### The solvabilities of eigenvalue optimization problems associated with p-order cone and circular cone

Wei-Ming Hsu ^{1}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan.

Xin-He Miao^{2}
School of Mathematics
Tianjin University, China

Tianjin 300072, China

Jein-Shan Chen ^{3}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan.

October 29, 2021

Abstract In this paper, we consider two kinds of conic eigenvalue complementarity problems, which are eigenvalue complementarity problems and quadratic eigenvalue com- plementarity problem associated with circular cone or p-order cone, respectively. In the setting of the circular cone, we establish the relation between the solution of the circular cone eigenvalue complementarity problems (or the solution of the circular cone quadratic eigenvalue complementarity problems) and the solution of the corresponding circular cone complementarity problems. The similar results in the setting of p-order cone are achieved as well. These results build up a theoretical basis for making further study on their corresponding eigenvalue complementarity problems in the future.

Keywords. Solvability, eigenvalue, p-order cone, circular cone.

AMS classifications. 26B05, 26B35, 90C33.

1E-mail: a32410@hotmail.com.

2E-mail: xinhemiao@tju.edu.cn. The author’s work is supported by National Natural Science Foun- dation of China (No. 11771328).

3Corresponding author. E-mail: jschen@math.ntnu.edu.tw. The author’s work is supported by Ministry of Science and Technology, Taiwan.

### 1 Introduction

The traditional eigenvalue complementarity problem (EiCP) consists of finding a scalar
λ ∈ IR and a vector 0 6= x ∈ IR^{n} such that

y = λBx − Cx,

x ≥ 0, y ≥ 0, x^{T}y = 0, (1)

where B, C ∈ IR^{n×n} and y ∈ IR^{n}. This problem is also called the Pareto eigenvalue
problem, which has been studied in [23] and possesses a wide variety of applications in
sciences and engineering, see [1, 21, 22]. In fact, the EiCP (1) is equivalent to seeking a
scalar λ ∈ IR and a vector 0 6= x ∈ IR^{n} such that

y = λBx − Cx,

x ≥ 0, y ≥ 0, x^{T}y = 0,
a^{T}x = 1,

with a 0. Usually, we choose a = (1, 1, · · · , 1)^{T} ∈ IR^{n}. Recently, an extension of
the EiCP has been introduced in [24], which is named quadratic eigenvalue complemen-
tarity problem (QEiCP). It differs from the EiCP because there exists an additional
quadratic term on λ attached to the eigenvalue problems. More specifically, given ma-
trices A, B, C ∈ IR^{n×n}, the QEiCP consists of finding (x, y, λ) ∈ IR^{n} × IR^{n}× IR such

that

y = λ^{2}Ax + λBx + Cx,
x ≥ 0, y ≥ 0, x^{T}y = 0,
a^{T}x = 1.

In the literature, further extensions of the EiCP and the QEiCP are introduced in [1, 3, 11, 15], which include second-order cone eigenvalue complementarity problem (SOCEiCP) and general closed convex cone eigenvalue complementarity problem. Their general format are as below:

y = λBx − Cx,

x ∈ K, y ∈ K^{∗}, x^{T}y = 0,
a^{T}x = 1,

or

y = λ^{2}Ax + λBx + Cx,
x ∈ K, y ∈ K^{∗}, x^{T}y = 0,
a^{T}x = 1,

(2)

where K is a closed convex cone, K^{∗} denotes its dual cone, and a ∈ int(K) is arbitrary
fixed point. When K represents the second-order cone (denoted by K^{n}), they become
second-order cone eigenvalue complementarity problem (SOCEiCP) and second-order
cone quadratic eigenvalue complementarity problem (SOCQEiCP) [17]. For the concept
and related properties of second-order cone, please refer to [2, 6, 7, 8, 9, 12, 13, 14, 26].

Roughly, there are two main research directions regarding the eigenvalue complemen- tarity problems. One is on the theoretical side in which their corresponding solution properties are investigated, see [3, 4, 5, 11, 21, 23, 24, 25]. The other one focuses on

the numerical algorithm for solving the problems, which include the Lattice projection method, the semismooth Newton methods, the RLT-based branch and bound method (BBRLT) and so on, see [1, 3, 4, 5, 15, 21, 22, 23, 24] and references therein. As seen in the literature, almost all the attention is paid to the standard eigenvalue complementar- ity problems and second-order cone eigenvalue complementarity problems [17]. However, the study about other special conic eigenvalue complementarity problems is very limited.

On the other hand, more and more nonsymmetric cones (for instance, circular cone,
p-order cone) appear in plenty of real applications. Hence, in this manuscript, we are
therefore motivated to extend the concepts and properties of the EiCP and the QEiCP
to the setting of the circular cone Lθ and p-order cone Kp. In other words, the cone K
in problem (2) is circular cone L_{θ} and p-order cone K_{p}. For the concepts of circular cone
L_{θ} and p-order cone K_{p}, we will review them in details in the next section.

In this paper, several results about the solutions of the eigenvalue complementar-
ity problem (EiCP) and the quadratic eigenvalue complementarity problems (QEiCP)
are extended to the version of circular cones L_{θ} and p-order cones K^{n}, respectively. In
the setting of circular cone, the relationship between the solution of the circular cone
eigenvalue complementarity problems (CCEiCP) and the solution of the corresponding
circular cone complementarity problems (or the solution of the circular cone quadratic
eigenvalue complementarity problems (CCQEiCP) and the solution of the corresponding
circular cone complementarity problems) are established. Besides, for the p-order cone
eigenvalue complementarity problems (POCEiCP) and p-order cone eigenvalue comple-
mentarity problems (POCQEiCP), we also sum up the corresponding theoretical results.

The solutions of the POCEiCP or the POCQEiCP and the solutions of the corresponding p-order cone complementarity problems correspond to each other. These results build up a theoretical basis for designing numerical solution algorithm or making further study about their corresponding eigenvalue complementarity problems in the future.

