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
R^{n}, including nonnegative octant R^{n}+ and second-order cone K^{n}, 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 ⊆ R^{n} 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 S^{n} be the collection of all real symmetric matrices
in the n dimensional matrix space R^{n×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 ∈ S^{n}with exact one negative
eigenvalue λ_{n}∈ R and corresponding eigenvector un∈ R^{n} such that

(1.1) K_{E} := {x ∈ R^{n}| x^{T}Qx ≤ 0, and u^{T}_{n}x ≥ 0},
where the matrix Q admits the orthogonal decomposition Q = Pn

i=1λiuiu^{T}_{i} with
eigen-pairs (λ_{i}, u_{i}) for i = 1, 2, · · · , n satisfying the conditions

(1.2) λ_{1} ≥ λ_{2} ≥ · · · ≥ λ_{n−1}> 0 > λ_{n} and u^{T}_{i} u_{j} =

1 if i = j, 0 if i 6= j.

Example 1.1. Let Q =

1 2 −√

2 −^{1}_{2}

−√

2 0 −√

2

−^{1}_{2} −√

2 ^{1}_{2}

∈ S^{3} and u_{n}=

1

√2 2 21 2

∈ R^{3}.

Figure 1 shows an ellipsoidal cone in R^{3} generated by these parameters Q and u_{n}.

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 ∈ R^{n×n}under
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]:

K^{n}:=(¯x, x_{n}) ∈ R^{n−1}× R | k¯xk ≤ x_{n} ,

where k¯xk stands for the Euclidean norm of ¯x ∈ R^{n−1}. Clearly, K^{n} 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 I_{n}.

(b) Circular cone [6, 21]:

L_{θ}:=(¯x, x_{n}) ∈ R^{n−1}× R | k¯xk ≤ x_{n}tan θ , where θ ∈ (0,π
2).

It is not hard to see that the circular cone L_{θ} is also a special case of ellipsoidal
cone with

Q =

I_{n−1} 0
0 − tan^{2}θ

and u_{n}= e_{n}.
(c) Elliptic cone [2]:

K_{M}^{n} :=(¯x, x_{n}) ∈ R^{n−1}× R | kM ¯xk ≤ x_{n} ,

where M is any nonsingular matrix of order n − 1. Obviously, the elliptic cone K^{n}_{M}
can be viewed as an ellipsoidal cone by letting

Q =

M^{T}M 0

0 −1

and u_{n}= e_{n}.

Remark 1.3. We elaborate more about the aforementioned convex cones. In fact,
there hold the relations ^{1} as follows:

K^{n}⊆ L_{θ}⊆ K^{n}_{M} ⊆ K_{E} ⊆ R^{n}.

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 K^{n}, L_{θ}, K^{n}_{M}, 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,

K_{E}^{∗} := {y ∈ R^{n}| hx, yi ≥ 0, ∀x ∈ K_{E}} ,

where h·, ·i stands for the standard Euclidean inner product defined on R^{n}. In what
follows, we write the matrices U ∈ R^{n×n} and Λ ∈ S^{n} to respectively represent
(2.1) U := u_{1} u_{2} · · · u_{n}, Λ := diag (λ_{1}, λ_{2}, · · · , λ_{n}) .

The orthogonal decomposition of Q given as in (1.2) implies
Q = U ΛU^{T}, and U^{T}U = U U^{T} = I_{n}.

For any given vector x ∈ KE, due to the orthogonal property of the sets {ui}^{n}_{i=1},
there exists a vector α := [α_{1}, α_{2}, · · · , α_{n}]^{T} ∈ R^{n} such that

x = U α, x^{T}Qx = α^{T}U^{T}QU α = α^{T}Λα =

n

X

i=1

λiα^{2}_{i}, u^{T}_{n}x = u^{T}_{n}

n

X

i=1

αiui

!

