1Introduction p -ordercone Projectionformulaandtwotypesofspectralfactorizationsassociatedwith

to appear in Optimization Letters, 2014

Projection formula and two types of spectral factorizations associated with p-order cone

Xinhe Miao ¹

Department of Mathematics School of Science Tianjin University Tianjin 300072, P.R. China

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

Taipei 11677, Taiwan

July 22, 2013

(1st revised on January 2, 2014) (2nd revised on March 21, 2014)

Abstract. In this short paper, we establish the projection formula associated with p-order cone and further discover two types of spectral factorizations associated with p-order cone. These expressions will be key bricks for further analysis and study about p-order cone optimization.

Key words. p-order cone, projection, spectral factorization.

1 Introduction

Recently, there has been much attention on symmetric cone optimization, see [5, 12, 13, 15, 16] and references therein, but not much on non-symmetric cone optimization.

In general, non-symmetric cones include p-order cone [1, 17], circular cone [3, 7, 18], L^p

1The author’s work is supported by National Young Natural Science Foundation (No. 11101302 and No. 61002027) and The Seed Foundation of Tianjin University (No. 60302041). E-mail: xin- [email protected]

2Corresponding author. Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is supported by Ministry of Science and Technology, Taiwan. E-mail:

[email protected]

cone [10], and copositive cone [8], etc. Unlike symmetric cone case in which the Euclidean Jordan algebra can unify the whole analysis, there has not been found a special unified Jordan algebra for non-symmetric cones until now. Nonetheless, analogous to tackling symmetric cone optimization, in which the spectral decomposition [9] plays a key role, we believe that in order to find out a way to deal with non-symmetric cone optimization problems, the first key step is to figure out their corresponding projection formulae and spectral factorizations.

A good spectral factorization, like the eigenvalue decomposition in linear algebra, provides an efficient way for computer software to compute some special function, for instance, projection function. Moreover, the efficiency of computing projection formulae can help on designing some algorithms for solving non-symmetric cone optimization prob- lems, for example, the so-called projection gradient method and merit function method, and so on. For circular cone case, its corresponding projection formula and spectral factorization are studied in [18]. However, there are no further investigations for other non-symmetric cone cases yet. In this paper, we characterize the projection formula of element z onto p-order cone, and establish two types of spectral factorizations associated with p-order cone. We believe that these expressions are key bricks for further analysis and study about p-order cone optimization.

The p-order cone in Rⁿ, which is a generalization of the second-order cone [4, 6], is defined as

K_p :=







x ∈ Rⁿ    

x₁ ≥

n

X

i=2

|x_i|^p

!¹_p





(p > 1). (1)

If we write x := (x₁, x₂) ∈ R × R⁽ⁿ⁻¹⁾, the p-order cone K_p can be equivalently expressed as

K_p =x = (x1, x₂) ∈ R × R⁽ⁿ⁻¹⁾| x₁ ≥ kx₂k_p , (p > 1).

The pictures of three different cones K_p in R³ are depicted in Figure 1.

(a) 2-order cone (b) 3-order cone (c) 10-order cone

Figure 1: Three different p-order cones in R³.

From (1) and Figure 1, it is clear to see that when p = 2, K₂is exactly the second-order cone Kⁿ = x = (x1, x2) ∈ R × R⁽ⁿ⁻¹⁾| x1 ≥ kx2k , which confirms that the second- order cone is a special case of p-order cone.

It is well known that K_p is a convex cone and its dual cone is given by

K^∗_p =







y ∈ Rⁿ    

y₁ ≥

n

X

i=2

|y_i|^q

!¹_q



 or equivalently

K^∗_p =y = (y1, y₂) ∈ R × R⁽ⁿ⁻¹⁾| y₁ ≥ ky₂k_q = Kq,

where q > 1 and satisfies ¹_p +¹_q = 1. In addition, the dual cone K^∗_p is also a convex cone.

For an application of p-order cone programming, we refer the readers to [17], in which a primal-dual potential reduction algorithm for p-order cone constrained optimization problems is studied. Besides, in [17], a special optimization problem called sum of p- norms is transformed into an p-order cone constrained optimization problems.