The remainder of this paper is organized as follows. In section 2, we recall some background materials regarding circular cone and p-order cone. In section 3, we study the properties of the solutions for the eigenvalue complementarity problems and quadratic eigenvalue complementarity problems in the setting of circular cone, respectively. In section 4, based on the cases of circular cone eigenvalue optimization problems, we study the corresponding properties of the solutions for p-order cone eigenvalue complementarity problems and p-order cone quadratic eigenvalue complementarity problems, respectively.

### 2 Preliminaries

In this section, we briefly review some basic concepts and background materials about the circular cone Lθ and the p-order cone Kp, which will be extensively used in the subsequent sections. More details can be found in [10, 16, 18, 20, 27, 28].

Let L_{θ} denote the circular cone (see [10, 16, 18, 28]) in IR^{n}, which is defined by
Lθ := {x = (x1, x2) ∈ IR × IR^{n−1}| kx2k ≤ x1tan θ}

with k · k denoting the Euclidean norm and θ ∈ (0,^{π}_{2}). It is well known that the dual
cone of L_{θ} can be expressed as

(L_{θ})^{∗} = {x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k ≤ x_{1}cot θ} = L^{π}

2−θ.

Moreover, Zhou and Chen [28] gave the below spectral decomposition of x = (x_{1}, x_{2}) ∈
IR × IR^{n−1} with respect to L_{θ}:

x = µ_{1}(x)v_{x}^{(1)}+ µ_{2}(x)v_{x}^{(2)},
where µ_{1}(x), µ_{2}(x), vx^{(1)}, and vx^{(2)} are expressed as

µ_{1}(x) = x_{1} − kx_{2}k cot θ, µ_{2}(x) = x_{1}+ kx_{2}k tan θ,
and

v^{(1)}_{x} = 1
1 + cot^{2}θ

1

− cot θ · w

, v^{(2)}_{x} = 1
1 + tan^{2}θ

1

tan θ · w

.
with w = _{kx}^{x}^{2}

2k if x_{2} 6= 0, or any vector in IR^{n−1} satisfying kwk = 1 if x_{2} = 0.

As for the p-order cone. Let K_{p} denote the p-order cone (p > 1) (see [19, 20, 27]) in
IR^{n}, which is defined by

Kp := {x = (x1, x2) ∈ IR × IR^{n−1}| kx2kp ≤ x1}

with k · k_{p} denoting the p-norm. It is clear to see that when p = 2, K_{2} is exactly the
second-order cone, which confirms that the second-order cone is a special case of p-order
cone. It is also well known that the dual cone of K_{p} can be expressed as

(K_{p})^{∗} = {x = (x_{1}, x_{2}) ∈ IR × IR^{n−1}| kx_{2}k_{q} ≤ x_{1}} = K_{q},

where q > 1 and satisfies ^{1}_{p} +^{1}_{q} = 1. Miao, Qi and Chen [20] gave the following spectral
decomposition of x = (x_{1}, x_{2}) ∈ IR × IR^{n−1} with respect to K_{p}:

x = α_{1}(x)v_{x}^{(1)}+ α_{2}(x)v_{x}^{(2)},
where α_{1}(x), α_{2}(x), v_{x}^{1}, and v^{2}_{x} are expressed as

α1(x) = (x_{1}+ kx_{2}k_{p})

2 , α2(x) = (x_{1}− kx_{2}k_{p})

2 ,

and

v_{x}^{(1)} =

1 w

, v_{x}^{(2)} =

1

−w

.
with w = _{kx}^{x}^{2}

2kp if x2 6= 0, or any vector in IR^{n−1} satisfying kwkp = 1 if x2 = 0.

### 3 Circular cone eigenvalue complementarity prob- lems

Now, we consider the eigenvalue complementarity problems and circular cone quadratic eigenvalue complementarity problems in the setting of the circular cone Lθ. We will sum up the relationship between the solution of circular cone eigenvalue complemen- tarity problems (or the solution of circular cone quadratic eigenvalue complementarity problems) and the solution of the corresponding circular cone complementarity problems, respectively.

### 3.1 Circular cone eigenvalue complementarity problems

Consider the circular cone eigenvalue complementarity problems (CCEiCP for short):

find (x, y, λ) ∈ IR^{n}× IR^{n}× IR such that

CCEiCP(B, C) :

y = λBx − Cx,

x ∈ L_{θ}, y ∈ (L_{θ})^{∗}, x^{T}y = 0,
a^{T}x = 1,

(3)

where B, C ∈ IR^{n×n} and a is an arbitrary fixed point with a ∈ int((L_{θ})^{∗}). We first study
the solvability of the CCEiCP that will be used in subsequent analysis.

Proposition 3.1. Suppose that x := (x_{1}, x_{2}) ∈ L_{θ}, y := (y_{1}, y_{2}) ∈ (L_{θ})^{∗}. Then, the
following hold.

(a) x^{T}y ≥ 0.

(b) If y ∈ int(Lθ), then x^{T}y > 0 if and only if x 6= 0.

(c) If x 6= 0 and y 6= 0, then

x^{T}y = 0 ⇐⇒ x_{1}tan θ = kx_{2}k y_{1}cot θ = ky_{2}k and y = α(x_{1}tan^{2}θ, −x_{2}),
where α is a positive constant, or x = β(y_{1}cot^{2}θ, −y_{2}) with a positive constant β.

Proof. (a) It is obvious by the definition of dual cone.

(b) The result that x^{T}y > 0 implies x 6= 0 is trivial. We only prove the other direction
as follows. Since x 6= 0, we know that x_{1} > 0. Otherwise, we have that kx_{2}k ≤ x_{1} = 0
implies x = 0. Then, it follows that

x^{T}y = x_{1}y_{1}+ x^{T}_{2}y_{2} ≥ x_{1}y_{1}− kx_{2}kky_{2}k > x_{1}y_{1}− x_{1}tan θ y_{1}cot θ = 0.

(c) The proof of this direction “⇐” is trivial. We only need to prove the direction “⇒”.

