 0 such that

 x₁− z₁ x₂− z₂

 + λ

(" 0 x₂ kx₂k

#

− tan θ 1 0

)

= 0,

which is equivalent to solving







x₁ = z₁+ λ tan θ,

x₂ = 1

1 + (λ/kx₂k) z₂. (15)

Thus,

kz₂k = (1 + (λ/kx₂k)) kx₂k = kx₂k + λ = kx₂k + x1 − z1

tan θ = x₁tan θ + x1− z1

tan θ , where the third equality is due to (15) and the last equality comes from the fact that kx2k = x1tan θ since the projection point of z /∈ Lθ must lie in the boundary of Lθ. Then, we have

x₁ = z₁+ kz₂k tan θ 1 + tan²θ . Substituting this into the first equation in (15) yields

λ = kz2k − z1tan θ 1 + tan²θ .

Therefore, according to the second equation in (15), we obtain x₂ = z₁+ kz₂k tan θ

1 + tan²θ tan θ

 z₂ kz₂k which says

ΠL_θ(z) =







z₁+ kz₂k tan θ 1 + tan²θ

 z₁+ kz₂k tan θ 1 + tan²θ tan θ

 z₂ kz₂k





 (16)

under this subcase.

From (13), we see that Π_L^◦

θ(z) = −Π_Lπ

2−θ(−z) which implies

Π_L^◦

θ(z) = −







−z₁+ kz₂kctanθ 1 + ctan²θ

 −z₁ + kz₂kctanθ

1 + ctan²θ ctanθ −z₂ kz₂k







=







z1− kz2kctanθ 1 + ctan²θ

 z₁− kz₂kctanθ

1 + ctan²θ ctanθ −z₂ kz₂k





. (17)

According to the above arguments, we obtain the following result, which is called the spectral factorization for z associated with circular cone.

Theorem 3.1. For any z ∈ IRⁿ, one has

z = λ₁(z) · u⁽¹⁾_z + λ₂(z) · u⁽²⁾_z (18)

where

λ₁(z) = z₁− kz₂kctanθ λ2(z) = z1+ kz2k tan θ and

u⁽¹⁾_z = 1 1 + ctan²θ

1 0

0 ctanθ

  1

−w



u⁽²⁾_z = 1 1 + tan²θ

1 0

0 tan θ

  1 w



with w = z₂

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

Proof. The case of z₂ = 0 is clear by simply calculating (18). The case of z₂ 6= 0 follows from (13), (16), and (17) because

z = ΠL_θ(z) + ΠL^◦_θ(z)

=







z1+ kz2k tan θ 1 + tan²θ

 z₁+ kz₂k tan θ 1 + tan²θ tan θ

 z₂ kz2k





+







z1− kz2kctanθ 1 + ctan²θ

 z₁ − kz₂kctanθ

1 + ctan²θ ctanθ −z₂ kz2k







= z₁ + kz₂k tan θ 1 + tan²θ

1 0

0 tan θ

" 1 z₂ kz₂k

#

+z₁− kz₂kctanθ 1 + ctan²θ

1 0

0 ctanθ

" 1

− z₂ kz₂k

# . 2

With Theorem 3.1, we could derive another expression for the projection shown as below.

Theorem 3.2. For any z ∈ IRⁿ, we have ΠL_θ(z) = λ₁(z)

+· u⁽¹⁾_z + λ₂(z)

+· u⁽²⁾_z , (19)

where (a)₊ := max{0, a}, λ_i(z) and uⁱ_z for i = 1, 2 are given as in Theorem 3.1.

Proof. The proof is divided into two cases, according to whether z2 = 0 or z2 6= 0.

Case 1: z₂ = 0. If z₁ ≥ 0, then z₁tan θ ≥ 0 = kz₂k and λ_i(z) = z₁ ≥ 0. Hence z ∈ L_θ and both sides of (19) are z by (14) and (18). If z₁ < 0, then −z₁ctanθ ≥ 0 = kz₂k and λ_i(z) = z₁ < 0 for i = 1, 2. Hence, z ∈ −L^π

2−θ = −L^∗_θ and both sides of (19) are 0 by (14).

Case 2: z₂ 6= 0. If z ∈ L_θ, then z₁tan θ ≥ kz₂k which implies z₁ ≥ 0. Therefore, λ_i(z) ≥ 0 for i = 1, 2 which gives Π_Lθ(z) = z = λ₁(z)u¹_z+λ₂(z)u²_z by (14) and (18). If z ∈ −L^∗_θ, then

−z ∈ L^π

2−θ, i.e., −z₁ctanθ ≥ kz₂k, which says z₁ ≤ 0. Hence, λ₁(z) = z₁− kz₂kctanθ ≤ 0 and λ₂(z) = z₁+kz₂k tan θ ≤ 0. This indicates that the right-hand side of (19) is zero and it coincides ΠL_θ(z) = 0 by (14) under this case. Other cases correspond to z₁tan θ < kz₂k and −z1ctanθ < kz2k, i.e., λ1(z) = z1− kz2kctanθ < 0 and λ2(z) = z1+ kz2k tan θ > 0.

Simplifying the right-hand side of (19) with this, we see that (14) is also satisfied under this case. Thus, all the above shows the validity of (19). 2

In particular, when θ = π/4, expressions (18) and (19) takes, respectively, the form of

z = (z1− kz₂k)1 2

 1

−w



+ (z1+ kz2k)1 2

 1 w



and

ΠL_θ(z) = (z₁ − kz₂k)₊1 2

 1

−w



+ (z₁+ kz₂k)₊1 2

 1 w



where w = z2

kz₂k if z₂ 6= 0, and any vector in IRⁿ⁻¹ satisfying kwk = 1 if z₂ = 0. These are exactly the well-known spectral factorization and projection associated with Kⁿ.

We believe that the spectral factorization given in Theorem 3.1 is very important for developing theory and algorithm for optimization associated with L_θ like the role played by the spectral factorization associated with Kⁿ in second-order cone optimization. We leave it for our future research topic.

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