To end this section, we say a few words about the notations used in this paper. We consider the Euclidean space Rⁿ equipped with the standard inner product h·, ·i. The Euclidean norm is defined as kzk := phz, zi. Let K be any closed convex cone. We denote its dual cone by

K^∗ = {y | hy, xi ≥ 0 ∀x ∈ K}, and denote its polar cone by

K^◦ = {y | hy, xi ≤ 0 ∀x ∈ K}.

Moreover, ∂K means the boundary of K and ΠK(z) is the projection of z onto K.

2 Projection formula and spectral factorization

In [18], we see that the spectral factorization associated with circular cone is figured out first and then the projection onto circular cone is characterized. For the p-order cone case, the procedure is totally opposite. More specifically, we need to characterize the projection onto such cone, and then figure out its corresponding spectral factorization.

In particular, two types of spectral factorizations associated with p-order cone are pro- vided.

First, we start with the general Orthogonal Projection Theorem associated with any closed convex cone in Hilbert space (see [14, Theorem II.3]). The Orthogonal Projection

Theorem is better known in the optimization community as the Moreau Decomposi- tion(see [11]), which says for any z ∈ Rⁿ, z can be decomposed as

z = ΠK(z) + ΠK^◦(z) = ΠK(z) + Π−K^∗(z) (2) where K is any closed convex cone with polar cone K^◦ and dual cone K^∗. When K represents the special structure of the p-order cone K_p, the explicit expression (2) is characterized in following theorem.

Theorem 2.1 Let z = (z₁, z₂) ∈ R × R⁽ⁿ⁻¹⁾. Then, the projection of z onto K_p is given by

ΠKp(z) =







z, z ∈ K_p

0, z ∈ −K^∗_p = −K_q

u, otherwise (i.e., −kz₂k_q < z₁ < kz₂k_p)

(3)

where

u =









 z₁+ kz₂k_q kz2kp+ kz2kq

 kz₂k_p

 z1+ kz2kq

kz₂k_p+ kz₂k_q

 z₂







 .

Proof. From Projection Theorem [2, Prop. 2.2.1], we know that, for every z ∈ Rⁿ, a vector u ∈ K_p is equal to the projection point ΠKp(z) if and only if

u ∈ K_p, z − u ∈ K_p^◦, and hz − u, ui = 0.

With this, the first two cases of (3) are obvious. Hence, we only need to consider the third case. Based on the expression of the element u, it is easy to verify that u ∈ ∂K_p. Moreover, we have

z − u =







 z₁−

 z₁+ kz₂k_q kz2kp+ kz2kq

 kz₂k_p z2−

 z₁+ kz₂k_q kz₂k_p+ kz₂k_q

 z2









=









 z₁− kz₂k_p kz2kp+ kz2kq

 kz₂k_q

 kz₂k_p− z₁ kz₂k_p+ kz₂k_q

 z2







 .

Noting that

 kz₂k_p− z₁ kz₂k_p + kz₂k_q



kz2kq =

 kz₂k_p − z₁ kz₂k_p+ kz₂k_q

 z2

q

,

we obtain −(z − u) ∈ K_q = K^∗_p which implies z − u ∈ K^◦_p. Hence, it remains to prove hz − u, ui = 0. We will proof it by contradiction. Suppose that hz − u, ui < 0. Since u ∈ K_p and z − u ∈ K^◦_p, there exists

z − u 6= w =





α

 kz₂k_p− z₁ kz₂k_p+ kz₂k_q

 z₂



∈ K^◦_p = −K^∗_p = −K_q

such that hw, ui = 0. Thus, we have 0 = hw, ui

= α (z₁+ kz₂k_q)kz₂k_p kz2kp+ kz2kq

+(kz₂k_p− z₁)(z₁+ kz₂k_q)kz₂k² (kz2kp+ kz2kq)²

which yields

α =

 z1− kz2kp

kz₂k_p+ kz₂k_q

 kz₂k² kz₂k_p. Moreover, from w ∈ −K_q, we also have

−α =

 kz₂k_p− z₁ kz₂k_p+ kz₂k_q

 kz₂k² kz₂k_p ≥

 kz₂k_p − z₁ kz₂k_p+ kz₂k_q

 kz2kq.