By the assumption x 6= 0, we know that x_{1} > 0. Similarly, we obtain that y_{1} > 0. Using

the condition x^{T}y = 0, the Schwarz’s inequality and the definition of circular cones , we
have

x_{1}y_{1} = |x^{T}_{2}y_{2}| ≤ kx_{2}kky_{2}k ≤ (x_{1}tan θ)(y_{1}cot θ) = x_{1}y_{1}. (4)
Thus, it follows that

(x_{1}tan θ)(y_{1}cot θ) = x_{1}y_{1} = kx_{2}kky_{2}k. (5)
By combining kx_{2}k ≤ x_{1}tan θ, ky_{2}k ≤ y_{1}cot θ and (5), it yields

x_{1}tan θ = kx_{2}k and y_{1}cot θ = ky_{2}k. (6)
Besides, by (4), (5) and (6) again, we have

x^{T}_{2}y_{2} = −x_{1}y_{1} = −kx_{2}kky_{2}k.

This implies the equality holds in the Schwarz’s inequality. Hence, there exists a constant k such that

y_{2} = kx_{2}. (7)

Since kky_{2}k^{2} = x^{T}_{2}y_{2} = −x_{1}y_{1} < 0, we have k < 0. Choosing α = −k, it leads to α > 0
and y2 = −αx2. Then, it follows from (6) and (7) that

y_{1} = ky_{2}k tan θ = k − αx_{2}k tan θ = | − α|kx_{2}k tan θ = αkx_{2}k tan θ = αx_{1}tan^{2}θ.

By this, we have y = α(x_{1}tan^{2}θ, −x_{2}). Thus, the proof is complete. 2

In the next theorem, we establish the relation between CCEiCP and the below special circular cone complementarity problem (CCCP):

CCCP(F ) : x ∈ L_{θ}, F (x) ∈ (L_{θ})^{∗}, x^{T}F (x) = 0, (8)
where

F (x) =

_{x}^{T}_{Cx}

x^{T}BxBx − Cx if x 6= 0,

0 if x = 0. (9)

Theorem 3.1. Consider the CCEiCP(B, C) given as in (3) where B is positive definite.

Let F : IR^{n} → IR^{n} be defined as in (9). Then, the following hold.

(a) If (x^{∗}, λ^{∗}) solves the CCEiCP(B, C), then x^{∗} solves the CCCP(F ) (8).

(b) If ¯x is a nonzero solution of the CCCP(F ) (8), then (x^{∗}, λ^{∗}) solves the CCEiCP(B, C)
with λ^{∗} = ^{x}_{x}^{¯}_{¯}^{T}T^{C ¯}B ¯^{x}x and x^{∗} = _{a}T^{x}^{¯}x¯.

Proof. Part(a) is trivial and we only need to prove part(b). Suppose that ¯x is a nonzero solution to the CCCP(F ) (8). Then, we have

¯

x ∈ L_{θ}, x¯^{T}C ¯x

¯

x^{T}B ¯x· B ¯x − C ¯x ∈ (L_{θ})^{∗}, and ¯x^{T} ¯x^{T}C ¯x

¯

x^{T}B ¯xB ¯x − C ¯x

= 0.

Using a ∈ int((Lθ)^{∗}) and ¯x ∈ Lθ, we have _{a}T^{1}x¯ > 0. In view of all the above, we conclude
that

x^{∗} := _{a}T^{1}x¯x ∈ L¯ _{θ},

w^{∗} := λ^{∗}Bx^{∗} − Cx^{∗} = _{a}T^{1}x¯

hx¯^{T}C ¯x

¯
x^{T}B ¯x

B ¯x − C ¯xi

∈ (L_{θ})^{∗},
(x^{∗})^{T}w^{∗} = _{a}T^{1}x¯

2h

¯
x^{T}

¯x^{T}C ¯x

¯

x^{T}B ¯xB ¯x − C ¯x

i

= 0,
a^{T}x^{∗} = ^{a}_{a}^{T}T^{x}x^{¯}¯ = 1.

Thus, (x^{∗}, λ^{∗}) is a solution to the CCEiCP(B, C). 2

Remark 3.1. For the CCCP(F ) (8), based on Proposition 3.1, we note that if x is a
solution of the CCCP(F ) and x has the spectral decomposition: x = µ_{1}(x)vx^{(1)}+ µ_{2}(x)vx^{(2)},
F (x) can be written by F (x) = µ_{1}(F (x))v^{(2)}x + µ_{2}(F (x))v^{(1)}x . Therefore, we only need to
check both x ∈ bd(L_{θ}) and F (x) ∈ bd((L_{θ})^{∗}). In addition, we define

H_{1}(x) =

µ_{1}(x)
...
µ1(x)
µ_{1}(F (x)).

∈ IR^{n} and g_{1}(z) := 1

2kH_{1}(z)k^{2} = 1

2H_{1}(z)^{T}H_{1}(z).

In fact, g is a merit function for the CCCP(F ). Then, we can obtain a solution of the
CCEiCP (3) by using Newton method for solving the equations H_{1}(x) = 0.

Next, we consider the three special kinds of the circular cone complementarity prob- lems, which have the following forms:

CCLCP(−C, 0) : x ∈ L_{θ}, −Cx ∈ (L_{θ})^{∗}, x^{T}(−Cx) = 0; (10)

CCCP(G_{1}) :

x ∈ L_{θ}, λ ≥ 0,

λBx − Cx ∈ (L_{θ})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(λBx − Cx) + λ(a^{T}x − 1) = 0;

(11) and

CCCP(G2) :

x ∈ L_{θ}, λ ≥ 0,

−λBx − Cx ∈ (Lθ)^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(−λBx − Cx) + λ(a^{T}x − 1) = 0.

(12) In fact, the solutions of the CCEiCP (3) have a close correspondence with the solutions of the above three kinds of complementarity problems. We shall show the relationship between the CCEiCP and the CCCP in the following theorems.