= αn.
The set KE can be rewritten as the form U ∆_{α} with

(2.2) ∆_{α}:=

(

α ∈ R^{n}

n

X

i=1

λ_{i}α^{2}_{i} ≤ 0, α_{n}≥ 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) K^{n}:=

(

α ∈ R^{n}

n−1

X

i=1

α^{2}_{i} ≤ α_{n}^{2}, αn≥ 0
)

.
For any α ∈ ∆_{α}, in light of the relation (1.2) for {λ_{i}}^{n}_{i=1}, we have

α ∈ ∆α ⇐⇒

n

X

i=1

λiα^{2}_{i} ≤ 0 and α_{n}≥ 0

⇐⇒

n−1

X

i=1

λ^{1/2}_{i} α_{i}2

≤

(−λ_{n})^{1/2}α_{n}2

and (−λ_{n})^{1/2}α_{n}≥ 0

⇐⇒

λ^{1/2}_{1} α_{1}, λ^{1/2}_{2} α_{2}, · · · , λ^{1/2}_{n−1}α_{n−1}, (−λ_{n})^{1/2}α_{n}T

∈ K^{n}

⇐⇒ α ∈ DK^{n}

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 K^{n} is described as ∆α = DK^{n}, which implies
(2.5) KE = U ∆α = U DK^{n}= T K^{n}, where T := U D.

It is clear that the matrix T ∈ R^{n×n} is nonsingular. The relation (2.5) between the
ellipsoidal cone KE and the second-order cone K^{n} 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 ∈ R^{n}| hx, yi ≥ 0, ∀x ∈ K_{E}}

= {y ∈ R^{n}| hT z, yi ≥ 0, ∀z ∈ K^{n}}

= y ∈ R^{n}| T^{T}y ∈ (K^{n})^{∗} = K^{n}

= (T^{T})^{−1}K^{n},

where

T^{T} = (U D)^{T} = D^{T}U^{T} = DU^{−1}= D^{2}D^{−1}U^{−1}= D^{2}T^{−1},
(T^{T})^{−1} = D^{2}T^{−1}−1

= T D^{−2} = U D^{−1}.

In addition, by denoting |Λ| := diag(|λ1|, |λ_{2}|, · · · , |λ_{n}|) ∈ S^{n}, which means |Λ| =
D^{−2}, then (T^{T})^{−1}= T |Λ| and the dual cone K_{E}^{∗} can be further expressed as
(2.6) K_{E}^{∗} = (T^{T})^{−1}K^{n}= U D^{−1}K^{n}= T |Λ|K^{n},

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 ∈ R^{n}| hx, yi ≥ 0, ∀y ∈ K^{∗}_{E}}

= {y ∈ R^{n}| hx, T |Λ|zi ≥ 0, ∀z ∈ K^{n}}

= y ∈ R^{n}: (T |Λ|)^{T}x ∈ (K^{n})^{∗} = K^{n}

= (T |Λ|)^{T}−1

K^{n},
where

(T |Λ|)^{T}^{−1}

= |Λ|T^{T}^{−1}

= (T^{T})^{−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 K^{n}= KE,

see Figure 5 for illustration.

To sum up these discussions, we state the relations among the ellipsoidal cone
K_{E}, its dual cones K^{∗}_{E}, K^{∗∗}_{E} , and the second-order cone K^{n}in the following theorem.

Theorem 2.1. Let KE and K^{n} be defined as in (1.1) and (2.3), respectively. Then,
we have

(a) KE = T K^{n} and K^{n}= T^{−1}K_{E};

(b) K^{∗}_{E} = T |Λ|K^{n} and K^{n}= |Λ|^{−1}T^{−1}K^{∗}_{E};
(c) K^{∗}_{E} = T |Λ|T^{−1}K_{E} and K_{E}^{∗∗}= 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 K_{E}^{∗}
is equivalently expressed by

(2.7) K_{E}^{∗} =y ∈ R^{n}| y^{T}Q^{−1}y ≤ 0, u^{T}_{n}y ≥ 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^{−1}K^{n}, where the matrix U is defined as in (2.1). Using
(1.2) for {λi}^{n}_{i=1}, for any given y ∈ R^{n} there exists a vector β ∈ R^{n} such that
y = U β, which yields

y^{T}Q^{−1}y ≤ 0 and u^{T}_{n}y ≥ 0

⇐⇒

n

X

i=1

λ^{−1}_{i} β_{i}^{2} ≤ 0 and β_{n}≥ 0

⇐⇒

n−1

X

i=1

λ^{−1/2}_{i} βi

2

≤

(−λn)^{−1/2}βn

2

and (−λn)^{−1/2}βn≥ 0

⇐⇒

λ^{−1/2}_{1} β1, λ^{−1/2}_{2} β2, · · · , λ^{−1/2}_{n−1} βn−1, (−λn)^{−1/2}βn

T

∈ K^{n}

⇐⇒ β ∈ D^{−1}K^{n}.

Then, the desired result follows.

