AND DECOMPOSITION ASSOCIATED WITH ELLIPSOIDAL CONES
YUE LU∗AND JEIN-SHAN CHEN†
Abstract. Non-symmetric cones have long been mysterious to optimization re- searchers because of no unified analysis technique to handle these cones. Nonethe- less, by looking into symmetric cones and non-symmetric cones, it is still possible to find relations between these kinds of cones. This paper tries an attempt to this aspect and focuses on an important class of convex cones, the ellipsoidal cone.
There are two main reasons for it. The ellipsoidal cone not only includes the well known second-order cone, circular cone and elliptic cone as special cases, but also it can be converted to a second-order cone by a transformation and vice versa.
With respect to the ellipsoidal cone, we characterize its dual cone, variational geometry, the projection mapping, and the decompositions. We believe these results may provide a fundamental approach on tackling with other unfamiliar non-symmetric cone optimization problems.
1. Introduction
During the past decades, symmetric cones associated with the Euclidean space Rn, including nonnegative octant Rn+ and second-order cone Kn, have been exten- sively studied from different views [1, 7, 8, 9, 10, 14]. With the developments of modern optimization, more and more non-symmetric cones appears in plenty of applications. However, due to the lack of a unified technical tool like the Euclidean Jordan Algebra (EJA) for symmetric cones, it seems no systematic study on non- symmetric cones. Until now, only a small group of them have been investigated thoroughly such as circular cone [6, 21] and p-order cone [3, 13, 20]. In this paper, we focus on another interesting type of non-symmetric cone, the ellipsoidal cone, which not only contains a few well known convex cones but also forms a bridge between symmetric cones and non-symmetric cones.
Before the formal discussion, we recall some definitions that will be used in the sequel. A set K ⊆ Rn is a cone if αK ⊆ K for all α ≥ 0. In addition, suppose that K is closed, convex, pointed (i.e., K ∩ (−K) = {0}) and has a nonempty interior, we call K a proper cone. Let Sn be the collection of all real symmetric matrices in the n dimensional matrix space Rn×n. The proper cone K is called ellipsoidal,
2010 Mathematics Subject Classification. 90C25.
Key words and phrases. Non-symmetric cones, Ellipsoidal cones, Variational geometry, Projection.
∗The author’s work is supported by National Natural Science Foundation of China (Grant Number: 11601389).
†Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan.
1
denoted by KE, if there exists a nonsingular matrix Q ∈ Snwith exact one negative eigenvalue λn∈ R and corresponding eigenvector un∈ Rn such that
(1.1) KE := {x ∈ Rn| xTQx ≤ 0, and uTnx ≥ 0}, where the matrix Q admits the orthogonal decomposition Q = Pn
i=1λiuiuTi with eigen-pairs (λi, ui) for i = 1, 2, · · · , n satisfying the conditions
(1.2) λ1 ≥ λ2 ≥ · · · ≥ λn−1> 0 > λn and uTi uj =
1 if i = j, 0 if i 6= j.
Example 1.1. Let Q =
1 2 −√
2 −12
−√
2 0 −√
2
−12 −√
2 12
∈ S3 and un=
1
√2 2 21 2
∈ R3.
Figure 1 shows an ellipsoidal cone in R3 generated by these parameters Q and un.
Figure 1. The graph of a 3-dimensional ellipsoidal cone.
The history of the ellipsoidal cone dates back to Stern and Wolkowicz’s research [18] on characterizing conditions for the spectrum of a given matrix A ∈ Rn×nunder the existence of an ellipsoidal cone. After that, they also provide an equivalent description on exponential nonnegativity for the second-order cone [19], which is related to the solution set of a linear autonomous system ˙ξ = Aξ and further applied to modelling rendezvous of the multiple agents system and measuring dispersion in directional datasets, see [4, 17] for more details.
On the other hand, the ellipsoidal cone KEincludes the second-order cone, circular cone and elliptic cone as special cases. To see this, we verify them as below.
Example 1.2. (a) Second-order cone [7, 8]:
Kn:=(¯x, xn) ∈ Rn−1× R | k¯xk ≤ xn ,
where k¯xk stands for the Euclidean norm of ¯x ∈ Rn−1. Clearly, Kn is an ellipsoidal cone with
Q =
In−1 0
0 −1
and un= en,
where In−1 denotes the identity matrix of order n − 1 and en is the n-th column vector of In.
(b) Circular cone [6, 21]:
Lθ:=(¯x, xn) ∈ Rn−1× R | k¯xk ≤ xntan θ , where θ ∈ (0,π 2).
