PROJECTION FORMULA AND ONE TYPE OF SPECTRAL FACTORIZATION ASSOCIATED WITH p-ORDER CONE

Volume 18, Number 9, 2017, 1699–1705

PROJECTION FORMULA AND ONE TYPE OF SPECTRAL FACTORIZATION ASSOCIATED WITH p-ORDER CONE

XINHE MIAO^∗, NUO QI, AND JEIN-SHAN CHEN^†

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

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 op- timization. In general, non-symmetric cones include p-order cone [1, 17], circular cone [3, 7, 18], L^p 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 spec- tral 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 factorization.

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 problems, for example, the so-called projection gradient method and merit function method, and so on. For circular cone case, its corresponding pro- jection 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 one type of spectral factorization associated with p-order cone. We believe that these expressions are key bricks for further analysis and study about p-order cone optimization.

2010 Mathematics Subject Classification. 49J52, 90C33, 65K05, 26B05.

Key words and phrases. p-Order cone, projection, spectral factorization.

∗The author’s work is supported by National Natural Science Foundation of China (No.

11471241).

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

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

(1.1) Kp :=



x∈ IRⁿ x1 ≥

( _n

∑

i=2

|xi|^p )¹

p



 (p > 1).

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

Kp = {

x = (x₁, x₂)∈ IR × IR⁽ⁿ⁻¹⁾| x1 ≥ ∥x2∥p

}

, (p > 1).

The pictures of three different conesKp in IR³ are depicted in Figure 1.

Figure 1. Three different p-order cones in IR³.

From (1.1) and Figure 1, it is clear to see that when p = 2, K2 is exactly the second-order cone Kⁿ={

x = (x1, x2)∈ IR × IR⁽ⁿ⁻¹⁾| x1 ≥ ∥x2∥}

, which confirms that the second-order cone is a special case of p-order cone.

It is well known that Kp is a convex cone and its dual cone is given by K^∗_p=



y∈ IRⁿ y1 ≥

( _n

∑

i=2

|yi|^q )¹

q



 or equivalently

K^∗p = {

y = (y₁, y₂)∈ IR × IR⁽ⁿ⁻¹⁾| y1 ≥ ∥y2∥q

}

=Kq,

where q > 1 and satisfies _p¹+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 IRⁿ equipped with the standard inner product

⟨·, ·⟩. The Euclidean norm is defined as ∥z∥ :=√

⟨z, z⟩. Let K be any closed convex cone. We denote its dual cone by

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

K^◦ ={y | ⟨y, x⟩ ≤ 0 ∀x ∈ K}.

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

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, one type of spectral factorization associated with p-order cone are provided.

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 also known in the optimization community as the Moreau Decomposition(see [11]), which says for any z∈ IRⁿ, z can be decomposed as (2.1) z = Π_K(z) + Π_K◦(z) = Π_K(z) + Π_−K∗(z)

whereK is any closed convex cone with polar cone K^◦ and dual cone K^∗. WhenK represents the special structure of the p-order coneKp, the explicit expression (2.1) is characterized in following theorem.

Theorem 2.1. Let z = (z1, z2) ∈ IR × IR⁽ⁿ⁻¹⁾. Then, the projection of z onto Kp

is given by

(2.2) Π_Kp(z) =





z, z∈ Kp

0, z∈ −K^∗p =−Kq

u, otherwise (i.e.,−∥z2∥q< z1 <∥z2∥p) where u = (u₁, ¯u) with ¯u = (u₂, u₃,· · · , un)^T ∈ IR⁽ⁿ⁻¹⁾ satisfying

u1=∥¯u∥p = (|u2|^p+|u3|^p+· · · + |un|^p)^p1 and

u_i− zi+u1− z1

u^p₁⁻¹ |ui|^p−2u_i = 0, ∀i = 2, · · · , n.

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

u∈ Kp, z− u ∈ K^◦_p and ⟨z − u, u⟩ = 0.

With this, the first two cases of (2.2) 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∈ ∂Kp. Moreover, we have

z− u =

[ z1− u1

z₂− ¯u ]

:=

[ z1− u1

h¯ ]

,

where ¯h = (h2, h3,· · · , hn)^T with hi = u₁− z1

u^p−1₁ |ui|^p−2ui, ∀i = 2, · · · , n.