Theorem 3.2. Let (x^{∗}, λ^{∗}) solves the CCEiCP(B,C) (3). Then, the following hold.

(a) If λ^{∗} > 0, then (x^{∗}, λ^{∗}) solves the CCCP(G_{1}) (11).

(b) If λ^{∗} < 0, then (x^{∗}, −λ^{∗}) solvesthe CCCP(G_{2}) (12).

(c) If λ^{∗} = 0, then x^{∗} solves the CCLCP(−C,0) (10).

Proof. From the assumption that (x^{∗}, λ^{∗}) solves the CCEiCP(B,C) (3), we have
x^{∗} ∈ Lθ, λ^{∗}Bx^{∗}− Cx^{∗} ∈ (Lθ)^{∗}, (x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) = 0 and a^{T}x^{∗} = 1.

To proceed, we discuss three cases of λ^{∗}.
(a) If λ^{∗} > 0, then we have

(x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) + λ^{∗}(a^{T}x^{∗}− 1) = (x^{∗})^{T}(λ^{∗}Bx^{∗} − Cx^{∗}) = 0,
which implies that (x^{∗}, λ^{∗}) solves the CCCP (G_{1}) (11).

(b) If λ^{∗} < 0, then we get that −λ^{∗} > 0 and

(x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) − λ^{∗}(a^{T}x^{∗}− 1) = (x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) = 0.

This indicates that (x^{∗}, −λ^{∗}) solves the CCCP(G_{2}) (12).

(c) If λ^{∗} = 0, then it follows that

λ^{∗}Bx^{∗}− Cx^{∗} = −Cx^{∗} ∈ (L_{θ})^{∗} and (x^{∗})^{T}(−Cx^{∗}) = (x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) = 0.

This implies that (x^{∗}, 0) solves the CCLCP(−C,0) (10).

Based on the above arguments, the proof is complete. 2

Theorem 3.3. (a) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G_{1}) (11), then (x^{∗}, λ^{∗}) solves
the CCEiCP(B,C) (3).

(b) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G_{2}) (12), then (x^{∗}, −λ^{∗}) solves the CCEiCP(B,C)
(3).

(c) If x^{∗} solves the CCLCP(−C, 0) (10) and x^{∗} 6= 0, then (_{a}T^{x}^{∗}x^{∗}, 0) solves the CCEiCP(B, C)
(3).

Proof. The proof is done by the below three cases..

Case (a): if λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G_{1}) (11), then we have

x^{∗} ∈ L_{θ}, λ^{∗} > 0,

λ^{∗}Bx^{∗}− Cx^{∗} ∈ (L_{θ})^{∗}, a^{T}x^{∗}− 1 ≥ 0,
(x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) + λ^{∗}(a^{T}x^{∗}− 1) = 0.

Combining with the definition of the dual cone again, it gives

(x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) = 0 and λ^{∗}(a^{T}x^{∗}− 1) = 0.

It follows from λ^{∗} > 0 that a^{T}x^{∗}− 1 = 0. Hence, (x^{∗}, λ^{∗}) solves the CCEiCP(B, C) (3).

Case (b): if λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G_{2}) (12), we have

x^{∗} ∈ L_{θ}, λ^{∗} > 0,

−λBx^{∗}− Cx^{∗} ∈ (L_{θ})^{∗}, a^{T}x^{∗}− 1 ≥ 0,
(x^{∗})^{T}(λ^{∗}Bx^{∗}− Cx^{∗}) + λ^{∗}(a^{T}x^{∗}− 1) = 0.

Then, it follows that (x^{∗})^{T}(−λ^{∗}Bx^{∗} − Cx^{∗}) = 0 and a^{T}x^{∗} − 1 = 0. Hence, (x^{∗}, −λ^{∗})
solves the CCEiCP(B, C) (3).

Case (c): if x^{∗} is a solution of CCLCP(−C, 0) (10) and x^{∗} 6= 0, then it is easy to see that
(_{a}T^{x}^{∗}x^{∗}, 0) solves the CCEiCP(B, C) (3).

Based on the above arguments, the proof is complete. 2

Remark 3.2. If let φ_{1} : IR^{n} × IR^{n} → IR be a circular cone complementarity function
[16], based on Theorem 3.2 and Theorem 3.3, then the CCEiCP(B, C) in (3) can be
reformulated as the following nonsmooth system of equations

Φ_{1}(z) = Φ_{1}(x, y, λ) :=

φ_{1}(x, y)
λBx − Cx − y

a^{T}x − 1

= 0. (13)

By this, we can solve the CCEiCP (3) via using semismooth Newton method to solve the nonsmooth system of equations (13).

Example 3.1. In the CCEiCP(B, C) (3) or the CCCP(G_{1}) (11) and the CCCP(G_{2})
(12), let θ = ^{π}_{3}. Suppose that the matrix B is the identity matrix, i.e., B := I, and the
matrix C takes the following matrices, respectively:

C1 := 1 −1 0 2

, C2 := −1 1 0 −2

and C3 :=

0 1

−1 0

.

For the CCEiCP(B, C_{1}), we find that w^{∗}_{1} := (x^{∗}_{1} = (α, 0)^{T}, λ^{∗}_{1} = 1) for any α > 0 are
the solutions of this problems. At this moment, λ^{∗}_{1} = 1 > 0, by Theorem 3.2, it follows
that w_{1}^{∗} = (x^{∗}_{1} = (α, 0)^{T}, λ^{∗}_{1} = 1) are also the solutions of CCCP(G1) (11).

For the CCEiCP(B, C_{2}), we know that w^{∗}_{2} := (x^{∗}_{2} = (α, 0)^{T}, λ^{∗}_{2} = −1) for any α > 0
are the solutions of this problems. Here, λ^{∗}_{2} = −1 < 0, by Theorem 3.2, it follows that
(x^{∗}_{2} = (α, 0)^{T}, −λ^{∗}_{2} = 1) are the solutions of the CCCP(G_{2}) (12).