Then, it follows that kz2k² ≥ kz2kpkz2kq. This together with the H¨older inequality imply kz₂k² = kz₂k_p· kz₂k_q which says

α =

 z1− kz2kp

kz₂k_p+ kz₂k_q

 kz₂k_q.

Hence, we obtain w = z−u, which contradicts z−u 6= w. This implies that hz−u, ui = 0.

Furthermore, the projection of z onto Kp is expressed as in (3). 2

In the sequel, for the sake of simplicity, we denote z⁺ := ΠKp(z). Because K^◦_p =

−K^∗_p = −K_q, we know

Π−K^∗_p(z) = Π−K_q(z) = −ΠK_q(−z).

This together with (3) gives

z⁻ := −ΠK_q(−z) =







z, −z ∈ K_q

0, −z ∈ −K^∗_q = −K_p

v, otherwise (i.e., −kz₂k_p < −z₁ < kz₂k_q)

(4)

where

v =









 z₁− kz₂k_p kz₂k_q+ kz₂k_p

 kz₂k_q

 z₁− kz₂k_p kz₂k_q+ kz₂k_p

 (−z₂)







 .

According to the Orthogonal Projection Theorem, i.e., z = ΠK(z)+ΠK^◦(z) = z⁺+z⁻, the following theorem presents the first type of factorization for z = (z₁, z₂) ∈ R×R⁽ⁿ⁻¹⁾, which is called the spectral factorization (type I) of z associated with p-order cone K_p.

Theorem 2.2 (Spectral factorization, type I) Let z = (z₁, z₂) ∈ R × R⁽ⁿ⁻¹⁾. Then, z can be decomposed as

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

 λ₁(z) = z₁+ kz₂k_q λ₂(z) = z₁− kz₂k_p and











u⁽¹⁾(z) =

 kz₂k_p kz₂k_p+ kz₂k_q

  1 w1



u⁽²⁾(z) =

 kz2kq

kz₂k_q+ kz₂k_p

  1 w₂



with w1 = _kz^z2

2k_p and w2 = −_kz^z2

2k_q when z2 6= 0; while w1 being an arbitrary element satisfying kw₁k_p = 1 and w₂ = −w₁ when z₂ = 0, and we set _kz ^kz2kp

2kp+kz2kq = _kz ^kz2kq

2kq+kz2kp =

1

2 under such case.

Proof. When z₂ = 0, the proof follows from simple calculation. When z₂ 6= 0, from expressions (3)-(4) and the formula z = ΠK(z) + ΠK^◦(z) = z⁺+ z⁻, we have

z = z⁺+ z⁻

=









 z₁+ kz₂k_q kz₂k_p+ kz₂k_q

 kz₂k_p

 z₁+ kz₂k_q kz2kp + kz2kq

 z₂







 +









 z₁− kz₂k_p kz₂k_q+ kz₂k_p

 kz₂k_q

 z₁− kz₂k_p kz2kq+ kz2kp

 (−z₂)









= (z₁+ kz₂k_q)kz₂k_p kz₂k_p+ kz₂k_q

"

1

z2

kz2kp

#

+(z₁− kz₂k_p)kz₂k_q kz₂k_q+ kz₂k_p

"

1

−_kz^z2

2kq

#

= (z₁ + kz₂k_q)

 kz2kp

kz₂k_p+ kz₂k_q

"

1

z2

kz2kp

#

+ (z₁− kz₂k_p)

 kz2kq

kz₂k_q+ kz₂k_p

"

1

−_kz^z2

2kq

#

= λ1(z) · u⁽¹⁾(z) + λ2(z) · u⁽²⁾(z).

Thus, the proof is complete. 2

Remark 2.1 From Theorem 2.2, we know that the properties of {u⁽¹⁾(z), u⁽²⁾(z)} are similar to the ones of the orthogonal system in the inner space Rⁿ, i.e.,

hu⁽¹⁾(z), u⁽²⁾(z)i = 0 and ku⁽¹⁾(z)kp+ ku⁽²⁾(z)kq= 2.