Noting that

∥¯h∥q =  

u1− z1

u^p₁⁻¹  

( _n

∑

i=2

|ui|^(p−1)q )¹

q

=  

u1− z1

u^p₁⁻¹  

(∥¯u∥^pp

)¹ (2.3) q

= |u1− z1|,

where the second equality holds due to ¹_p+¹_q = 1, and the last equality holds because of ∥¯u∥p = u₁. Noting that

⟨z − u, u⟩ = (z1− u1)u1+⟨¯h, ¯u⟩

= (z₁− u1)u₁+u1− z1

u^p₁⁻¹ ( _n

∑

i=2

|ui|^p )

= (z₁− u1)u₁+u₁− z1

u^p₁⁻¹ ∥¯u∥^pp

= (z₁− u1)u₁+ (u₁− z1)u₁

= 0, On the other hand,

⟨z − u, u⟩ = (z1− u1)z1+⟨¯h, ¯u⟩

= (z₁− u1)u₁− ∥¯h∥q∥¯u∥p

= (z₁− u1)u₁− |z1− u1|∥¯u∥p

= ((z1− u1)− |z1− u1|)∥¯u∥p,

where the second equality holds due to the equal case of H¨older inequality, This implies that (z₁− u1)− |z1− u1| = 0. Hence, we have z1− u1 < 0. Together with (2.3) again, this leads to ∥¯h∥q = u1 − z1, which implies z− u ∈ K^◦p. Hence, the desired result is obtained. Furthermore, the projection of z ontoKp is expressed as

in (2.2). 

In the sequel, for the sake of simplicity, we denote z⁺ := Π_Kp(z). Moreover, because K^◦_p =−K_p^∗=−Kq, we know

Π_−K∗

p(z) = Π_−Kq(z) =−Π_Kq(−z).

This together with (2.2) and the proof of Theorem 2.1 gives

(2.4) z⁻:=−ΠKq(−z) =





z, −z ∈ Kq

0, −z ∈ −K^∗_q=−Kp

v, otherwise (i.e.,−∥z2∥p <−z1 <∥z2∥q)

where v = (v1, ¯v) with ¯v = (v2, v3,· · · , vn)^T ∈ IR⁽ⁿ⁻¹⁾ satisfying

−v1 =∥¯v∥q= (|v2|^q+|v3|^q+· · · + |vn|^q)^1q and

v_i− zi− (−1)^q−1v₁− z1

v^q₁⁻¹ |vi|^q−2v_i = 0, ∀i = 2, · · · , n.

By the definition of z⁺and z⁻, it follows that⟨z⁺, z⁻⟩ = 0. Together the expression of u in (2.2) with the expression of v in (2.4) again, we obtain

(2.5)





v₁ = z₁− u1

vi = zi− ui = u1− z1

u^q₁⁻¹ |ui|^p−2ui, ∀i = 2, 3, · · · , n.

Remark 2.2. Unfortunately, from the formula (2.2) in Theorem 2.1 and the for- mula (2.4), we can not obtain the spectral factorization for z = (z₁, z₂) ∈ IR × IR⁽ⁿ⁻¹⁾. This is different from the case of second-order cone. In order to get the goal, we develop one type of factorization for z as below. Such factorization is called the spectral factorization.

Theorem 2.3 (Spectral factorization). Let z = (z1, z2) ∈ IR × IR⁽ⁿ⁻¹⁾. Then, z can be decomposed as

z = α1(z)· v⁽¹⁾(z) + α2(z)· v⁽²⁾(z),

where 





α1(z) = z1+∥z2∥p

2 α₂(z) = z₁− ∥z2∥p

2

and 









v⁽¹⁾(z) =

[ 1 w2

]

v⁽²⁾(z) =

[ 1

−w2

]

with w2 = _∥z^z2

2∥^p when z2 ̸= 0; while w2 being an arbitrary element satisfying

∥w2∥p = 1 when z2 = 0.

Proof. For z₂̸= 0, we define eu(z) :=

[ τ∥z2∥p

τ z2

]

∈ ∂Kp such that eu(z) − z ∈ ∂Kp, where τ is an undetermined coefficient. From eu(z) − z ∈ Kp, we have

τ∥z2∥p− z1 =∥(τ − 1)z2∥p

which yields

τ = z1+∥z2∥p

2∥z2∥p

. This further implies

eu(z) =







(z₁+∥z2∥p

2∥z2∥p

)

∥z2∥p

(z₁+∥z2∥p

2∥z2∥p

) z₂





 .

Therefore, we can rewrite z as z = eu(z) + (z − eu(z))

=







(z1+∥z2∥p

2∥z2∥p

)

∥z2∥p

(z₁+∥z2∥p

2∥z2∥p

) z₂





 +







(z1− ∥z2∥p

2∥z2∥p

)

∥z2∥p

(∥z2∥p− z1

2∥z2∥p

) z₂







=

(z1+∥z2∥p

2

) [ 1

z2

∥z²∥^p

] +

(z1− ∥z2∥p

2

) [ 1

−_∥z^z₂²_∥p ]

:= 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 ∥w2∥p = 1. Then, the desired factor-

ization holds. 