For the CCEiCP(B, C_{3}), it is easy to see that w_{3}^{∗} := (x^{∗}_{3} = (1, −√

3)^{T}, λ^{∗}_{3} = 0) is a
solution of this problems. Now, λ^{∗}_{3} = 0, by Theorem 3.2 again, we have x^{∗}_{3} = (1, −√

3)^{T}
is a solution of the CCCP(−C_{3}, 0) (10).

Conversely, we know that w^{∗}_{1}, w^{∗}_{2} and x^{∗}_{3} solve the CCCP(G1) (11), the CCCP(G2) (12)
and the CCCP(−C_{3}, 0) (10), respectively. By Theorem 3.3, based on the value of λ^{∗}_{i} for
i = 1, 2, 3, it follows that w^{∗}_{1}, (x^{∗}_{2}, −λ^{∗}_{2}) and (_{a}T^{x}^{∗}^{3}x^{∗}_{3}, 0) are the solutions of CCEiCP(B, C_{i})
for i = 1, 2, 3.

### 3.2 Circular cone quadratic eigenvalue complementarity prob- lems

In this subsection, we consider the following circular cone quadratic eigenvalue comple-
mentarity problems (CCQEiCP for short). Given matrices A, B, C ∈ IR^{n×n}, the CCQE-
iCP seeks to find (x, y, λ) ∈ IR^{n}× IR^{n}× IR such that

CCQEiCP(A, B, C) :

y = λ^{2}Ax + λBx + Cx,
x ∈ L_{θ}, y ∈ (L_{θ})^{∗}, x^{T}y = 0,
a^{T}x = 1,

(14)

where a is an arbitrary fixed point with a ∈ int((L_{θ})^{∗}).

Before discussing the properties of the CCQEiCP (14), we first introduce the definition
of L_{θ}-hyperbolic that will be used later.

Definition 3.1. A triple (A, B, C), with A, B, C ∈ IR^{n×n} is called Lθ-hyperbolic if
(x^{T}Bx)^{2} ≥ 4(x^{T}Ax)(x^{T}Cx),

for all nonzero x ∈ L_{θ}.

Again, similar to Theorem 3.1, we build up the relation between the CCQEiCP (14) and the CCCP (8).

Theorem 3.4. Consider the CCQEiCP(A, B, C) given as in (14) where A is positive
definite and a triple (A, B, C) is L_{θ}-hyperbolic. Let F_{i} : IR^{n}→ IR^{n} be defined as follows:

F_{i}(x) = λ^{2}_{i}(x)Ax + λ_{i}(x)Bx + Cx if x 6= 0,

0 if x = 0, (15)

where λ_{i}(x) = ^{−(x}^{T}^{Bx)+(−1)}^{i+1}

√

(x^{T}Bx)^{2}−4(x^{T}Ax)(x^{T}Cx)

2(x^{T}Ax) for i = 1, 2. Then the following hold.

(a) If (x^{∗}, λ^{∗}) is a solution to the CCQEiCP(A, B, C), then x^{∗} solves either the CCCP(F_{1})
or the CCCP(F_{2}).

(b) If ¯x is a nonzero solution to the CCCP(F_{1}) or the CCCP(F_{2}), then (x^{∗}, λ^{∗}) is a
solution of the CCQEiCP(A, B, C) with x^{∗} = _{a}T^{x}^{¯}x¯ and λ^{∗} = λ1(¯x) or λ2(¯x).

Proof. The result of part(a) is trivial. We only need to prove part(b). Suppose that ¯x
is a nonzero solution to the CCCP(F_{i}) with F_{i} as in (15) for i = 1, 2. Then, for each i,
we know that

¯

x ∈ L_{θ}, λ^{2}_{i}(¯x)A¯x + λ_{i}(¯x)B ¯x + C ¯x ∈ L_{θ} and ¯x^{T}(λ^{2}_{i}(¯x)A¯x + λ_{i}(¯x)B ¯x + C ¯x) = 0.

Since a ∈ int((L_{θ})^{∗}) and ¯x ∈ L_{θ}, we have _{a}T^{1}x¯ > 0. From all the above, we conclude that
x^{∗} := _{a}T^{1}x¯x ∈ L¯ _{θ},

w^{∗} := (λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗}+ Cx^{∗} = _{a}T^{1}x¯ (λ^{2}_{i}(¯x)A¯x + λ_{i}(¯x)B ¯x + C ¯x) ∈ (L_{θ})^{∗},
(x^{∗})^{T}w^{∗} = _{a}T^{1}¯x

2

¯

x^{T}(λ^{2}_{i}(¯x)A¯x + λ_{i}(¯x)B ¯x + C ¯x)

= 0,
a^{T}x^{∗} = ^{a}_{a}^{T}T^{x}x^{¯}¯ = 1.

Thus, (λ^{∗}, x^{∗}) solves the CCQEiCP(A, B, C) and the proof is complete. 2

Remark 3.3. Similar to Remark 3.1, for the CCCP(F_{i}), where F_{i} is defined as in (15)
(i = 1, 2), we use the same skills to define

H_{i1}(x) =

µ1(x)
...
µ_{1}(x)
µ1(Fi(x)).

∈ IR^{n}

with i = 1, 2. Then, we obtain a solution of the CCQEiCP (14) via using Newton method
for solving the equation H_{i1}(x) = 0 (i = 1, 2).

Next, we consider two special classes of the circular cone complementarity problems, which have the following forms:

CCCP(G_{4}) :

x ∈ L_{θ}, λ ≥ 0,

λ^{2}Ax + λBx + Cx ∈ (L_{θ})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(λ^{2}Ax + λBx + Cx) + λ(a^{T}x − 1) = 0,

(16)

and

CCCP(G_{5}) :

x ∈ L_{θ}, λ ≥ 0,

λ^{2}Ax − λBx + Cx ∈ (L_{θ})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(λ^{2}Ax − λBx + Cx) + λ(a^{T}x − 1) = 0,

(17)

Similar to the cases of the CCEiCP (3), We will show the relationship between the
CCQEiCP(A, B, C), the CCLCP(C, 0) and the CCCP(G_{i}) for i = 4, 5.