As a byproduct, we denote K_{E}^{◦} the polar of the ellipsoidal cone KE. Applying
K_{E}^{◦} = −K^{∗}_{E} and (2.7), the exact form of the polar cone K_{E}^{◦} is given by

(2.8) K^{◦}_{E} :=y ∈ R^{n}| y^{T}Q^{−1}y ≤ 0, u^{T}_{n}y ≤ 0 .

Remark 2.3. By applying (2.7), the duals of the circular cone and the elliptic cone,
denoted by L^{∗}_{θ} and (K^{n}_{M})^{∗} respectively, can be characterized as

L^{∗}_{θ} :=(¯y_{n−1}, y_{n}) ∈ R^{n−1}× R | k¯y_{n−1}k ≤ y_{n}cot θ = L^{π}

2−θ, with θ ∈ (0,π
2),
(K^{n}_{M})^{∗} :=

n

(¯yn−1, yn) ∈ R^{n−1}× R

M^{−1}T

¯ yn−1

≤ y_{n}o

= K^{n}_{(M}−1)^{T},

where ¯yn−1 := (y1, y2, · · · , yn−1)^{T} ∈ R^{n−1}. Same arguments can be applied to the
polar cone K_{E}^{◦} given as in (2.8). In other words, the polar of the circular cone L^{◦}_{θ}
and the elliptic cone (K^{n}_{M})^{◦} are described as

L^{◦}_{θ} :=(¯y_{n−1}, y_{n}) ∈ R^{n−1}× R | k¯y_{n−1}k ≤ −y_{n}cot θ ,
(K^{n}_{M})^{◦}:=n

(¯y_{n−1}, y_{n}) ∈ R^{n−1}× R

M^{−1}T

¯
y_{n−1}

≤ −y_{n}o
.
3. The variational geometry of the ellipsoidal cone

In this section, we pay attention to the variational geometry of the ellipsoidal cone
K_{E}, 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

T_{K}_{E}(x) := {d ∈ R^{n}| ∃ t_{n}↓ 0, dist(x + t_{n}d, KE) = o(t_{n})} ,
N_{K}_{E}(x) := {v ∈ R^{n}| hv, di ≤ 0, ∀d ∈ T_{K}_{E}(x)} ,

where dist(x, S) denotes the distance from x ∈ R^{n}to 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 K^{n}.

Theorem 3.1. Let T ∈ R^{n×n} be a nonsingular matrix defined as in (2.5). For any
x ∈ KE, there exists a vector α = T^{−1}x ∈ K^{n} such that

T_{K}_{E}(x) = T TK^{n}(α) and N_{K}_{E}(x) = T |Λ| NK^{n}(α).

Proof. For any d ∈ R^{n}, we denote p := T^{−1}d. Then, applying Theorem 2.1(a)
yields

kT k^{−1}dist (x + t_{n}d, KE)

= kT k^{−1}dist (x + tnd, T K^{n})

= kT k^{−1} min

y∈K^{n}kx + t_{n}d − T yk

= kT k^{−1} min

y∈K^{n}

T (T^{−1}x + t_{n}T^{−1}d − y)

≤ min

y∈K^{n}kα + t_{n}p − yk

= dist (α + t_{n}p, K^{n})

= dist α + t_{n}p, T^{−1}K_{E}

≤ kT^{−1}k min

w∈KE

kx + t_{n}d − wk

= kT^{−1}k dist (x + t_{n}d, KE) .

On the other hand, from definition, there exists tn↓ 0 such that dist(x + t_{n}d, KE) =
o(t_{n}) if and only if d ∈ TKE(x). Thus, we know dist(α + t_{n}p, K^{n}) = o(t_{n}), which
yields T^{−1}d = p ∈ TK^{n}(α). The opposite inclusion can be achieved in the similar
way. In summary, we have shown TK_{E}(x) = T TK^{n}(α).

As for the part of NKE(x), we have

N_{K}_{E}(x) = {v ∈ R^{n}| hv, di ≤ 0, ∀d ∈ T_{K}_{E}(x)}

= {v ∈ R^{n}| hv, T pi ≤ 0, ∀p ∈ T_{K}n(α)}

= v ∈ R^{n}| hT^{T}v, pi ≤ 0, ∀p ∈ TK^{n}(α)

= v ∈ R^{n}| T^{T}v ∈ NK^{n}(α)

= n

v ∈ R^{n}| v ∈ T^{T}−1

N_{K}^{n}(α)
o

.