It is not hard to see that the circular cone Lθ is also a special case of ellipsoidal cone with
Q =
In−1 0 0 − tan2θ
and un= en. (c) Elliptic cone [2]:
KMn :=(¯x, xn) ∈ Rn−1× R | kM ¯xk ≤ xn ,
where M is any nonsingular matrix of order n − 1. Obviously, the elliptic cone KnM can be viewed as an ellipsoidal cone by letting
Q =
MTM 0
0 −1
and un= en.
Remark 1.3. We elaborate more about the aforementioned convex cones. In fact, there hold the relations 1 as follows:
Kn⊆ Lθ⊆ KnM ⊆ KE ⊆ Rn.
Hence, the ellipsoidal cone is a natural generalization of the second-order cone, circular cone and elliptic cone, see Figure 2 for illustration.
Figure 2. The relations among Kn, Lθ, KnM, and KE.
Unlike symmetric cone optimization, there is no unified framework for dealing with non-symmetric cone optimization. The experience and techniques for non- symmetric cone optimization are very limited. The paper aims to find a way which can help understanding more about non-symmetric cones. With this goal, we focus on the ellipsoidal cone KE given as in (1.1). There are two main reasons for it. The first reason is that the ellipsoidal cone includes the well known second-order cone,
1The first inclusion comes from [21], the second one is established in [2].
circular cone and elliptic cone as special cases, as mentioned above. The second rea- son is indeed more important, through a transformation, the ellipsoidal cone and the second-order cone can be converted to each other, see Theorem 2.1 in Section 2 for more details. This is a key which may open a new vision because it connects symmetric cones and non-symmetric cones together. In order to pave a way to its corresponding non-symmetric cone optimization, we explore the interior and bound- ary sets, the dual cone, variational geometry including the tangent cone and the normal cone, the projection mapping and the decompositions with respect to the ellipsoidal cone. We believe these contexts will provide some fundamental bricks to build a systematic optimization theory related to the ellipsoidal cone. Moreover, with the connection (see Theorem 2.1 in Section 2) to second-order cone, some analysis techniques may be carried to the territory of mysterious non-symmetric cones. In other words, the links between these two types of cones may provide a new perspective view on how to deal with unfamiliar non-symmetric cones thor- oughly, which is an important contribution to the development of non-symmetric cone optimization.
The remainder of this paper is organized as follows. In Section 2, we develop the theory on the dual of the ellipsoidal cone. In Section 3, we proceed with the study on its variational geometry including the tangent cone and the normal cone.
As a byproduct, the explicit expressions of its interior and boundary sets are also established. Sections 4 and 5 are devoted to discovering a detailed exposition of the projection mapping and the decompositions with respect to the ellipsoidal cone, respectively. Finally, we have some concluding remarks and say a few words about future directions in Section 6.
2. The dual of the ellipsoidal cone
In this section, we develop the theory regarding the dual of the ellipsoidal cone KE, which is denoted by K∗E, in other words,
KE∗ := {y ∈ Rn| hx, yi ≥ 0, ∀x ∈ KE} ,
where h·, ·i stands for the standard Euclidean inner product defined on Rn. In what follows, we write the matrices U ∈ Rn×n and Λ ∈ Sn to respectively represent (2.1) U := u1 u2 · · · un, Λ := diag (λ1, λ2, · · · , λn) .
The orthogonal decomposition of Q given as in (1.2) implies Q = U ΛUT, and UTU = U UT = In.
For any given vector x ∈ KE, due to the orthogonal property of the sets {ui}ni=1, there exists a vector α := [α1, α2, · · · , αn]T ∈ Rn such that
x = U α, xTQx = αTUTQU α = αTΛα =
n
X
i=1
λiα2i, uTnx = uTn
n
X
i=1
αiui
!
= αn. The set KE can be rewritten as the form U ∆α with
(2.2) ∆α:=
(
α ∈ Rn
n
X
i=1
λiα2i ≤ 0, αn≥ 0 )
.
If we take λi = 1 for i = 1, 2, · · · , n − 1 and λn = −1, then the set ∆α reduces to second-order cone
(2.3) Kn:=
(
α ∈ Rn
n−1
X
i=1
α2i ≤ αn2, αn≥ 0 )
. For any α ∈ ∆α, in light of the relation (1.2) for {λi}ni=1, we have
α ∈ ∆α ⇐⇒
n
X
i=1
λiα2i ≤ 0 and αn≥ 0
⇐⇒
n−1
X
i=1
λ1/2i αi2
≤
(−λn)1/2αn2
and (−λn)1/2αn≥ 0
⇐⇒
λ1/21 α1, λ1/22 α2, · · · , λ1/2n−1αn−1, (−λn)1/2αnT
∈ Kn
⇐⇒ α ∈ DKn
where D is a n × n diagonal matrix in the form of (2.4) D := diag
(λ1)−1/2, (λ2)−1/2, · · · , (λn−1)−1/2, (−λn)−1/2
.