Theorem 3.5. Let (x^{∗}, λ^{∗}) solves the CCQEiCP(A,B,C) (14). Then, the following hold.

(a) If λ^{∗} > 0, then (x^{∗}, λ^{∗}) solves the CCCP(G_{4}) (16).

(b) If λ^{∗} < 0, then (x^{∗}, −λ^{∗}) solves the CCCP(G_{5}) (17).

(c) If λ^{∗} = 0, then x^{∗} solves the CCLCP(C,0) (10).

Proof. From the assumption that (x^{∗}, λ^{∗}) solves the CCQEiCP(A,B,C) (14), we have
x^{∗} ∈ Lθ, (λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗}+ Cx^{∗} ∈ (Lθ)^{∗}, (x^{∗})^{T}(λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗}+ Cx^{∗} = 0
and a^{T}x^{∗} = 1. To proceed, we discuss three cases.

Case (a): if λ^{∗} > 0, then we have

(x^{∗})^{T}[(λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗}+ Cx^{∗}] + λ^{∗}(a^{T}x^{∗}− 1) = 0.

This implies that (x^{∗}, λ^{∗}) solves the CCCP(G_{4}) (16).

Case (b): if λ^{∗} < 0, then we have −λ^{∗} > 0 and

(x^{∗})^{T}(λ^{∗})^{2}Ax^{∗}− λ^{∗}Bx^{∗}+ Cx^{∗} − λ^{∗}(a^{T}x^{∗}− 1) = 0.

This indicates that (x^{∗}, −λ^{∗}) solves the CCCP(G_{5}) (17).

Case (c): if λ^{∗} = 0, then we have

(λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗}+ Cx^{∗} = Cx^{∗} ∈ L_{θ} and (x^{∗})^{T}Cx^{∗} = 0.

This says that (x^{∗}, 0) solves the CCLCP(C,0) (10).

Based on the above arguments, we prove the desired result. 2

Theorem 3.6. (a) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G_{4}) (16), then (x^{∗}, λ^{∗}) solves
the CCQEiCP(A,B,C) (14).

(b) If (x^{∗}, λ^{∗}) solves the CCCP(G_{5}) (17) and λ^{∗} 6= 0, then (x^{∗}, −λ^{∗}) solves the CCQEiCP(A,B,C)
(14).

(c) If x^{∗} solves CCLCP(−C, 0) and x^{∗} 6= 0, then (_{a}T^{x}^{∗}x^{∗}, 0) is a solution to the CCQEiCP(A,B,C)
(14).

Proof. Again, the proof is done by discussing three cases.

Case (a): if λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G_{4}) (16), then we have

x^{∗} ∈ L_{θ}, λ^{∗} > 0,

(λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗}+ Cx^{∗} ∈ (L_{θ})^{∗}, a^{T}x^{∗}− 1 ≥ 0,
(x^{∗})^{T}(λ^{∗})^{2}Ax^{∗}+ λ^{∗}Bx^{∗} + Cx^{∗} + λ^{∗}(a^{T}x^{∗} − 1) = 0.

By the definition of the dual cone, this implies (x^{∗})^{T}[(λ^{∗})^{2}Ax^{∗} + λ^{∗}Bx^{∗}+ Cx^{∗}] = 0 and
λ^{∗}(a^{T}x^{∗} − 1) = 0. In addition, it follows from λ^{∗} > 0 that a^{T}x^{∗} − 1 = 0. Hence, we
obtain that (x^{∗}, λ^{∗}) solves the CCQEiCP(A,B,C) (14).

Case (b): if λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the CCCP(G5) (17), then we have

x^{∗} ∈ L_{θ}, λ^{∗} > 0,

(−λ^{∗})^{2}Ax^{∗}+ (−λ^{∗})Bx^{∗}+ Cx^{∗} ∈ (L_{θ})^{∗}, a^{T}x^{∗} − 1 ≥ 0,
(x^{∗})^{T}(−λ^{∗})^{2}Ax^{∗}+ (−λ^{∗})Bx^{∗}+ Cx^{∗} + λ^{∗}(a^{T}x^{∗}− 1) = 0.

This says that (x^{∗})^{T}[(−λ^{∗})^{2}Ax^{∗} + (−λ^{∗})Bx^{∗} + Cx^{∗}] = 0 and a^{T}x^{∗} − 1 = 0. Hence, we
obtain that (x^{∗}, −λ^{∗}) solves the CCQEiCP(A,B,C) (14).

Case (c): if x^{∗} solves the CCLCP(−C, 0) (10) and x^{∗} 6= 0, then it is easy to see that
(_{a}T^{x}^{∗}x^{∗}, 0) solves the CCQEiCP(A,B,C) (14).

Based on the above arguments, the proof is complete. 2

Remark 3.4. Similar to Remark 3.2, the CCQEiCP(A, B, C) in (14) can be reformulated as the following nonsmooth system of equations

Ψ_{1}(z) = Ψ_{1}(x, y, λ) :=

φ1(x, y)

λ^{2}Ax + λBx + Cx − y
a^{T}x − 1

= 0, (18)

From this, we can solve the CCQEiCP(A, B, C) (14) via using semismooth Newton method to solve the nonsmooth system of equations (18).

### 4 p-order cone eigenvalue complementarity problems

In this section, we consider p-order cone eigenvalue complementarity problems and p- order cone quadratic eigenvalue complementarity problems. Similar to the cases of the circular cone, we will show the relationship between the solution of p-order cone eigen- value complementarity problems (or the solution of p-order cone quadratic eigenvalue complementarity problems) and the solution of the corresponding p-order cone comple- mentarity problems, respectively.

