to appear in Journal of Nonlinear and Convex Analysis, 2016
Projection formula and one type of spectral factorization associated with p-order cone
Xinhe Miao 1
Department of Mathematics School of Science Tianjin University Tianjin 300072, P.R. China
Nuo Qi 2
Department of Mathematics School of Science Tianjin University Tianjin 300072, P.R. China
Jein-Shan Chen 3 Department of Mathematics National Taiwan Normal University
Taipei 11677, Taiwan
Abstract. In this short paper, we establish the projection formula associated with p- order cone and further discover one type of spectral factorization 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.
1The author’s work is supported by National Natural Science Foundation of China (No. 11471241).
E-mail: [email protected]
2E-mail: [email protected]
3Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan. E-mail: [email protected]
In general, non-symmetric cones include p-order cone [1, 17], circular cone [3, 7, 18], Lp 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 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 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 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.
The p-order cone in Rn, which is a generalization of the second-order cone [4, 6], is defined as
Kp :=
x ∈ Rn
x1 ≥
n
X
i=2
|xi|p
!1p
(p > 1). (1)
If we write x := (x1, x2) ∈ R × R(n−1), the p-order cone Kp can be equivalently expressed as
Kp =x = (x1, x2) ∈ R × R(n−1)| x1 ≥ kx2kp , (p > 1).
The pictures of three different cones Kp in R3 are depicted in Figure 1.
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 Kp is a convex cone and its dual cone is given by
K∗p =
y ∈ Rn
y1 ≥
n
X
i=2
|yi|q
!1q
or equivalently
K∗p =y = (y1, y2) ∈ R × R(n−1)| y1 ≥ ky2kq = Kq,
(a) 2-order cone (b) 3-order cone (c) 10-order cone
Figure 1: Three different p-order cones in R3.
where q > 1 and satisfies 1p +1q = 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, 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 ∈ 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 Kp, 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 Kp is given by
ΠKp(z) =
z, z ∈ Kp
0, z ∈ −K∗p = −Kq
u, otherwise (i.e., −kz2kq < z1 < kz2kp)
(3)
where u = (u1, ¯u) with ¯u = (u2, u3, · · · , un)T ∈ R(n−1) satisfying u1 = k¯ukp = (|u2|p+ |u3|p+ · · · + |un|p)1p and
ui− zi+u1− z1
up−11 |ui|p−2ui = 0, ∀i = 2, · · · , n.
Proof. From Projection Theorem [2, Prop. 2.2.1], we know that, for every z ∈ Rn, a vector u ∈ Kp is equal to the projection point ΠKp(z) if and only if
u ∈ Kp, z − u ∈ Kp◦ 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 ∈ ∂Kp. Moreover, we have
z − u = z1− u1 z2− ¯u
:= z1− u1 h¯
, where ¯h = (h2, h3, · · · , hn)T with
hi = u1− z1
up−11 |ui|p−2ui, ∀i = 2, · · · , n.
Noting that
k¯hkq =
u1− z1 up−11
n
X
i=2
|ui|(p−1)q
!1q
=
u1− z1
up−11
k¯ukpp1q
(4)
= |u1− z1|,
where the second equality holds due to 1p +1q = 1, and the last equality holds because of k¯ukp = u1. Noting that
hz − u, ui = (z1− u1)u1+ h¯h, ¯ui
= (z1− u1)u1+u1− z1 up−11
n
X
i=2
|ui|p
!
= (z1− u1)u1+u1− z1 up−11 k¯ukpp
= (z1− u1)u1+ (u1− z1)u1
= 0, On the other hand,
hz − u, ui = (z1− u1)z1 + h¯h, ¯ui
= (z1− u1)u1− k¯hkqk¯ukp
= (z1− u1)u1− |z1 − u1|k¯ukp
= ((z1− u1) − |z1− u1|)k¯ukp,
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 (4) again, this leads to k¯hkq = u1− z1, which implies z − u ∈ Kp◦. Hence, the desired result is obtained.
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). Moreover, because K◦p = −Kp∗ = −Kq, we know
Π−K∗p(z) = Π−Kq(z) = −ΠKq(−z).
This together with (3) and the proof of Theorem 2.1 gives
z− := −ΠKq(−z) =
z, −z ∈ Kq
0, −z ∈ −K∗q = −Kp
v, otherwise (i.e., −kz2kp < −z1 < kz2kq)
(5)
where v = (v1, ¯v) with ¯v = (v2, v3, · · · , vn)T ∈ R(n−1) satisfying
−v1 = k¯vkq = (|v2|q+ |v3|q+ · · · + |vn|q)1q and
vi− zi− (−1)q−1v1− z1
v1q−1 |vi|q−2vi = 0, ∀i = 2, · · · , n.
By the definition of z+ and z−, it follows that hz+, z−i = 0. Together the expression of u in (3) with the expression of v in (5) again, we obtain
v1 = z1− u1
vi = zi− ui = u1− z1
uq−11 |ui|p−2ui, ∀i = 2, 3, · · · , n. (6) Remark 2.1 Unfortunately, from the formula (3) in Theorem 2.1 and the formula (5), we can not obtain the spectral factorization for z = (z1, z2) ∈ R × R(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 2.2 (Spectral factorization) 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+ kz2kp 2 α2(z) = z1− kz2kp
2 and
v(1)(z) =
1 w2
v(2)(z) =
1
−w2
with w2 = kzz2
2kp when z2 6= 0; while w2 being an arbitrary element satisfying kw2kp = 1 when z2 = 0.
Proof. For z2 6= 0, we define eu(z) := τ kz2kp
τ z2
∈ ∂Kp such that u(z) − z ∈ ∂Ke p, where τ is an undetermined coefficient. From u(z) − z ∈ Ke p, we have
τ kz2kp− z1 = k(τ − 1)z2kp which yields
τ = z1+ kz2kp 2kz2kp . This further implies
u(z) =e
z1+ kz2kp 2kz2kp
kz2kp
z1+ kz2kp 2kz2kp
z2
.
Therefore, we can rewrite z as
z = eu(z) + (z −u(z))e
=
z1+ kz2kp 2kz2kp
kz2kp
z1+ kz2kp 2kz2kp
z2
+
z1− kz2kp 2kz2kp
kz2kp
kz2kp− z1 2kz2kp
z2
= z1+ kz2kp 2
"
1
z2
kz2kp
#
+ z1− kz2kp 2
"
1
−kzz2
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 kw2kp = 1. Then, the desired factorization holds. 2
Remark 2.2 Theorem 2.2 can be proved by verifying the equality directly. Nonetheless, we provide the constructive way to show how to obtain v1(z), v2(z) and α1(z), α2(z).
Moreover, from Theorem 2.2, we also know that α1(z) ≥ α2(z).
As a consequence of Theorem 2.2 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 2.2. 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 Rn can be expressed by two vectors in p-order cone Kp. 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.
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