Thus, the relation between ∆α and Kn is described as ∆α = DKn, which implies (2.5) KE = U ∆α = U DKn= T Kn, where T := U D.
It is clear that the matrix T ∈ Rn×n is nonsingular. The relation (2.5) between the ellipsoidal cone KE and the second-order cone Kn is depicted in Figure 3.
Figure 3. The graphs of a 3-dimensional ellipsoidal cone and a 3- dimensional second-order cone.
In fact, similar idea has been used in [18, Proposition 2.3] and [19, Lemma 2.2].
According to the relation (2.5), we can derive
K∗E = {y ∈ Rn| hx, yi ≥ 0, ∀x ∈ KE}
= {y ∈ Rn| hT z, yi ≥ 0, ∀z ∈ Kn}
= y ∈ Rn| TTy ∈ (Kn)∗ = Kn
= (TT)−1Kn,
where
TT = (U D)T = DTUT = DU−1= D2D−1U−1= D2T−1, (TT)−1 = D2T−1−1
= T D−2 = U D−1.
In addition, by denoting |Λ| := diag(|λ1|, |λ2|, · · · , |λn|) ∈ Sn, which means |Λ| = D−2, then (TT)−1= T |Λ| and the dual cone KE∗ can be further expressed as (2.6) KE∗ = (TT)−1Kn= U D−1Kn= T |Λ|Kn,
which is displayed in Figure 4.
Figure 4. The graphs of the dual of a 3-dimensional ellipsoidal cone and a 3-dimensional second-order cone.
Likewise, we deduce the double dual K∗∗E of the ellipsoidal cone KE as follows:
K∗∗E = {x ∈ Rn| hx, yi ≥ 0, ∀y ∈ K∗E}
= {y ∈ Rn| hx, T |Λ|zi ≥ 0, ∀z ∈ Kn}
= y ∈ Rn: (T |Λ|)Tx ∈ (Kn)∗ = Kn
= (T |Λ|)T−1
Kn, where
(T |Λ|)T−1
= |Λ|TT−1
= (TT)−1|Λ|−1 = T |Λ||Λ|−1= T.
This implies that the connection between KE and its double dual K∗∗E is K∗∗E = T Kn= KE,
see Figure 5 for illustration.
To sum up these discussions, we state the relations among the ellipsoidal cone KE, its dual cones K∗E, K∗∗E , and the second-order cone Knin the following theorem.
Theorem 2.1. Let KE and Kn be defined as in (1.1) and (2.3), respectively. Then, we have
(a) KE = T Kn and Kn= T−1KE;
(b) K∗E = T |Λ|Kn and Kn= |Λ|−1T−1K∗E; (c) K∗E = T |Λ|T−1KE and KE∗∗= KE.
The next theorem presents an explicit description of the dual cone K∗E.
Figure 5. The graphs of a 3-dimensional ellipsoidal cone and its dual.
Theorem 2.2. Let KE be an ellipsoidal cone defined as in (1.1). The dual cone KE∗ is equivalently expressed by
(2.7) KE∗ =y ∈ Rn| yTQ−1y ≤ 0, uTny ≥ 0 .
Proof. In view of (2.6), it suffices to show that the set of the right-hand side in (2.7) is equal to the set U D−1Kn, where the matrix U is defined as in (2.1). Using (1.2) for {λi}ni=1, for any given y ∈ Rn there exists a vector β ∈ Rn such that y = U β, which yields
yTQ−1y ≤ 0 and uTny ≥ 0
⇐⇒
n
X
i=1
λ−1i βi2 ≤ 0 and βn≥ 0
⇐⇒
n−1
X
i=1
λ−1/2i βi
2
≤
(−λn)−1/2βn
2
and (−λn)−1/2βn≥ 0
⇐⇒
λ−1/21 β1, λ−1/22 β2, · · · , λ−1/2n−1 βn−1, (−λn)−1/2βn
T
∈ Kn
⇐⇒ β ∈ D−1Kn.
Then, the desired result follows.
As a byproduct, we denote KE◦ the polar of the ellipsoidal cone KE. Applying KE◦ = −K∗E and (2.7), the exact form of the polar cone KE◦ is given by
(2.8) K◦E :=y ∈ Rn| yTQ−1y ≤ 0, uTny ≤ 0 .