### 4.1 p-order cone eigenvalue complementarity problems

Consider the p-order cone eigenvalue complementarity problems (POCEiCP for short):

find (x, y, λ) ∈ IR^{n}× IR^{n}× IR such that

POCEiCP(B, C) :

y = λBx − Cx,

x ∈ K_{p}, y ∈ (K_{p})^{∗}, x^{T}y = 0,
a^{T}x = 1,

(19)

where B, C ∈ IR^{n×n} and a is an arbitrary fixed point with a ∈ int((K_{p})^{∗}). First, we
introduce some notations for the sake of convenience. If x = (x1, x2) ∈ IR × IR^{n−1} and
x2 = (x^{(1)}_{2} , ..., x^{(n−1)}_{2} )^{T}, then we denote

|x^{(1)}_{2} |^{p}, · · · , |x^{(n−1)}_{2} |^{p}T

by |x2|^{p}. Similar to the
CCEiCP (Proposition 3.1), by H¨older’s inequality, we achieve the version of p-order cone
as the following proposition.

Proposition 4.1. Suppose that x := (x_{1}, x_{2}) ∈ K_{p}, y := (y_{1}, y_{2}) ∈ (K_{p})^{∗}. Then, the
following hold.

(a) x^{T}y ≥ 0.

(b) If y ∈ int(K_{p}), then x^{T}y > 0 if and only if x 6= 0.

(c) If x 6= 0 and y 6= 0, then

x^{T}y = 0 ⇒ x_{1} = kx_{2}k_{p} and |y_{2}|^{q} = α|x_{2}|^{p},
where α is a positive constant. Similarly, if x 6= 0 and y 6= 0, then

x^{T}y = 0 ⇒ y_{1} = ky_{2}k_{q} and |x_{2}|^{p} = β|y_{2}|^{q},
where β is a positive constant.

Proof. By the definition of dual cones, part(a) is obvious. For part(b), it is also trivial
that x^{T}y > 0 implies x 6= 0. We only prove the other direction as follows. Since x 6= 0,
we know that x_{1} > 0. Otherwise, we have that kx_{2}k_{p} ≤ x_{1} = 0 implies x = 0. Thus, it
yields

x^{T}y = x_{1}y_{1}+ x^{T}_{2}y_{2} ≥ x_{1}y_{1}− kx_{2}k_{p}ky_{2}k_{q} > x_{1}y_{1}− x_{1}y_{1} = 0.

Hence, we prove the result of part(b).

(c) “⇐” The proof of this direction is trivial. “⇒” By the assumption x 6= 0, we know
that x_{1} > 0. Similarly, we know that y_{1} > 0. Using the condition of x^{T}y = 0, the
H¨older’s inequality and the definition of circular cones, we have

x_{1}y_{1} = |x^{T}_{2}y_{2}| ≤ kx_{2}k_{p}ky_{2}k_{q}≤ x_{1}y_{1} (20)
which leads to

x_{1}y_{1} = kx_{2}k_{p}ky_{2}k_{q}. (21)
From x_{1} > 0, y_{1} > 0, x_{1} ≥ kx_{2}k_{p}, y_{1} ≥ ky_{2}k_{q} and the equality (21), it follows that

x_{1} = kx_{2}k_{p} and y_{1} = ky_{2}k_{q}. (22)
Moreover, by (20), (21) and (22) again, we have

x^{T}_{2}y_{2} = −x_{1}y_{1} = −kx_{2}kky_{2}k.

This implies the equality holds in the H¨older’s inequality. Thus, there exists a constant α such that

|y_{2}|^{q} = α|x_{2}|^{p}.
Then, the proof is complete. 2

Based on the cases of circular cone, we build up the relation between the POCEiCP and the below p-order cone complementarity problem (POCCP):

POCCP(F ) : x ∈ K_{p}, F (x) ∈ (K_{p})^{∗}, x^{T}F (x) = 0, (23)
where F : IR^{n}→ IR^{n} be defined as in (9), i.e.,

F (x) =

_{x}^{T}_{Cx}

x^{T}BxBx − Cx if x 6= 0,

0 if x = 0.

Similar to Theorem 3.1, we use the same technique to establish the relation between the POCEiCP and the POCCP.

Theorem 4.1. Consider the POCEiCP(B, C) given as in (19) where B is positive defi- nite. Then, the following hold.

(a) If (x^{∗}, λ^{∗}) solves the POCEiCP(B, C), then x^{∗} solves the POCCP(F ) (23).

(b) If ¯x is a nonzero solution to the POCCP(F ) (23), then (x^{∗}, λ^{∗}) solves the POCEiCP(B, C)
with λ^{∗} = ^{x}_{x}^{¯}_{¯}^{T}T^{C ¯}B ¯^{x}x and x^{∗} = _{a}T^{x}^{¯}x¯.

Proof. The proof is similar to Theorem 3.1. Hence, we omit it. 2

Likewise, we consider the following three special classes of the p-order cone comple- mentarity problems:

POCLCP(−C, 0) : x ∈ Kp, −Cx ∈ (Kp)^{∗}, x^{T}(−Cx) = 0, (24)

POCCP(G_{1}) :

x ∈ Kp, λ ≥ 0,

λBx − Cx ∈ (K_{p})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(λBx − Cx) + λ(a^{T}x − 1) = 0,

(25)

and

POCCP(G_{2}) :

x ∈ K_{p}, λ ≥ 0,

−λBx − Cx ∈ (K_{p})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(−λBx − Cx) + λ(a^{T}x − 1) = 0,

(26)

where the matrixes B, C are given as in the POCEiCP (19).

Analogous to Theorem 3.2 and Theorem 3.3, we have the relation between the POCE- iCP and the POCCP.

Theorem 4.2. Let (x^{∗}, λ^{∗}) solves thePOCEiCP(B,C) (19). Then, the following hold.

(a) If λ^{∗} > 0, then (x^{∗}, λ^{∗}) solves the POCCP(G1) (25).

(b) If λ^{∗} < 0, then (x^{∗}, −λ^{∗}) solves the POCCP(G_{2}) (26).

