(1)2013 (2)Considerable attentions have been devoted to the second-order cone Kn [5–7], a special case of self-dual cone

2013

Considerable attentions have been devoted to the second-order cone Kⁿ [5–7], a special case of self-dual cone. However, the study on the circular cone Lθ, a non- self-dual (or non-symmetric cone) is rather limited. In this paper, we show that there exists a close relationship between Kⁿ and Lθ by establishing an inequality regrading distance between Kⁿ and Lθ. This nice property plays an essential role in our subsequence analysis and give us more information and insight on Lθ. In particular, we develop the formulae of tangent cone, normal cone, and second-order tangent cone of Lθ in terms of Kⁿ (the formula of the latter has been given by different scholars). Furthermore, we show that Lθ, 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-definite 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 ofLθ by studying the projection onLθ which will be useful in dealing with optimization associated circular cone.

In fact, it is not hard to see that Lθ = {

(x1, x2)^T ∈ IR × IRⁿ⁻¹ | x1 ≥ 0, ∥x∥²cos²θ≤ x²1

}

= {

(x1, x2)^T ∈ IR × IRⁿ⁻¹ | x1 ≥ 0, (x²1+∥x2∥²) cos²θ≤ x²1

}

= {

(x1, x2)^T ∈ IR × IRⁿ⁻¹ | x1 ≥ 0, ∥x2∥² ≤ x²1tan²θ}

= {

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

(1.3)

[x₁ x2

]

∈ Lθ ⇐⇒

[tan θx₁ x2

]

∈ Kⁿ ⇐⇒

[ tan θ 0

0 I

] [x₁ x2

]

∈ Kⁿ. For simplicity, let us denote

A :=

[ tan θ 0

0 I

] . Then, the above expression (1.3) is equivalent to (1.4)

[x₁ x₂ ]

∈ Lθ ⇐⇒ A [x₁

x₂ ]

∈ Kⁿ.

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

[ ctanθ 0

0 I

]

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 | ⟨v, x⟩ ≥ 0, ∀x ∈ K} , while its polar cone is given by

(K)^◦ ={v | ⟨v, x⟩ ≤ 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 ofLθ, including characterizing its tangle cone, normal cone, second-order tangent cone, etc.. To this end, we first describe the relationship between Kⁿ and Lθ.

Theorem 2.1. Let Lθ and Kⁿ be defined as in (1.1) and (1.2), respectively. Then, we have

(a) Lθ= A⁻¹Kⁿ and Kⁿ= ALθ. (b) AKⁿ=L^π_2−θ and L^π_2−θ= A²Lθ.

(c) L^∗_θ=L^π

2−θ and (L^∗_θ)^∗ =Lθ.

Proof. (a) This follows from equivalence (1.4) because Lθ = {x | x ∈ Lθ}

= {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 I

] Kⁿ=

[ tan θ 0

0 I

]

Kⁿ= AKⁿ= A(ALθ) = A²Lθ

which is the desired result.

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

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

= {v | ⟨v, Az⟩ ≥ 0, ∀z ∈ Lθ}

= {v | ⟨Av, z⟩ ≥ 0, ∀z ∈ Lθ}

= {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. 

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

(2.1) ∥A∥⁻¹ dist(Az,Kⁿ)≤ dist(z, Lθ)≤ ∥A⁻¹∥ dist(Az, Kⁿ) and

(2.2) ∥A⁻¹∥⁻¹ dist(A⁻¹x,Lθ)≤ dist(x, Kⁿ)≤ ∥A∥ dist(A⁻¹x,Lθ).

Proof. First, we observe the following:

dist(x,Kⁿ) = min

k∈Kⁿ∥x − k∥ = min

k∈ALθ

∥x − k∥

= min

z∈Lθ

∥x − Az∥ = min

z∈Lθ

∥A(A⁻¹x)− Az∥

(2.3)

= min

z∈Lθ

∥A(A⁻¹x− z)∥ ≤ ∥A∥ min

z∈Lθ

∥A⁻¹x− z∥

= ∥A∥ dist(

A⁻¹x,Lθ

),

dist(z,Lθ) = min

u∈Lθ

∥z − u∥ = min

u∈A⁻¹Kⁿ∥z − u∥

= min

k∈Kⁿ∥z − A⁻¹k∥ = min

k∈Kⁿ∥A⁻¹(Az)− A⁻¹k∥ (2.4)

= min

k∈Kⁿ∥A⁻¹(Az− k)∥ ≤ ∥A⁻¹∥ min

k∈Kⁿ∥Az − k∥

= ∥A⁻¹∥ dist(Az, Kⁿ).