Remark 2.3. By applying (2.7), the duals of the circular cone and the elliptic cone, denoted by L∗θ and (KnM)∗ respectively, can be characterized as
L∗θ :=(¯yn−1, yn) ∈ Rn−1× R | k¯yn−1k ≤ yncot θ = Lπ
2−θ, with θ ∈ (0,π 2), (KnM)∗ :=
n
(¯yn−1, yn) ∈ Rn−1× R
M−1T
¯ yn−1
≤ yno
= Kn(M−1)T,
where ¯yn−1 := (y1, y2, · · · , yn−1)T ∈ Rn−1. Same arguments can be applied to the polar cone KE◦ given as in (2.8). In other words, the polar of the circular cone L◦θ and the elliptic cone (KnM)◦ are described as
L◦θ :=(¯yn−1, yn) ∈ Rn−1× R | k¯yn−1k ≤ −yncot θ , (KnM)◦:=n
(¯yn−1, yn) ∈ Rn−1× R
M−1T
¯ yn−1
≤ −yno . 3. The variational geometry of the ellipsoidal cone
In this section, we pay attention to the variational geometry of the ellipsoidal cone KE, which includes the tangent cone TKE(x) and the normal cone NKE(x). From the convexity of KE and the definitions of variational geometry in convex analysis [15], we have
TKE(x) := {d ∈ Rn| ∃ tn↓ 0, dist(x + tnd, KE) = o(tn)} , NKE(x) := {v ∈ Rn| hv, di ≤ 0, ∀d ∈ TKE(x)} ,
where dist(x, S) denotes the distance from x ∈ Rnto the set S, that is, dist(x, S) := min
y∈Skx − yk.
The following theorem presents characterizations of TKE(x) and NKE(x) in terms of those tangent cone and normal cone for the second-order cone Kn.
Theorem 3.1. Let T ∈ Rn×n be a nonsingular matrix defined as in (2.5). For any x ∈ KE, there exists a vector α = T−1x ∈ Kn such that
TKE(x) = T TKn(α) and NKE(x) = T |Λ| NKn(α).
Proof. For any d ∈ Rn, we denote p := T−1d. Then, applying Theorem 2.1(a) yields
kT k−1dist (x + tnd, KE)
= kT k−1dist (x + tnd, T Kn)
= kT k−1 min
y∈Knkx + tnd − T yk
= kT k−1 min
y∈Kn
T (T−1x + tnT−1d − y)
≤ min
y∈Knkα + tnp − yk
= dist (α + tnp, Kn)
= dist α + tnp, T−1KE
≤ kT−1k min
w∈KE
kx + tnd − wk
= kT−1k dist (x + tnd, KE) .
On the other hand, from definition, there exists tn↓ 0 such that dist(x + tnd, KE) = o(tn) if and only if d ∈ TKE(x). Thus, we know dist(α + tnp, Kn) = o(tn), which yields T−1d = p ∈ TKn(α). The opposite inclusion can be achieved in the similar way. In summary, we have shown TKE(x) = T TKn(α).
As for the part of NKE(x), we have
NKE(x) = {v ∈ Rn| hv, di ≤ 0, ∀d ∈ TKE(x)}
= {v ∈ Rn| hv, T pi ≤ 0, ∀p ∈ TKn(α)}
= v ∈ Rn| hTTv, pi ≤ 0, ∀p ∈ TKn(α)
= v ∈ Rn| TTv ∈ NKn(α)
= n
v ∈ Rn| v ∈ TT−1
NKn(α) o
.
Together with the fact (TT)−1= T |Λ|, it follows that NKE(x) = T |Λ| NKn(α). For convenience, we also denote int KE and bd KE the interior and the boundary of the ellipsoidal cone KE, respectively. Then, it follows from Theorem 2.1 and [15, Theorem 6.6] that
(3.1) int KE = T (int Kn) and bd KE = T (bd Kn).
This together with the definition of Kn implies that
int Kn:=α ∈ Rn| αTQnα < 0, eTnα > 0 , bd Kn:=α ∈ Rn| αTQnα = 0, eTnα > 0 ∪ {0}, where the matrix Qn is given by
(3.2) Qn:=
In−1 0
0 −1
∈ Sn.
For any given x ∈ KE and its corresponding vector α = T−1x ∈ Kn, from (3.1), we obtain
int KE =n
x ∈ Rn
T−1xT
Qn(T−1x) < 0, eTnT−1x > 0o , bd KE =n
x ∈ Rn
T−1xT
Qn(T−1x) = 0, eTnT−1x > 0o
∪ {0}.
Due to the definitions of T and Λ as in (2.1) and (2.4), we also have some useful transformations
T−1T
QnT−1= D−1U−1T
QnD−1U−1= U D−1QnD−1UT = U ΛUT = Q, eTnT−1= eTn(U D)−1= eTnD−1U−1= eTnD−1UT = (−λn)1/2uTn.
With the above discussions, we provide the explicit expressions for int KE and bd KE.