Remark 2.4. Theorem 2.3 can be proved by verifying the equality directly. Nonethe- less, we provide the constructive way to show how to obtain v¹(z), v²(z) and α1(z), α₂(z). Moreover, from Theorem 2.3, we also know that α₁(z)≥ α2(z).

As a consequence of Theorem 2.3 and Remark 2.4, we have the following corollary.

Corollary 2.5. 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 2.3. Then, v⁽ⁱ⁾(z)∈ Kp for i = 1, 2. Moreover, the following hold

z∈ Kp ⇐⇒ α2(z)≥ 0.

3. Concluding remarks

In this short paper, we have characterized the projection formula of any element z onto p-order cone, and have established one type of spectral factorization associated with p-order cone. As mentioned, this expression will be key bricks for further analysis and study about p-order cone optimization.

One may ask what the advantages and disadvantages of the spectral factorization are? To answer this question, we say a few words for this point. The advantage of the spectral factorization is that the vectors v⁽ⁱ⁾(z) (i = 1, 2) both lie in Kp, which implies that any z in IRⁿcan be expressed by two vectors in p-order coneKp. However, to the contrast, this factorization for z is not an orthogonal decomposition, which is different from the case in the second-order cone setting.

References

[1] E. D. Andersen, C. Roos and T. Terlaky, Notes on duality in second order and p-order cone optimization, Optimization 51 (2002), 627–643.

[2] D. P. Bertsekas, A. Ned´c and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.

[3] Y.-L. Chang, C.-Y. Yang and J.-S. Chen, Smooth and nonsmooth analysis of vector-valued functions associated with circular cones, Nonlinear Anal. TMA 85 (2013), 160–173.

[4] J.-S. Chen, Conditions for error bounds and bounded level sets of some merit functions for the second-order cone complementarity problem, J. Optim. Theory Appl. 135 (2007), 459–473.

[5] J.-S. Chen and S.-H. Pan, A entropy-like proximal algorithm and the exponential multiplier method for symmetric cone programming, Comput. Optim. Appl. 47 (2010), 477–499.

[6] J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of second-order cone complementarity problem, Math. Program., Ser. B 104 (2005), 293–327.

[7] A. Pinto Da Costa and A. Seeger, Numerical resolution of cone-constrained eigenvalue prob- lems, Comp. Appl. Math. 28 (2009), 37–61.

[8] M. D¨ur, Copositive programming- a survey, in Recent Advances in Optimization and its Ap- plications in Engineering, edited by M. Diehl et al, Springer-Verlag, Berlin, 2010.

[9] U. Faraut and A. Kor´anyi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, 1994.

[10] F. Glineur and T. Terlaky, Conic formulation for lp-norm optimization, J. Optim. Theory Appl. 122 (2004) 285–307.

[11] J.-J. Moreau, D´ecomposition orthogonale d’un espace hilbertien selon deux cˆones mutuellement polaires, Comptes Rendus de l’Acad´emie des Sciences 255 (1962) 238–240.

[12] S.-H. Pan and J.-S. Chen, An R-linearly convergent nonmonotone derivative-free method for symmetric cone complementarity problems, Advan. Model. Optim. 13 (2011) 185–211.

[13] S.-H. Pan, Y.-L. Chang, and J.-S. Chen, Stationary point conditions for the FB merit function associated with symmetric cones, Oper. Res. Lett. 38 (2010) 372–377.

[14] M. Reed and B. Simon, Functional Analysis, Academic Press, New York, NY, 1980.

[15] S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to sym- metric cones, Math. Program. 96 (2003) 409–438.

[16] D. Sun and J. Sun, L¨owner’s operator and spectral functions in Euclidean Jordan algebras, Math. Oper. Res 33 (2008) 421–445.

[17] G. L. Xue and Y. Y. Ye, An efficient algorithm for minimizing a sum of P-norm, SIAM J.

Optim. 10 (2000) 551–579.

[18] J.-C. Zhou and J.-S. Chen, Properties of circular cone and spectral factorization associated with circular cone, J. Nonlinear Convex Anal. 14 (2013) 807–816.

Manuscript received April 1, 2016

X.-H. Miao

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P.R. China E-mail address: xinhemiao@tju.edu.cn

N. Qi

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P.R. China E-mail address: qinuo@163.co

J.-S. Chen

Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan E-mail address: jschen@math.ntnu.edu.tw