Combing Theorem 2.1 with Theorem 2.2, the following theorem provides another expression for the projection of z onto Kp, which is a very useful working formula.

Theorem 2.3 Let z = (z₁, z₂) ∈ R × R⁽ⁿ⁻¹⁾ have the spectral factorization (type I) as in Theorem 2.2. Then, we have

z⁺= ΠKp(z) = (λ₁(z))₊· u⁽¹⁾(z) + (λ₂(z))₊· u⁽²⁾(z) where (a)₊ := max{0, a}.

Proof. We proceed the arguments by discussing two cases.

Case 1: z₂ = 0. Under this case, we go further by discussing two subcases.

• For z1 ≥ 0, we have z1 ≥ 0 = kz2kp and λi(z) = z1 ≥ 0 (i = 1, 2), which leads to z ∈ K_p. Then, applying Theorem 2.1 gives

z⁺ = z

= λ₁(z) · u⁽¹⁾(z) + λ₂(z) · u⁽²⁾(z)

= (λ₁(z))₊· u⁽¹⁾(z) + (λ₂(z))₊· u⁽¹⁾(z).

• For z₁ < 0, we have −z₁ > 0 = kz₂k_q, which says z ∈ −K_q and λ_i(z) = z₁ < 0.

Using Theorem 2.1 again, we obtain

z⁺= 0 = (λ₁(z))₊· u⁽¹⁾(z) + (λ₂(z))₊· u⁽¹⁾(z).

Case 2: z₂ 6= 0. Under this case, we still have three subcases.

• For z₁ ≥ kz₂k_p, we know z ∈ K_p and z₁ ≥ 0, which imply λ_i(z) ≥ 0 for i = 1, 2.

Then, applying Theorem 2.1 yields z⁺ = z

= λ₁(z) · u⁽¹⁾(z) + λ₂(z) · u⁽²⁾(z)

= (λ₁(z))₊· u⁽¹⁾(z) + (λ₂(z))₊· u⁽¹⁾(z).

• For z₁ ≤ −kz₂k_q, it follows that z ∈ −K_q and z₁ ≤ 0, which gives λ_i(z) ≤ 0 for i = 1, 2. Using Theorem 2.1 again, we have

z⁺= 0 = (λ1(z))+· u⁽¹⁾(z) + (λ2(z))+· u⁽¹⁾(z).

• For −kz₂k_q < z1 < kz2k_p, we have λ1(z) = z1+ kz2k_q > 0 and λ2(z) = z1− kz₂k_p <

0. Hence,

(λ₁(z))₊· u⁽¹⁾(z) + (λ₂(z))₊· u⁽¹⁾(z)

= λ₁(z) · u⁽¹⁾(z)

= (z₁+ kz₂k_q)

 kz₂k_p kz₂k_p+ kz₂k_q

"

1

z2

kz2kp

# . Using Theorem 2.1 again, we obtain

z⁺= (λ₁(z))₊· u⁽¹⁾(z) + (λ₂(z))₊· u⁽¹⁾(z).

From all the above discussion, the proof is complete. 2

From Theorem 2.3, we observe that λ_i(z) ≥ 0 (i = 1, 2) for any z ∈ K_p. Conversely, if both λi(z) ≥ 0 in the spectral factorization of z, it is easy to verify that z ∈ Kp. However, the vectors u⁽ⁱ⁾(z) (i = 1, 2) associated with the first type of spectral factorization of z are not both in K_p. To see this, observing from the expression of u⁽ⁱ⁾(z) (i = 1, 2) in Theorem 2.2, we have u⁽¹⁾(z) ∈ ∂K_p and u⁽²⁾(z) ∈ ∂K_q. However, when q > p, it can be verified that u⁽²⁾(z) /∈ Kp. To conquer this inconvenience, we develop another type of spectral factorization for z as below which guarantees u⁽ⁱ⁾(z) (i = 1, 2) both lie in the p-order cone K_p. Such factorization is called the spectral factorization of type II.