Together with the fact (T^{T})^{−1}= T |Λ|, it follows that NKE(x) = T |Λ| NK^{n}(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 K^{n}) and bd KE = T (bd K^{n}).

This together with the definition of K^{n} implies that

int K^{n}:=α ∈ R^{n}| α^{T}Qnα < 0, e^{T}_{n}α > 0 ,
bd K^{n}:=α ∈ R^{n}| α^{T}Qnα = 0, e^{T}_{n}α > 0 ∪ {0},
where the matrix Q_{n} is given by

(3.2) Qn:=

In−1 0

0 −1

∈ S^{n}.

For any given x ∈ KE and its corresponding vector α = T^{−1}x ∈ K^{n}, from (3.1), we
obtain

int KE =n

x ∈ R^{n}

T^{−1}xT

Q_{n}(T^{−1}x) < 0, e^{T}_{n}T^{−1}x > 0o
,
bd KE =n

x ∈ R^{n}

T^{−1}xT

Q_{n}(T^{−1}x) = 0, e^{T}_{n}T^{−1}x > 0o

∪ {0}.

Due to the definitions of T and Λ as in (2.1) and (2.4), we also have some useful transformations

T^{−1}T

Q_{n}T^{−1}= D^{−1}U^{−1}T

Q_{n}D^{−1}U^{−1}= U D^{−1}Q_{n}D^{−1}U^{T} = U ΛU^{T} = Q,
e^{T}_{n}T^{−1}= e^{T}_{n}(U D)^{−1}= e^{T}_{n}D^{−1}U^{−1}= e^{T}_{n}D^{−1}U^{T} = (−λ_{n})^{1/2}u^{T}_{n}.

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 ∈ R^{n}| x^{T}Qx < 0, u^{T}_{n}x > 0 ,
bd KE =x ∈ R^{n}| x^{T}Qx = 0, u^{T}_{n}x > 0 ∪ {0}.

Remark 3.3. Similar to Remark 2.3, we conclude from Theorem 3.2 that the
interior and the boundary of L_{θ} and K^{n}_{M} are described by

int L_{θ} =(¯x_{n−1}, x_{n}) ∈ R^{n−1}× R | k¯x_{n−1}k < x_{n}tan θ ,

bd Lθ =(¯xn−1, xn) ∈ R^{n−1}× R | k¯xn−1k = x_{n}tan θ > 0 ∪ {0},
int K^{n}_{M} =(¯x_{n−1}, x_{n}) ∈ R^{n−1}× R | kM ¯x_{n−1}k < x_{n} ,

bd K^{n}_{M} =(¯xn−1, xn) ∈ R^{n−1}× R | kM ¯xn−1k = x_{n}> 0 ∪ {0},
where ¯x_{n−1}:= (x_{1}, x_{2}, · · · , x_{n−1})^{T} ∈ R^{n−1}.

To present the tangent cone and normal cone, we first recall their counterparts
for second-order cone K^{n}, which can be found in [5]:

T_{K}^{n}(α) =

R^{n} if α ∈ int K^{n},

K^{n} if α = 0,

{p ∈ R^{n}| p^{T}Q_{n}α ≤ 0} if α ∈ bd K^{n}\ {0},
N_{K}^{n}(α) =

{0} if α ∈ int K^{n},

−K^{n} if α = 0,

R+(Q_{n}α) if α ∈ bd K^{n}\ {0},

where Q_{n} is defined as in (3.2) and R+(Q_{n}α) stands for the set {ηQ_{n}α | η ≥ 0}.

Combining Theorem 3.1, Theorem 3.2 with the definitions of TK^{n}(α) and NK^{n}(α),
we present the expressions of tangent cone and normal cone regarding KE as below.

Theorem 3.4. For any given x ∈ R^{n}, the tangent cone and normal cone with
respect to the ellipsoidal cone KE at x are described by

TK_{E}(x) =

R^{n} if x ∈ int KE,

K_{E} if x = 0,

{d ∈ R^{n}| d^{T}Qx ≤ 0} if x ∈ bd KE\ {0},
N_{K}_{E}(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
L_{θ} or K_{M}^{n} . In fact, if take

Q =

In−1 0
0 − tan^{2}θ

or Q =

M^{T}M 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 K_{M}^{n} are respectively given by