These prove the second inequality in (2.1) and (2.2), respectively. Next, plugging z = A⁻¹x and x = Az in (2.3) and (2.4), respectively, yields the first inequality in (2.1) and (2.2), respectively. Thus, the proof is complete.  Theorem 2.2 indicates that the distances of arbitrary points to Kⁿ and Lθ are equivalent. This is an essential property for analyzing the tangent cone and normal cone ofLθ. 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 TS(x) and inner tangent cone T_Sⁱ(x) of S at x are defined respectively as

TS(x) :={d ∈ IRⁿ| ∃tn↓ 0, dist (x + tnd, S) = o(tn)}

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 bNS(x), is defined as

NbS(x) :={v ∈ IRⁿ| ⟨v, z − x⟩ ≤ o(∥z − x∥), 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 | ⟨v, d⟩ ≤ 0, ∀d ∈ TS(x)} . Theorem 2.3. For any z∈ Lθ, we have

(a) T_Lθ(z) = A⁻¹T_Kⁿ(Az), (b) N_Lθ(z) = AN_Kⁿ(Az).

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

(2.5) dist(z + td,Lθ) = o(t).

Plugging x = A(z + td) into (2.2) yields

∥A⁻¹∥⁻¹ dist(z + td,Lθ)≤ dist (A(z + td), Kⁿ)≤ ∥A∥ dist(z + td, Lθ).

This together with (2.5) implies dist(Az + tAd,Kⁿ) = o(t). Thus, Ad ∈ T_Kⁿ(Az), which says d∈ A⁻¹T_Kⁿ(Az).

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

(2.6) dist(Az + tAd,Kⁿ) = o(t).

Replacing z in (2.1) by z + td gives

∥A∥⁻¹ dist(Az + tAd,Kⁿ)≤ dist(z + td, Lθ)≤ ∥A⁻¹∥ dist(Az + tAd, Kⁿ).

This together with (2.6) implies dist(z + td,Lθ) = o(t), which says d∈ TLθ(z).

(b) The desired result follows from

N_Lθ(z) = {v ∈ IRⁿ | ⟨v, d⟩ ≤ 0, ∀d ∈ T_Lθ(z)}

= {

v∈ IRⁿ | ⟨v, A⁻¹w⟩ ≤ 0, ∀w ∈ TKⁿ(Az)}

= {

v∈ IRⁿ | ⟨A⁻¹v, w⟩ ≤ 0, ∀w ∈ T_Kⁿ(Az)}

= {

v∈ IRⁿ | A⁻¹v ∈ N_Kⁿ(Az)}

= AN_Kⁿ(Az).

 Theorem 2.3 tells us that the explicit formula of tangent cone T_Lθ(z) can be established by T_Kⁿ(Az), which has been given in [3].

It is well known that in the study of second order analysis for optimization prob- lems, 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 2.4 ( [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 2.5 ( [4, Definition 3.32]). We say that the set S is second order direc- tionally differentiable at a point x∈ S in a direction d ∈ TS(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 dif- ferentiable at a point x ∈ S if it is second order directionally differentiable in all directions d∈ TS(x).

Theorem 2.6. Let z∈ Lθ and d∈ TLθ(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. 

Definition 2.7 ( [4, Definition 3.85]). We say that a subset S⊂ IRⁿis second order regular at x if it satisfies

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

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

nlim→∞dist(

r_n, T_S²(x, d))

= 0.

Theorem 2.8. The circular cone Lθ is second order regular.

Proof. Let z ∈ Lθ and d ∈ T_Lθ(z). According to Theorem 2.6, it suffices to show that for any sequence z + tnd +¹₂t²_nrn∈ Lθ with tnrn→ 0, there holds

(2.7) lim

n→∞dist( r_n, T_L²

θ(z, d))

= 0.

We will complete the proof by using the relationship between Lθ and Kⁿ. Since z + t_nd +¹₂t²_nr_n∈ Lθ, we know Az + t_nAd +¹₂t²_nAr_n∈ Kⁿby Theorem 2.1(a). Note that tnArn → 0 because ∥tnArn∥ ≤ ∥A∥ · ∥tnrn∥. In addition, Kⁿ is second order regular (see [10] for detailed proof), from Definition 2.7, we have

(2.8) lim

n→∞dist(

Ar_n, T_K²n(Az, Ad))

= 0.

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))

≤ ∥A⁻¹∥ dist(

Arn, T_K²ⁿ(Az, Ad)) .

This together with (2.8) implies the validity of (2.7).  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 onLθ, i.e.,

Π_Lθ(z) := arg min

x∈Lθ

∥z − x∥ = {x ∈ Lθ | ∥z − x∥ ≤ ∥z − u∥, ∀u ∈ Lθ} . It should be mentioned that the projection cannot be obtained by using the relationship betweenLθ and Kⁿ because

∥A⁻¹x∥ ≤ ∥A⁻¹y∥ ; ∥x∥ ≤ ∥y∥ whenever θ ̸= π/4.