Theorem 2.4 (Spectral factorization, type II) Let z = (z₁, z₂) ∈ R × R⁽ⁿ⁻¹⁾. Then, z can be decomposed as

z = α₁(z) · v⁽¹⁾(z) + α₂(z) · v⁽²⁾(z), where







α₁(z) = z₁+ kz₂k_p 2 α₂(z) = z₁− kz₂k_p

2 and











v⁽¹⁾(z) =

 1 w₂



v⁽²⁾(z) =

 1

−w₂



with w₂ = _kz^z2

2kp when z₂ 6= 0; while w₂ being an arbitrary element satisfying kw₂k_p = 1 when z₂ = 0.

Proof. For z₂ 6= 0, we define eu(z) :=  τ kz₂kp

τ z₂



∈ ∂K_p such that u(z) − z ∈ ∂Ke _p, where τ is an undetermined coefficient. From u(z) − z ∈ Ke _p, we have

τ kz₂k_p− z₁ = k(τ − 1)z₂k_p which yields

τ = z₁+ kz₂k_p 2kz₂k_p . This further implies

u(z) =e









 z₁+ kz₂k_p 2kz₂k_p

 kz₂k_p

 z₁+ kz₂k_p 2kz2kp

 z₂







 .

Therefore, we can rewrite z as

z = eu(z) + (z −u(z))e

=









 z₁+ kz₂k_p 2kz₂k_p

 kz₂k_p

 z₁+ kz₂k_p 2kz2kp

 z₂







 +









 z₁− kz₂k_p 2kz₂k_p

 kz₂k_p

 kz₂k_p− z₁ 2kz2kp

 z₂









=  z₁+ kz₂k_p 2

"

1

z2

kz2kp

#

+ z₁− kz₂k_p 2

"

1

−_kz^z2

2kp

#

:= α₁(z) · v⁽¹⁾(z) + α₂(z) · v⁽²⁾(z) which gives the desired spectral factorization.

For z₂ = 0, it is easy to verify that z = α₁(z) · v⁽¹⁾(z) + α₂(z) · v⁽²⁾(z) with v⁽¹⁾(z) =

 1 w₂



and v⁽²⁾(z) =

 1

−w₂

 ,

where w₂ is an arbitrary element satisfying kw₂k_p = 1. Then, the desired factorization holds. 2

Remark 2.2 As pointed out by one referee, Theorem 2.2 and Theorem 2.4 can be proved by verifying the equality directly, which is easier. Nonetheless, we provide the constructive way to show how to obtain v¹(z), v²(z) and α1(z) α2(z). Moreover, from Theorem 2.4, we know that α₁(z) ≥ α₂(z).

As a consequence of Theorem 2.4 and Remark 2.2, we have the following corollary.

Corollary 2.1 Let z = α₁(z) · v⁽¹⁾(z) + α₂(z) · v⁽²⁾(z) be the spectral factorization of type II for z given as in Theorem 2.4. Then, v⁽ⁱ⁾(z) ∈ K_p. Moreover, the following hold

z ∈ K_p, ⇐⇒ α₂(z) ≥ 0.

3 Concluding Remarks

In this short paper, we have characterized the projection formula of element z onto p-order cone, and have established two types of spectral factorizations associated with p-order cone. As mentioned, these expressions will be key bricks for further analysis and study about p-order cone optimization.

One may ask what the advantages and disadvantages of each are? To answer this question, we say a few words for this point. In general, the advantages of the type I factorization is that the spectral factorization for z is an orthogonal decomposition, which means

u⁽¹⁾(z), u⁽²⁾(z) = 0.

This is in favor of calculating the inner product (when it is needed) and further studying p-order cone optimization. But, the vectors u⁽ⁱ⁾(z) (i = 1, 2) are not both in K_p, which is different from the case of symmetric cone. On the other hand, the advantages of the type II factorization is that the vectors v⁽ⁱ⁾(z) (i = 1, 2) both lie in Kp, which implies that any z in Rⁿ can be expressed by two vectors in p-order cone K_p. To the contrast, this factorization for z is not an orthogonal decomposition.

Acknowledgments. The authors are very grateful to the referee for the constructive comments, which have considerably improved the paper.

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