Theorem 3.2. Let KE be an ellipsoidal cone defined as in (1.1). Then, the interior and the boundary of KE are respectively given by
int KE =x ∈ Rn| xTQx < 0, uTnx > 0 , bd KE =x ∈ Rn| xTQx = 0, uTnx > 0 ∪ {0}.
Remark 3.3. Similar to Remark 2.3, we conclude from Theorem 3.2 that the interior and the boundary of Lθ and KnM are described by
int Lθ =(¯xn−1, xn) ∈ Rn−1× R | k¯xn−1k < xntan θ ,
bd Lθ =(¯xn−1, xn) ∈ Rn−1× R | k¯xn−1k = xntan θ > 0 ∪ {0}, int KnM =(¯xn−1, xn) ∈ Rn−1× R | kM ¯xn−1k < xn ,
bd KnM =(¯xn−1, xn) ∈ Rn−1× R | kM ¯xn−1k = xn> 0 ∪ {0}, where ¯xn−1:= (x1, x2, · · · , xn−1)T ∈ Rn−1.
To present the tangent cone and normal cone, we first recall their counterparts for second-order cone Kn, which can be found in [5]:
TKn(α) =
Rn if α ∈ int Kn,
Kn if α = 0,
{p ∈ Rn| pTQnα ≤ 0} if α ∈ bd Kn\ {0}, NKn(α) =
{0} if α ∈ int Kn,
−Kn if α = 0,
R+(Qnα) if α ∈ bd Kn\ {0},
where Qn is defined as in (3.2) and R+(Qnα) stands for the set {ηQnα | η ≥ 0}.
Combining Theorem 3.1, Theorem 3.2 with the definitions of TKn(α) and NKn(α), we present the expressions of tangent cone and normal cone regarding KE as below.
Theorem 3.4. For any given x ∈ Rn, the tangent cone and normal cone with respect to the ellipsoidal cone KE at x are described by
TKE(x) =
Rn if x ∈ int KE,
KE if x = 0,
{d ∈ Rn| dTQx ≤ 0} if x ∈ bd KE\ {0}, NKE(x) =
{0} if x ∈ int KE, K◦E if x = 0,
R+(Qx) if x ∈ bd KE\ {0}, where R+(Qx) := {ηQx | η ≥ 0}.
Remark 3.5. We also present the following two special cases when KE reduces to Lθ or KMn . In fact, if take
Q =
In−1 0 0 − tan2θ
or Q =
MTM 0
0 −1
,
where M is any given nonsingular matrix of order n − 1 as in Example 1.2(c). Then, the tangent cone and normal cone of Lθ and KMn are respectively given by
TLθ(x) =
Rn if x ∈ int Lθ, Lθ if x = 0,
ΞLθ if x ∈ bd Lθ\ {0}, NLθ(x) =
{0} if x ∈ int Lθ,
L◦θ if x = 0,
R+(¯x, −xntan2θ) if x ∈ bd Lθ\ {0}
where ΞLθ :=( ¯dn−1, dn) ∈ Rn−1× R | h ¯dn−1, ¯xn−1i − dnxntan2θ ≤ 0 and ¯dn−1 :=
(d1, d2, · · · , dn−1)T ∈ Rn−1. Similarly, we also obtain TKn
M(x) =
Rn if x ∈ int KnM, KnM if x = 0,
ΞKnM if x ∈ bd KnM \ {0}, NKn
M(x) =
{0} if x ∈ int KnM,
(KnM)◦ if x = 0,
R+ MTM ¯xn−1, −xn
if x ∈ bd KMn \ {0}, where ΞKnM :=( ¯dn−1, dn) ∈ Rn−1× R | hM ¯dn−1, M ¯xn−1i − dnxn≤ 0 .
4. The projection onto the ellipsoidal cone
In this section, we focus on the projection of any vector y ∈ Rnonto the ellipsoidal cone KE. In other words, the following minimization problem is considered:
(4.1) min 12kx − yk2
s.t. x ∈ KE.
From the first-order optimality condition (e.g. [16, Theorem 6.12]), it is known that 0 ∈ x − y + NKE(x), which implies
x = (I + NKE)−1(y) := ΠKE(y), where ΠKE(y) denotes the projection of y onto KE.
On the other hand, from the orthogonal property (2.1) for the set {ui}ni=1, there exist α ∈ IRn and β ∈ IRnsuch that x = U α and y = U β. For simplicity, we write
Λ := diag( ¯Λn−1, λn) ∈ Sn, ¯Λn−1:= diag(λ1, λ2, · · · , λn−1) ∈ Sn−1, α := ( ¯αn−1, αn) ∈ Rn−1× R, ¯αn−1:= (α1, α2, · · · , αn−1)T ∈ Rn−1, β := ( ¯βn−1, βn) ∈ Rn−1× R, ¯βn−1:= (β1, β2, · · · , βn−1)T ∈ Rn−1.