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

∥A⁻¹x∥ =√ 2 <√

17/2 =∥A⁻¹y∥, but ∥x∥ =√

65 >√

20 =∥y∥.

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

Π_−Ω(x) =−ΠΩ(−x).

In fact, letting a = Π_−Ω(x) yields

∥(−x) − (−a)∥ = ∥x − a∥ ≤ ∥x − (−y)∥ = ∥(−x) − y∥ ∀y ∈ Ω,

where the inequality comes from the fact that a = Π_−Ω(x) by definition of projec- tion. This means that −a = ΠΩ(−x). Besides, it is well known that any vector z∈ IRⁿcan be written as

z = Π_Ω(z) + Π_Ω◦(z).

Hence,

z = Π_Lθ(z) + Π_L◦

θ(z) = Π_Lθ(z) + Π_−L^∗

θ(z)

= Π_Lθ(z)− Π_L^∗_θ(−z) = Π_Lθ(z)− Π_Lπ

2 −θ(−z).

(3.1)

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

(3.2) Π_Lθ(z) =





z, if z∈ Lθ

0, if z∈ −L^∗_θ u, otherwise, where

u =





z1+∥z2∥ tan θ 1 + tan²θ (z1+∥z2∥ tan θ

1 + tan²θ tan θ ) z₂

∥z2∥



 .

In fact, formula (3.2) 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 (3.2), nonetheless, by a different approach from that in [2, Theorem 3.3.6], which leads us to establish the spectral factorization associated withLθ.

The first two cases in (3.2) follow from (3.1) directly. Now, consider the third case. Note that it corresponds to z1tan θ <∥z2∥ and −z1ctanθ <∥z2∥. Hence we must have z₂ ̸= 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

2∥x − z∥² s.t. x∈ Lθ

which is equivalent to

min 1

2∥x − z∥²

s.t. ∥x2∥ − x1tan θ≤ 0.

The KKT point of the above convex programming is to find x∈ Lθ (noting x̸= 0 since z̸= Lθ and z ̸= −L^∗_θ) and λ≥ 0 such that

[ x₁− z1

x2− z2

] + λ

{[ 0

x2

∥x2∥ ]

− tan θ [ 1

0 ]}

= 0,

which is equivalent to solving (3.3)





x₁= z₁+ λ tan θ,

x₂= 1

1 + (λ/∥x2∥) z₂. Thus,

∥z2∥ = (1 + (λ/∥x2∥)) ∥x2∥ = ∥x2∥ + λ = ∥x2∥ +x1− z1

tan θ = x1tan θ +x1− z1

tan θ , where the third equality is due to (3.3) and the last equality comes from the fact that ∥x2∥ = x1tan θ since the projection point of z /∈ Lθ must lie in the boundary of Lθ. Then, we have

x1= z₁+∥z2∥ tan θ 1 + tan²θ . Substituting this into the first equation in (3.3) yields

λ = ∥z2∥ − z1tan θ 1 + tan²θ .

Therefore, according to the second equation in (3.3), we obtain x₂ =

(z₁+∥z2∥ tan θ 1 + tan²θ tan θ

) z₂

∥z2∥ which says

(3.4) Π_Lθ(z) =





z₁+∥z2∥ tan θ 1 + tan²θ (z1+∥z2∥ tan θ

1 + tan²θ tan θ ) z2

∥z2∥





under this subcase.

From (3.1), we see that Π_L◦

θ(z) =−Π_Lπ

2 −θ(−z) which implies Π_L◦

θ(z) = −





−z1+∥z2∥ctanθ 1 + ctan²θ (−z1+∥z2∥ctanθ

1 + ctan²θ ctanθ ) −z2

∥z2∥





=





z1− ∥z2∥ctanθ 1 + ctan²θ (z1− ∥z2∥ctanθ

1 + ctan²θ ctanθ ) −z2

∥z2∥



 . (3.5)

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

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

λ₁(z) = z₁− ∥z2∥ctanθ λ₂(z) = z₁+∥z2∥ 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

∥z2∥ if z2̸= 0, and any vector in IRⁿ⁻¹ satisfying ∥w∥ = 1 if z2 = 0.

Proof. The case of z2 = 0 is clear by simply calculating (3.6). The case of z2 ̸= 0 follows from (3.1), (3.4), and (3.5) because

z = Π_Lθ(z) + Π_L◦ θ(z)

=





z1+∥z2∥ tan θ 1 + tan²θ (z₁+∥z2∥ tan θ

1 + tan²θ tan θ ) z₂

∥z2∥



 +





z1− ∥z2∥ctanθ 1 + ctan²θ (z₁− ∥z2∥ctanθ

1 + ctan²θ ctanθ ) −z2

∥z2∥





= z1+∥z2∥ tan θ 1 + tan²θ

[1 0 0 tan θ

] [ 1 z2

∥z2∥ ]

+z1− ∥z2∥ctanθ 1 + ctan²θ

[1 0 0 ctanθ

] [ 1

− z2

∥z2∥ ]

.