T_{L}_{θ}(x) =

R^{n} if x ∈ int L_{θ},
L_{θ} if x = 0,

ΞL_{θ} if x ∈ bd L_{θ}\ {0},
N_{L}_{θ}(x) =

{0} if x ∈ int L_{θ},

L^{◦}_{θ} if x = 0,

R+(¯x, −x_{n}tan^{2}θ) if x ∈ bd L_{θ}\ {0}

where ΞLθ :=( ¯dn−1, dn) ∈ R^{n−1}× R | h ¯dn−1, ¯xn−1i − d_{n}xntan^{2}θ ≤ 0 and ¯dn−1 :=

(d_{1}, d_{2}, · · · , d_{n−1})^{T} ∈ R^{n−1}. Similarly, we also obtain
T_{K}^{n}

M(x) =

R^{n} if x ∈ int K^{n}_{M},
K^{n}_{M} if x = 0,

ΞK^{n}_{M} if x ∈ bd K^{n}_{M} \ {0},
N_{K}^{n}

M(x) =

{0} if x ∈ int K^{n}_{M},

(K^{n}_{M})^{◦} if x = 0,

R+ M^{T}M ¯xn−1, −xn

if x ∈ bd K_{M}^{n} \ {0},
where ΞK^{n}_{M} :=( ¯dn−1, dn) ∈ R^{n−1}× R | hM ¯dn−1, M ¯xn−1i − d_{n}xn≤ 0 .

4. The projection onto the ellipsoidal cone

In this section, we focus on the projection of any vector y ∈ R^{n}onto the ellipsoidal
cone KE. In other words, the following minimization problem is considered:

(4.1) min ^{1}_{2}kx − yk^{2}

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 ΠK_{E}(y) denotes the projection of y onto KE.

On the other hand, from the orthogonal property (2.1) for the set {ui}^{n}_{i=1}, there
exist α ∈ IR^{n} and β ∈ IR^{n}such that x = U α and y = U β. For simplicity, we write

Λ := diag( ¯Λn−1, λn) ∈ S^{n}, ¯Λn−1:= diag(λ1, λ2, · · · , λn−1) ∈ S^{n−1},
α := ( ¯αn−1, αn) ∈ R^{n−1}× R, ¯αn−1:= (α1, α2, · · · , αn−1)^{T} ∈ R^{n−1},
β := ( ¯β_{n−1}, β_{n}) ∈ R^{n−1}× R, ¯β_{n−1}:= (β_{1}, β_{2}, · · · , β_{n−1})^{T} ∈ R^{n−1}.

The problem (4.1) is equivalent to solving the elliptic optimization problem with
respect to the variables ( ¯αn−1, αn) ∈ IR^{n−1}× IR, that is,

(4.2) min ^{1}_{2} k ¯α_{n−1}− ¯β_{n−1}k^{2}+ (α_{n}− β_{n})^{2}
s.t.

¯M ¯α_{n−1}
≤ α_{n},

where ¯M is a diagonal matrix of order n − 1 in the form of (4.3) M := diag¯

s λ_{1}
(−λn),

s
λ_{2}

(−λn), · · · , s

λ_{n−1}
(−λn)

!
.
It is easy to verify that the matrix ¯M also satisfies the equation
(4.4) Λ¯n−1+ λnM¯^{T}M = 0.¯

Theorem 4.1. Let KE be an ellipsoidal cone defined as in (1.1) and y ∈ R^{n}. 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 ∈ R^{n×n} is defined as in (2.1), the vector α = ( ¯αn−1, αn) ∈
R^{n−1}× R is the optimal solution of (4.2) and has the following forms:

(a) If βn= 0, then

¯

α_{n−1}= I_{n−1}+ ¯M^{T}M¯^{−1}β¯_{n−1}, α_{n}= k ¯M I_{n−1}+ ¯M^{T}M¯^{−1}β¯_{n−1}k.

(b) If β_{n}6= 0, then

¯

α_{n−1}= I_{n−1}− η_{0}λ_{n}M¯^{T}M¯−1β¯_{n−1}, α_{n}= βn

1 + η_{0}λ_{n},

where the matrix ¯M ∈ S^{n−1} is defined as in (4.3) and the scalar η_{0} ∈ R
satisfies the relations

(4.5) η_{0}∈

(0, −1/λ_{n}) if β_{n}> 0,

(−1/λn, +∞) if βn< 0 and

n

X

i=1

λ_{i}β_{i}^{2}

(1 + η0λi)^{2} = 0

with β ∈ R^{n} lying outside of the setβ ∈ R^{n}| k ¯M ¯βn−1k ≤ β_{n} and its polar

β ∈ R^{n}| k ¯M^{−1}β¯n−1k ≤ β_{n} .