The problem (4.1) is equivalent to solving the elliptic optimization problem with respect to the variables ( ¯αn−1, αn) ∈ IRn−1× IR, that is,
(4.2) min 12 k ¯αn−1− ¯βn−1k2+ (αn− βn)2 s.t.
¯M ¯αn−1 ≤ αn,
where ¯M is a diagonal matrix of order n − 1 in the form of (4.3) M := diag¯
s λ1 (−λn),
s λ2
(−λn), · · · , s
λn−1 (−λn)
! . It is easy to verify that the matrix ¯M also satisfies the equation (4.4) Λ¯n−1+ λnM¯TM = 0.¯
Theorem 4.1. Let KE be an ellipsoidal cone defined as in (1.1) and y ∈ Rn. Then, the projection of y onto KE is given by
ΠKE(y) =
y if y ∈ KE, 0 if y ∈ K◦E, U α otherwise,
where the matrix U ∈ Rn×n is defined as in (2.1), the vector α = ( ¯αn−1, αn) ∈ Rn−1× R is the optimal solution of (4.2) and has the following forms:
(a) If βn= 0, then
¯
αn−1= In−1+ ¯MTM¯−1β¯n−1, αn= k ¯M In−1+ ¯MTM¯−1β¯n−1k.
(b) If βn6= 0, then
¯
αn−1= In−1− η0λnM¯TM¯−1β¯n−1, αn= βn
1 + η0λn,
where the matrix ¯M ∈ Sn−1 is defined as in (4.3) and the scalar η0 ∈ R satisfies the relations
(4.5) η0∈
(0, −1/λn) if βn> 0,
(−1/λn, +∞) if βn< 0 and
n
X
i=1
λiβi2
(1 + η0λi)2 = 0
with β ∈ Rn lying outside of the setβ ∈ Rn| k ¯M ¯βn−1k ≤ βn and its polar
β ∈ Rn| k ¯M−1β¯n−1k ≤ βn .
Proof. By checking the definition of KE or K◦E, it is trivial to obtain the first two cases. It remains to discuss the case of y /∈ KE ∪ KE◦. From Theorem 3.4, there exists a scalar η0 > 0 such that
(4.6) x = ΠKE(y) ∈ bdKE\ {0} and 0 = x − y + η0Qx.
We set x = U α and y = U β as earlier, where U ∈ Rn×n is defined as in (2.1). Then, the relations (4.6) are equivalent to the system with respect to the variables α ∈ Rn and η0 ∈ R as follows:
(4.7)
β = (I + η0Λ)α, αTΛα = 0, αn> 0, η0> 0.
It turns out that the system (4.7) can be rewritten in the following form
(4.8)
β¯n−1= (In−1+ η0Λ¯n−1) ¯αn−1, βn= (1 + η0λn)αn,
¯
αTn−1Λ¯n−1α¯n−1+ λnα2n= 0, αn> 0, η0 > 0.
Next, we proceed to show that the following two subcases hold for the system (4.8).
(a) If βn = 0, then η0 = −λ1
n > 0. From the system (4.8) and the equation (4.4), we have
¯ αn−1=
In−1− 1 λnΛ¯n−1
−1
β¯n−1= In−1+ ¯MTM¯−1β¯n−1,
αn= α¯Tn−1Λ¯n−1α¯n−1
−λn
!1/2
=
M I¯ n−1+ ¯MTM¯−1β¯n−1
.
(b) If βn 6= 0, from the second and fourth relations in (4.8), we know η0 6= −λ1
n. The first two relations in (4.8) and the equation (4.4) further imply that
¯
αn−1= In−1+ η0Λ¯n−1
−1β¯n−1= In−1− η0λnM¯TM¯−1β¯n−1, αn= βn
1 + η0λn
, where η0 ∈ R satisfies the condition
η0 ∈
(0, −1/λn) if βn> 0, (−1/λn, +∞) if βn< 0.
In addition, we have
¯
αTn−1Λ¯n−1α¯n−1+ λnα2n
= β¯n−1T In−1+ η0Λ¯n−1−1Λ¯n−1 In−1+ η0Λ¯n−1−1β¯n−1+ λn
βn
1 + η0λn
2
=
n
X
i=1
λiβi2 (1 + η0λi)2
and the third relation in (4.8) reduces to the equation (4.5). Since y /∈ KE∪ K◦E and y = U β, from the definitions of KE and K◦E, we obtain
β /∈ (
β ∈ Rn|
n
X
i=1
λiβi2≤ 0, βn≥ 0 )
∪ (
β ∈ Rn|
n
X
i=1
λ−1i βi2≤ 0, βn≤ 0 )
, which means that β /∈ β ∈ Rn| k ¯M ¯βn−1k ≤ βn ∪ β ∈ Rn| k ¯M−1β¯n−1k ≤ βn .