(c) If λ^{∗} = 0, then x^{∗} solves the POCLCP(−C,0)(24).

Proof. The proof is similar to Theorem 3.2. Hence, we omit it. 2

Theorem 4.3. (a) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the POCCP(G_{1}) (25), then (x^{∗}, λ^{∗})
solves the POCEiCP(B,C) (19).

(b) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the POCCP(G_{2}) (26), then (x^{∗}, −λ^{∗}) solves the POCEiCP(B,C)
(19).

(c) If x^{∗} solves the POCLCP(−C, 0)(24) and x^{∗} 6= 0, then (_{a}T^{x}^{∗}x^{∗}, 0) solves the POCEiCP(B, C)
(19).

Proof. The proof is similar to Theorem 3.3. Hence, we omit it. 2

Remark 4.1. If let φ_{2} : IR^{n}×IR^{n} → IR be a p-order cone complementarity function, based
on Theorem 4.2 and Theorem 4.3, then the POCEiCP(B, C) in (19) can be reformulated
as the following nonsmooth system of equations:

Φ_{2}(z) = Φ_{2}(x, y, λ) :=

φ2(x, y) λBx − Cx − y

a^{T}x − 1

= 0. (27)

Accordingly, we can solve the POCEiCP (19) via using semismooth Newton method to solve the nonsmooth system of equations (27).

### 4.2 p-order Cone quadratic eigenvalue complementarity prob- lems

Now, we consider the p-order cone quadratic eigenvalue complementarity problems (POC-
QEiCP for short) as follows: given matrices A, B, C ∈ IR^{n×n}, the POCQEiCP seeks to
find (x, y, λ) ∈ IR^{n}× IR^{n}× IR such that

POCQEiCP(A, B, C) :

y = λ^{2}Ax + λBx + Cx,
x ∈ Kp, y ∈ (Kp)^{∗}, x^{T}y = 0,
a^{T}x = 1,

(28)

where a is an arbitrary fixed point with a ∈ int((K_{p})^{∗}). We first introduce the definition
of K_{p}-hyperbolic, which will be used later.

Definition 4.1. A triple (A, B, C), with A, B, C ∈ IR^{n×n} is called K_{p}-hyperbolic if
(x^{T}Bx)^{2} ≥ 4(x^{T}Ax)(x^{T}Cx),

for all nonzero x ∈ K_{p}.

Again, analogous to Theorem 3.4, we build up the relation between POCQEiCP and the POCCP.

Theorem 4.4. Consider the POCQEiCP(A, B, C) given as in (28) where A is positive
definite and the triple (A, B, C) is Kp-hyperbolic. Let Fi : IR^{n} → IR^{n} be defined as in
(15). Then, the following hold.

(a) If (x^{∗}, λ^{∗}) solves the POCQEiCP(A, B, C), then x^{∗} solves either the POCCP(F_{1}) or
the POCCP(F_{2}).

(b) If ¯x is a nonzero solution to the POCCP(F_{1}) or the POCCP(F_{2}), then (x^{∗}, λ^{∗}) solves
the POCQEiCP(A, B, C) with x^{∗} = _{a}T^{x}^{¯}x¯ and λ^{∗} = λ_{1}(¯x) or λ_{2}(¯x), where

λi(x) = −(x^{T}Bx) + (−1)^{i+1}p(x^{T}Bx)^{2}− 4(x^{T}Ax)(x^{T}Cx)
2(x^{T}Ax)

for i = 1, 2.

Proof. The proof is similar to Theorem 3.4. Hence, we omit it. 2

Lastly, we consider the following two special kinds of p-order cone complementarity problems:

POCCP(G4) :

x ∈ K_{p}, λ ≥ 0,

λ^{2}Ax + λBx + Cx ∈ (K_{p})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(λ^{2}Ax + λBx + Cx) + λ(a^{T}x − 1) = 0,

(29)

and

POCCP(G_{5}) :

x ∈ K_{p}, λ ≥ 0,

λ^{2}Ax − λBx + Cx ∈ (K_{p})^{∗}, a^{T}x − 1 ≥ 0,
x^{T}(λ^{2}Ax − λBx + Cx) + λ(a^{T}x − 1) = 0.

(30)

Accordingly, we have the relation between the POCEiCP and the POCCP.

Theorem 4.5. Let (x^{∗}, λ^{∗}) solves the POCQEiCP(A,B,C) (28). Then, the following
hold.

(a) If λ^{∗} > 0, then (x^{∗}, λ^{∗}) solves the POCCP (G_{4}) (29).

(b) If λ^{∗} < 0, then (x^{∗}, −λ^{∗}) solves the POCCP(G_{5}) (30).

(c) If λ^{∗} = 0, then x^{∗} solves the POCLCP(C,0).

Proof. The proof is similar to Theorem 3.5. Hence, we omit it. 2

Theorem 4.6. (a) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the POCCP(G_{4}) (29), then (x^{∗}, λ^{∗})
solves the POCQEiCP(A,B,C) (28).

(b) If λ^{∗} 6= 0 and (x^{∗}, λ^{∗}) solves the POCCP(G_{5}) (30), then (x^{∗}, −λ^{∗}) solves POCQEiCP(A,B,C)
(28).

(c) If x^{∗} solves POCLCP(−C, 0) and x^{∗} 6= 0, then (_{a}T^{x}^{∗}x^{∗}, 0) is a solution of the POCLCP(−C,0).

Proof. The proof is similar to Theorem 3.6. Hence, we omit it. 2

Remark 4.2. Similar to Remark 3.4, the POCQEiCP(A, B, C) in (28) can be reformu- lated as the following nonsmooth system of equations:

Ψ_{2}(z) = Ψ_{2}(x, y, λ) :=

φ2(x, y)

λ^{2}Ax + λBx + Cx − y
a^{T}x − 1

= 0. (31)

From this, we can solve the POCQEiCP(A, B, C) (28) vis using semismooth Newton method to solve the nonsmooth system of equations (31).

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