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 /∈ K_{E} ∪ K_{E}^{◦}. From Theorem 3.4, there
exists a scalar η_{0} > 0 such that

(4.6) x = ΠK_{E}(y) ∈ bdKE\ {0} and 0 = x − y + η_{0}Qx.

We set x = U α and y = U β as earlier, where U ∈ R^{n×n} is defined as in (2.1). Then,
the relations (4.6) are equivalent to the system with respect to the variables α ∈ R^{n}
and η_{0} ∈ R as follows:

(4.7)

β = (I + η_{0}Λ)α,
α^{T}Λα = 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}= (I_{n−1}+ η_{0}Λ¯_{n−1}) ¯α_{n−1},
βn= (1 + η0λn)αn,

¯

α^{T}_{n−1}Λ¯n−1α¯n−1+ λnα^{2}_{n}= 0,
α_{n}> 0, η_{0} > 0.

Next, we proceed to show that the following two subcases hold for the system (4.8).

(a) If β_{n} = 0, then η_{0} = −_{λ}^{1}

n > 0. From the system (4.8) and the equation (4.4), we have

¯
α_{n−1}=

I_{n−1}− 1
λ_{n}Λ¯_{n−1}

−1

β¯_{n−1}= I_{n−1}+ ¯M^{T}M¯^{−1}β¯_{n−1},

αn= α¯^{T}_{n−1}Λ¯n−1α¯n−1

−λ_{n}

!1/2

=

M I¯ n−1+ ¯M^{T}M¯−1β¯n−1

.

(b) If βn 6= 0, from the second and fourth relations in (4.8), we know η_{0} 6= −_{λ}^{1}

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− η_{0}λnM¯^{T}M¯−1β¯n−1, αn= βn

1 + η0λn

,
where η_{0} ∈ R satisfies the condition

η_{0} ∈

(0, −1/λn) if βn> 0,
(−1/λ_{n}, +∞) if β_{n}< 0.

In addition, we have

¯

α^{T}_{n−1}Λ¯n−1α¯n−1+ λnα^{2}_{n}

= β¯_{n−1}^{T} I_{n−1}+ η_{0}Λ¯_{n−1}−1Λ¯_{n−1} I_{n−1}+ η_{0}Λ¯_{n−1}−1β¯_{n−1}+ λ_{n}

βn

1 + η_{0}λ_{n}

2

=

n

X

i=1

λ_{i}β_{i}^{2}
(1 + η_{0}λ_{i})^{2}

and the third relation in (4.8) reduces to the equation (4.5). Since y /∈ K_{E}∪ K^{◦}_{E} and
y = U β, from the definitions of KE and K^{◦}_{E}, we obtain

β /∈ (

β ∈ R^{n}|

n

X

i=1

λiβ_{i}^{2}≤ 0, β_{n}≥ 0
)

∪ (

β ∈ R^{n}|

n

X

i=1

λ^{−1}_{i} β_{i}^{2}≤ 0, β_{n}≤ 0
)

,
which means that β /∈ β ∈ R^{n}| k ¯M ¯β_{n−1}k ≤ β_{n} ∪ β ∈ R^{n}| k ¯M^{−1}β¯_{n−1}k ≤ β_{n} .

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 η_{0}∈ R in general. However, under some
special cases, the equation (4.5) has closed-form solutions. For example, if we set

U = I_{n}, λ_{i} = 1(i = 1, 2, · · · , n − 1), λ_{n}= −1 or λ_{n}= − tan^{2}θ,

which correspond to the cases of the second-order cone K^{n} and the circular cone
L_{θ}. 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 K^{n}_{M}_{¯} be an elliptic cone with the matrix ¯M defined as in (4.3),
i.e.,

(5.1) K^{n}_{M}_{¯} :=(¯αn−1, αn) ∈ R^{n−1}× R | k ¯M ¯αn−1k ≤ α_{n} .
According to [11, Remark 2.2], the dual cone of K^{n}_{M}_{¯} is defined by

(5.2) (K_{M}^{n}_{¯})^{∗} = {( ¯βn−1, βn) ∈ R^{n−1}× R | k ¯M^{−1}β¯n−1k ≤ β_{n}} = K^{n}_{M}_{¯}−1.

It is easy to see that the ellipsoidal cone KE and its dual cone (KE)^{∗} can be
described in terms of K^{n}_{M}_{¯} and its dual cone K^{n}_{M}_{¯}−1.