Remark 4.2. For the projection onto the ellipsoidal cone KE, we emphasize that this projection is not yet an explicit expression because it is hard to solve the equation (4.5) with respect to the variable η0∈ R in general. However, under some special cases, the equation (4.5) has closed-form solutions. For example, if we set
U = In, λi = 1(i = 1, 2, · · · , n − 1), λn= −1 or λn= − tan2θ,
which correspond to the cases of the second-order cone Kn and the circular cone Lθ. For more details about their projections, we refer the readers to [9, Proposition 3.3] and [21, Theorem 3.2].
5. The decompositions of the ellipsoidal cone
In this section, we try to express out the decompositions with respect to the ellipsoidal cone. Let KnM¯ be an elliptic cone with the matrix ¯M defined as in (4.3), i.e.,
(5.1) KnM¯ :=(¯αn−1, αn) ∈ Rn−1× R | k ¯M ¯αn−1k ≤ αn . According to [11, Remark 2.2], the dual cone of KnM¯ is defined by
(5.2) (KMn¯)∗ = {( ¯βn−1, βn) ∈ Rn−1× R | k ¯M−1β¯n−1k ≤ βn} = KnM¯−1.
It is easy to see that the ellipsoidal cone KE and its dual cone (KE)∗ can be described in terms of KnM¯ and its dual cone KnM¯−1.
Theorem 5.1. Let KE be an ellipsoidal cone defined as in (1.1) and KnM¯ be an elliptic cone defined as in (5.1). Then, we have
KE = U KMn¯, K∗E = U KnM¯−1.
Proof. For any given x ∈ KE, since {ui}ni=1 are orthogonal to each other, there exists a vector α = ( ¯αn−1, αn) ∈ Rn−1× R such that x = Uα. From the definition of KE, we have
x ∈ KE
⇐⇒ xTQx ≤ 0, uTnx ≥ 0
⇐⇒ αTΛα ≤ 0, αn≥ 0
⇐⇒ α¯Tn−1Λ¯n−1α¯n−1+ λnα2n≤ 0, αn≥ 0
⇐⇒ α¯Tn−1M¯TM ¯¯αn−1≤ α2n, αn≥ 0
⇐⇒ x = U α, α ∈ KnM¯,
which implies the relation KE = U KnM¯. One the other hand, for any given y ∈ K∗E, due to the orthogonal property of {ui}ni=1, there exists a vector β = ( ¯βn−1, βn) ∈ Rn−1× R such that y = Uβ. It follows from above that
y ∈ K∗E
⇐⇒ yTQ−1y ≤ 0, uTny ≥ 0
⇐⇒ βTΛ−1β ≤ 0, βn≥ 0
⇐⇒ β¯n−1T Λ¯−1n−1β¯n−1+ λ−1n βn2 ≤ 0, βn≥ 0
⇐⇒
n−1
X
i=1
(−λn) λi
βi2≤ βn2, βn≥ 0
⇐⇒ β¯n−1T ( ¯M−1)TM¯−1β¯n−1≤ β2n, βn≥ 0
⇐⇒ y = U β, β ∈ KnM¯−1. Therefore, we obtain K∗E = U KnM¯−1.
Inspired by recent studies on spectral factorization associated with p-order cone in [13, Theorem 2.3] or [12, Theorem 3.2], there exists one type of the decomposition for a point ( ¯αn−1, αn) ∈ Rn−1× R with respect to the elliptic cone KnM¯.
Type I:
α¯n−1 αn
=
αn+ k ¯M ¯αn−1k 2
¯ αn−1
k ¯M ¯αn−1k 1
+αn− k ¯M ¯αn−1k 2
− ¯αn−1
k ¯M ¯αn−1k 1
if ¯αn−16= 0,
αn
2
" w k ¯M wk
1
# +αn
2
−w k ¯M wk
1
if ¯αn−1= 0,
where w is any given nonzero vector in Rn−1. Focusing on the right-hand side of the Type I decomposition, we observe that the vectors
¯ αn−1
k ¯M ¯αn−1k 1
,
− ¯αn−1
k ¯M ¯αn−1k 1
,
" w k ¯M wk
1
# ,
−w k ¯M wk
1
all belong to the set KnM¯, which is different from the decomposition with respect to the circular cone Lθ established in [21, Theorem 3.1], since its associated dual cone L∗θ is involved in.
In contrast to the Type I decomposition, through importing the information of its dual cone KnM¯−1 defined as in (5.2), we present another type of decomposition for any given point ( ¯αn−1, αn) ∈ Rn−1× R with respect to the elliptic cone KMn¯ and its dual cone KnM¯−1.