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

Theorem 3.2. For any z∈ IRⁿ, we have

(3.7) Π_Lθ(z) =(

λ₁(z))

+· u⁽¹⁾z +( λ₂(z))

+· u⁽²⁾z ,

where (a)+:= max{0, a}, λi(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 ̸= 0.

Case 1: z2 = 0. If z1≥ 0, then z1tan θ≥ 0 = ∥z2∥ and λi(z) = z1 ≥ 0. Hence z ∈ Lθ

and both sides of (3.7) are z by (3.2) and (3.6). If z₁ < 0, then−z1ctanθ≥ 0 = ∥z2∥ and λi(z) = z1 < 0 for i = 1, 2. Hence, z ∈ −L^π

2−θ =−L^∗_θ and both sides of (3.7) are 0 by (3.2).

Case 2: z2 ̸= 0. If z ∈ Lθ, then z1tan θ ≥ ∥z2∥ 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 (3.2) and (3.6). If z ∈ −L^∗_θ, then −z ∈ L^π_2−θ, i.e., −z1ctanθ ≥ ∥z2∥, which says z1 ≤ 0.

Hence, λ₁(z) = z₁− ∥z2∥ctanθ ≤ 0 and λ2(z) = z₁+∥z2∥ tan θ ≤ 0. This indicates that the right-hand side of (3.7) is zero and it coincides Π_Lθ(z) = 0 by (3.2) under this case. Other cases correspond to z1tan θ < ∥z2∥ and −z1ctanθ < ∥z2∥, i.e., λ₁(z) = z₁− ∥z2∥ctanθ < 0 and λ2(z) = z₁+∥z2∥ tan θ > 0. Simplifying the right- hand side of (3.7) with this, we see that (3.2) is also satisfied under this case. Thus,

all the above shows the validity of (3.7). 

In particular, when θ = π/4, expressions (3.6) and (3.7) takes, respectively, the form of

z = (z1− ∥z2∥)1 2

[ 1

−w ]

+ (z1+∥z2∥)1 2

[1 w ]

and

Π_Lθ(z) = (z1− ∥z2∥)+

1 2

[ 1

−w ]

+ (z1+∥z2∥)+

1 2

[1 w ]

where w = z₂

∥z2∥ if z₂ ̸= 0, and any vector in IRⁿ⁻¹ satisfying ∥w∥ = 1 if z2 = 0.

These are exactly the well-known spectral factorization and projection associated withKⁿ.

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

References

[1] H. H. Bauschke and S. G. Kruk, Reflection-projection method for convex feasibility problems with an obtuse cone, J. Optim. Theory Appl. 120 (2004), 503–531.

[2] H. H. Bauschke, Projection Algorithms and Monotone Operators, PhD Thesis, Simon Fraster University, 1996.

[3] J. F. Bonnans and H. R. Cabrera, Perturbation analysis of second-order cone programming problems, Math. Program. 104 (2005), 205–227.

[4] J. F. Bonnans and A. Shapiro, Perturbation Analyisis of Optimization Problems, Springer- Verlag, New York, 2000.

[5] J.-S. Chen, The convex and monotone functions associated with second-order cone, Optimiza- tion 55 (2006), 363–385.

[6] J.-S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second-oredr cone, Math. Program. 101 (2004), 95–117.

[7] J.-S. Chen, T.-K. Liao and S.-H. Pan, Using Schur Complement Theorem to prove convexity of some SOC-functions, J. Nonlinear Convex Anal. 13 (2012), 421–431.

[8] A. Pinto Da Costa and A. Seeger, Numberical resolultion of cone-constrained eigenvalue prob- lems, Com. Appl. Math. 28 (2009), 37–61.

[9] J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo Publishing, Cal- ifornia, 2005.

[10] J.-C. Zhou and J.-S. Chen, Saddle points of nonlinear second-order cone programming and second order regularity of second-order cone, submitted manuscript, 2012.

Manuscript received October 5, 2012 revised November 15, 2012

Jinchuan Zhou

Department of Mathematics, School of Science, Shandong University of Technology, Zibo 255049, P.R. China

E-mail address: [email protected]

Jein-Shan Chen

Department of Mathematics, National Taiwan Normal University Taipei 11677, Taiwan; and Mem- ber of Mathematics Division, National Center for Theoretical Sciences, Taipei Office

E-mail address: [email protected]