Theorem 5.1. Let KE be an ellipsoidal cone defined as in (1.1) and K^{n}_{M}_{¯} be an
elliptic cone defined as in (5.1). Then, we have

KE = U KM^{n}¯, K^{∗}_{E} = U K^{n}M¯^{−1}.

Proof. For any given x ∈ KE, since {u_{i}}^{n}_{i=1} are orthogonal to each other, there
exists a vector α = ( ¯α_{n−1}, α_{n}) ∈ R^{n−1}× R such that x = Uα. From the definition
of KE, we have

x ∈ KE

⇐⇒ x^{T}Qx ≤ 0, u^{T}_{n}x ≥ 0

⇐⇒ α^{T}Λα ≤ 0, αn≥ 0

⇐⇒ α¯^{T}_{n−1}Λ¯_{n−1}α¯_{n−1}+ λ_{n}α^{2}_{n}≤ 0, α_{n}≥ 0

⇐⇒ α¯^{T}_{n−1}M¯^{T}M ¯¯αn−1≤ α^{2}_{n}, αn≥ 0

⇐⇒ x = U α, α ∈ K^{n}_{M}_{¯},

which implies the relation KE = U K^{n}_{M}_{¯}. One the other hand, for any given y ∈ K^{∗}_{E},
due to the orthogonal property of {u_{i}}^{n}_{i=1}, there exists a vector β = ( ¯β_{n−1}, β_{n}) ∈
R^{n−1}× R such that y = Uβ. It follows from above that

y ∈ K^{∗}_{E}

⇐⇒ y^{T}Q^{−1}y ≤ 0, u^{T}_{n}y ≥ 0

⇐⇒ β^{T}Λ^{−1}β ≤ 0, β_{n}≥ 0

⇐⇒ β¯_{n−1}^{T} Λ¯^{−1}_{n−1}β¯n−1+ λ^{−1}_{n} β_{n}^{2} ≤ 0, β_{n}≥ 0

⇐⇒

n−1

X

i=1

(−λn) λi

β_{i}^{2}≤ β_{n}^{2}, βn≥ 0

⇐⇒ β¯_{n−1}^{T} ( ¯M^{−1})^{T}M¯^{−1}β¯_{n−1}≤ β^{2}_{n}, β_{n}≥ 0

⇐⇒ y = U β, β ∈ K^{n}_{M}_{¯}−1.
Therefore, we obtain K^{∗}_{E} = U K^{n}_{M}_{¯}−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) ∈ R^{n−1}× R with respect to the elliptic cone K^{n}M¯.

Type I:

α¯_{n−1}
α_{n}

=

αn+ k ¯M ¯αn−1k 2

¯ αn−1

k ¯M ¯αn−1k 1

+α_{n}− k ¯M ¯α_{n−1}k
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 ¯α_{n−1}= 0,

where w is any given nonzero vector in R^{n−1}. Focusing on the right-hand side of
the Type I decomposition, we observe that the vectors

¯ αn−1

k ¯M ¯α_{n−1}k
1

,

− ¯αn−1

k ¯M ¯α_{n−1}k
1

,

" w k ¯M wk

1

# ,

−w k ¯M wk

1

all belong to the set K^{n}_{M}_{¯}, 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 K^{n}_{M}_{¯}−1 defined as in (5.2), we present another type of decomposition
for any given point ( ¯αn−1, αn) ∈ R^{n−1}× R with respect to the elliptic cone KM^{n}¯ and
its dual cone K^{n}_{M}_{¯}−1.

Type II:

α¯_{n−1}
α_{n}

=

α_{n}+ k ¯M^{−1}α¯_{n−1}k

k ¯M^{−1}α¯n−1k + k ¯M ¯αn−1k · k ¯M ¯α_{n−1}k ·

¯ αn−1

k ¯M ¯α_{n−1}k
1

+ αn− k ¯M ¯αn−1k

k ¯M^{−1}α¯n−1k + k ¯M ¯αn−1k· k ¯M^{−1}α¯n−1k ·

− ¯αn−1

k ¯M^{−1}α¯_{n−1}k
1

if ¯α_{n−1}6= 0,

α_{n}

k ¯M^{−1}wk + k ¯M wk· k ¯M wk ·

" w k ¯M wk

1

#

+ αn

k ¯M^{−1}wk + k ¯M wk· k ¯M^{−1}wk ·

−w
k ¯M^{−1}wk

1

if ¯α_{n−1}= 0,

where w is any given nonzero vector in R^{n−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 K_{M}^{n}_{¯} , whereas the vectors