Type II:
α¯n−1 αn
=
αn+ k ¯M−1α¯n−1k
k ¯M−1α¯n−1k + k ¯M ¯αn−1k · k ¯M ¯αn−1k ·
¯ αn−1
k ¯M ¯αn−1k 1
+ αn− k ¯M ¯αn−1k
k ¯M−1α¯n−1k + k ¯M ¯αn−1k· k ¯M−1α¯n−1k ·
− ¯αn−1
k ¯M−1α¯n−1k 1
if ¯αn−16= 0,
αn
k ¯M−1wk + k ¯M wk· k ¯M wk ·
" w k ¯M wk
1
#
+ αn
k ¯M−1wk + k ¯M wk· k ¯M−1wk ·
−w k ¯M−1wk
1
if ¯αn−1= 0,
where w is any given nonzero vector in Rn−1. In contrast to the Type I decompo- sition, these vectors
¯ αn−1
k ¯M ¯αn−1k 1
,
" w k ¯M wk
1
#
belong to the set KMn¯ , whereas the vectors
− ¯αn−1 k ¯M−1α¯n−1k
1
,
−w k ¯M−1wk
1
belong to its dual cone KnM¯−1.
The following theorem presents the decompositions regarding the ellipsoidal cone.
Theorem 5.2. Let KE be an ellipsoidal cone defined as in (1.1) and K∗E be its dual cone defined as in (2.7). For any given x ∈ Rn, it has two types of decompositions, namely Type I and Type II.
Type I:
x =
s(1)I
a (x) · v(1)I
a (x) + s(2)I
a (x) · v(2)I
a (x) if U¯n−1T x 6= 0, s(1)I
b (x) · v(1)I
b (x) + s(2)I
b (x) · v(2)I
b (x) if U¯n−1T x = 0, where s(1)Ia (x), s(2)Ia (x), s(1)I
b (x), s(2)I
b (x) and v(1)Ia (x), vI(2)a (x), v(1)I
b (x), vI(2)
b (x) have the following expressions
s(1)I
a(x) := uTnx + k ¯M ¯Un−1T xk, v(1)I
a(x) := 1 2 ·
U¯n−1U¯n−1T x k ¯M ¯Un−1T xk + un
∈ KE,
s(2)I
a(x) := uTnx − k ¯M ¯Un−1T xk, v(2)I
a(x) := 1 2 ·
−
U¯n−1U¯n−1T x k ¯M ¯Un−1T xk + un
∈ KE,
s(1)I
b(x) := uTnx, vI(1)
b (x) := 1 2·
U¯n−1w k ¯M wk + un
∈ KE,
s(2)I
b(x) := uTnx, vI(2)
b (x) := 1 2·
−U¯n−1w k ¯M wk + un
∈ KE
with any given nonzero vector w ∈ Rn−1 and a diagonal matrix ¯M looks like
(5.3) M =¯
" ¯Un−1T (Q − λnunuTn) ¯Un−1 (−λn)
#1/2
.
Type II:
x =
s(1)II
a(x) · v(1)II
a(x) + s(2)II
a(x) · vII(2)
a(x) if U¯n−1T x 6= 0, s(1)II
b(x) · v(1)II
b(x) + s(2)II
b(x) · v(2)II
b(x) if U¯n−1T x = 0, where s(1)IIa(x), s(2)IIa(x), s(1)II
b(x), s(2)II
b(x) and vII(1)a, vII(2)a, vII(1)
b, vII(2)
b are defined as follows:
s(1)II
a(x) := uTnx + k ¯M−1U¯n−1T xk, vII(1)
a(x) :=
U¯n−1U¯n−1T x + k ¯M ¯Un−1T xk · un
k ¯M−1U¯n−1T xk + k ¯M ¯Un−1T xk ∈ KE,
s(2)II
a(x) := uTnx − k ¯M ¯Un−1T xk, v(2)II
a(x) := − ¯Un−1U¯n−1T x + k ¯M−1U¯n−1T xk · un
k ¯M−1U¯n−1T xk + k ¯M ¯Un−1T xk ∈ K∗E,
s(1)II
b(x) := uTnx, v(1)II
b(x) :=
U¯n−1w + k ¯M wk · un
k ¯M−1wk + k ¯M wk ∈ KE,
s(2)II
b(x) := uTnx, v(2)II
b(x) := − ¯Un−1w + k ¯M−1wk · un k ¯M−1wk + k ¯M wk ∈ K∗E.
Proof. For any given x ∈ Rn, due to the orthogonal property of {ui}ni=1, there exists a vector α = ( ¯αn−1, αn) ∈ Rn−1× R such that x = Uα, therefore we obtain