− ¯α_{n−1}
k ¯M^{−1}α¯n−1k

1

,

−w
k ¯M^{−1}wk

1

belong to its dual cone K^{n}_{M}_{¯}−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 ∈ R^{n}, 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−1}^{T} 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−1}^{T} x = 0,
where s^{(1)}_{I}_{a} (x), s^{(2)}_{I}_{a} (x), s^{(1)}_{I}

b (x), s^{(2)}_{I}

b (x) and v^{(1)}_{I}_{a} (x), v_{I}^{(2)}_{a} (x), v^{(1)}_{I}

b (x), v_{I}^{(2)}

b (x) have the following expressions

s^{(1)}_{I}

a(x) := u^{T}_{n}x + k ¯M ¯U_{n−1}^{T} xk, v^{(1)}_{I}

a(x) := 1 2 ·

U¯n−1U¯_{n−1}^{T} x
k ¯M ¯U_{n−1}^{T} xk + un

∈ KE,

s^{(2)}_{I}

a(x) := u^{T}_{n}x − k ¯M ¯U_{n−1}^{T} xk, v^{(2)}_{I}

a(x) := 1 2 ·

−

U¯n−1U¯_{n−1}^{T} x
k ¯M ¯U_{n−1}^{T} xk + un

∈ KE,

s^{(1)}_{I}

b(x) := u^{T}_{n}x, v_{I}^{(1)}

b (x) := 1 2·

U¯n−1w k ¯M wk + un

∈ K_{E},

s^{(2)}_{I}

b(x) := u^{T}_{n}x, v_{I}^{(2)}

b (x) := 1 2·

−U¯_{n−1}w
k ¯M wk + un

∈ KE

with any given nonzero vector w ∈ R^{n−1} and a diagonal matrix ¯M looks like

(5.3) M =¯

" ¯U_{n−1}^{T} (Q − λ_{n}u_{n}u^{T}_{n}) ¯U_{n−1}
(−λ_{n})

#1/2

.

Type II:

x =

s^{(1)}_{II}

a(x) · v^{(1)}_{II}

a(x) + s^{(2)}_{II}

a(x) · v_{II}^{(2)}

a(x) if U¯_{n−1}^{T} 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−1}^{T} x = 0,
where s^{(1)}_{II}_{a}(x), s^{(2)}_{II}_{a}(x), s^{(1)}_{II}

b(x), s^{(2)}_{II}

b(x) and v_{II}^{(1)}_{a}, v_{II}^{(2)}_{a}, v_{II}^{(1)}

b, v_{II}^{(2)}

b are defined as follows:

s^{(1)}_{II}

a(x) := u^{T}_{n}x + k ¯M^{−1}U¯_{n−1}^{T} xk, v_{II}^{(1)}

a(x) :=

U¯n−1U¯_{n−1}^{T} x + k ¯M ¯U_{n−1}^{T} xk · un

k ¯M^{−1}U¯_{n−1}^{T} xk + k ¯M ¯U_{n−1}^{T} xk ∈ KE,

s^{(2)}_{II}

a(x) := u^{T}_{n}x − k ¯M ¯U_{n−1}^{T} xk, v^{(2)}_{II}

a(x) := − ¯Un−1U¯_{n−1}^{T} x + k ¯M^{−1}U¯_{n−1}^{T} xk · un

k ¯M^{−1}U¯_{n−1}^{T} xk + k ¯M ¯U_{n−1}^{T} xk ∈ K^{∗}_{E},

s^{(1)}_{II}

b(x) := u^{T}_{n}x, v^{(1)}_{II}

b(x) :=

U¯n−1w + k ¯M wk · un

k ¯M^{−1}wk + k ¯M wk ∈ K_{E},

s^{(2)}_{II}

b(x) := u^{T}_{n}x, v^{(2)}_{II}

b(x) := − ¯U_{n−1}w + k ¯M^{−1}wk · u_{n}
k ¯M^{−1}wk + k ¯M wk ∈ K^{∗}_{E}.

Proof. For any given x ∈ R^{n}, due to the orthogonal property of {ui}^{n}_{i=1}, there
exists a vector α = ( ¯α_{n−1}, α_{n}) ∈ R^{n−1}× R such that x = Uα, therefore we obtain