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

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 Miao² 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 ∈ IRⁿ such that

 y = λBx − Cx,

x ≥ 0, y ≥ 0, x^Ty = 0, (1)

where B, C ∈ IR^n×n and y ∈ IRⁿ. 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 ∈ IRⁿ such that







y = λBx − Cx,

x ≥ 0, y ≥ 0, x^Ty = 0, a^Tx = 1,

with a  0. Usually, we choose a = (1, 1, · · · , 1)^T ∈ IRⁿ. 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) ∈ IRⁿ × IRⁿ× IR such

that 





y = λ²Ax + λBx + Cx, x ≥ 0, y ≥ 0, x^Ty = 0, a^Tx = 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^Ty = 0, a^Tx = 1,

or







y = λ²Ax + λBx + Cx, x ∈ K, y ∈ K^∗, x^Ty = 0, a^Tx = 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 Kⁿ), 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 Kⁿ, 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 IRⁿ, which is defined by Lθ := {x = (x1, x2) ∈ IR × IRⁿ⁻¹| kx2k ≤ x1tan θ}

with k · k denoting the Euclidean norm and θ ∈ (0,^π₂). It is well known that the dual cone of L_θ can be expressed as

(L_θ)^∗ = {x = (x₁, x₂) ∈ IR × IRⁿ⁻¹| kx₂k ≤ x₁cot θ} = L^π

2−θ.

Moreover, Zhou and Chen [28] gave the below spectral decomposition of x = (x₁, x₂) ∈ IR × IRⁿ⁻¹ with respect to L_θ:

x = µ₁(x)v_x⁽¹⁾+ µ₂(x)v_x⁽²⁾, where µ₁(x), µ₂(x), vx⁽¹⁾, and vx⁽²⁾ are expressed as

µ₁(x) = x₁ − kx₂k cot θ, µ₂(x) = x₁+ kx₂k tan θ, and

v⁽¹⁾_x = 1 1 + cot²θ

 1

− cot θ · w



, v⁽²⁾_x = 1 1 + tan²θ

 1

tan θ · w

 . with w = _kx^x2

2k if x₂ 6= 0, or any vector in IRⁿ⁻¹ satisfying kwk = 1 if x₂ = 0.

As for the p-order cone. Let K_p denote the p-order cone (p > 1) (see [19, 20, 27]) in IRⁿ, which is defined by

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

with k · k_p denoting the p-norm. It is clear to see that when p = 2, K₂ 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 = (x₁, x₂) ∈ IR × IRⁿ⁻¹| kx₂k_q ≤ x₁} = K_q,

where q > 1 and satisfies ¹_p +¹_q = 1. Miao, Qi and Chen [20] gave the following spectral decomposition of x = (x₁, x₂) ∈ IR × IRⁿ⁻¹ with respect to K_p:

x = α₁(x)v_x⁽¹⁾+ α₂(x)v_x⁽²⁾, where α₁(x), α₂(x), v_x¹, and v²_x are expressed as

α1(x) = (x₁+ kx₂k_p)

2 , α2(x) = (x₁− kx₂k_p)

2 ,

and

v_x⁽¹⁾ =

 1 w



, v_x⁽²⁾ =

 1

−w

 . with w = _kx^x2

2kp if x2 6= 0, or any vector in IRⁿ⁻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) ∈ IRⁿ× IRⁿ× IR such that

CCEiCP(B, C) :







y = λBx − Cx,

x ∈ L_θ, y ∈ (L_θ)^∗, x^Ty = 0, a^Tx = 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 := (x₁, x₂) ∈ L_θ, y := (y₁, y₂) ∈ (L_θ)^∗. Then, the following hold.

(a) x^Ty ≥ 0.

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

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

x^Ty = 0 ⇐⇒ x₁tan θ = kx₂k y₁cot θ = ky₂k and y = α(x₁tan²θ, −x₂), where α is a positive constant, or x = β(y₁cot²θ, −y₂) with a positive constant β.

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

(b) The result that x^Ty > 0 implies x 6= 0 is trivial. We only prove the other direction as follows. Since x 6= 0, we know that x₁ > 0. Otherwise, we have that kx₂k ≤ x₁ = 0 implies x = 0. Then, it follows that

x^Ty = x₁y₁+ x^T₂y₂ ≥ x₁y₁− kx₂kky₂k > x₁y₁− x₁tan θ y₁cot 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 x₁ > 0. Similarly, we obtain that y₁ > 0. Using

the condition x^Ty = 0, the Schwarz’s inequality and the definition of circular cones , we have

x₁y₁ = |x^T₂y₂| ≤ kx₂kky₂k ≤ (x₁tan θ)(y₁cot θ) = x₁y₁. (4) Thus, it follows that

(x₁tan θ)(y₁cot θ) = x₁y₁ = kx₂kky₂k. (5) By combining kx₂k ≤ x₁tan θ, ky₂k ≤ y₁cot θ and (5), it yields

x₁tan θ = kx₂k and y₁cot θ = ky₂k. (6) Besides, by (4), (5) and (6) again, we have

x^T₂y₂ = −x₁y₁ = −kx₂kky₂k.

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

y₂ = kx₂. (7)

Since kky₂k² = x^T₂y₂ = −x₁y₁ < 0, we have k < 0. Choosing α = −k, it leads to α > 0 and y2 = −αx2. Then, it follows from (6) and (7) that

y₁ = ky₂k tan θ = k − αx₂k tan θ = | − α|kx₂k tan θ = αkx₂k tan θ = αx₁tan²θ.

By this, we have y = α(x₁tan²θ, −x₂). 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^TF (x) = 0, (8) where

F (x) =

 _x^T_Cx

x^TBxBx − 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 : IRⁿ → IRⁿ 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^¯_¯^TT^{C ¯}B ¯^xx and x^∗ = _aT^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¯^TC ¯x

¯

x^TB ¯x· B ¯x − C ¯x ∈ (L_θ)^∗, and ¯x^T  ¯x^TC ¯x

¯

x^TB ¯xB ¯x − C ¯x



= 0.

Using a ∈ int((Lθ)^∗) and ¯x ∈ Lθ, we have _aT¹x¯ > 0. In view of all the above, we conclude that

x^∗ := _aT¹x¯x ∈ L¯ _θ,

w^∗ := λ^∗Bx^∗ − Cx^∗ = _aT¹x¯

hx¯^TC ¯x

¯ x^TB ¯x



B ¯x − C ¯xi

∈ (L_θ)^∗, (x^∗)^Tw^∗ = _aT¹x¯

2h

¯ x^T

¯x^TC ¯x

¯

x^TB ¯xB ¯x − C ¯x

i

= 0, a^Tx^∗ = ^a_a^TT^xx^¯¯ = 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 = µ₁(x)vx⁽¹⁾+ µ₂(x)vx⁽²⁾, F (x) can be written by F (x) = µ₁(F (x))v⁽²⁾x + µ₂(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

H₁(x) =











µ₁(x) ... µ1(x) µ₁(F (x)).











∈ IRⁿ and g₁(z) := 1

2kH₁(z)k² = 1

2H₁(z)^TH₁(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 H₁(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(G₁) :







x ∈ L_θ, λ ≥ 0,

λBx − Cx ∈ (L_θ)^∗, a^Tx − 1 ≥ 0, x^T(λBx − Cx) + λ(a^Tx − 1) = 0;

(11) and

CCCP(G2) :







x ∈ L_θ, λ ≥ 0,

−λBx − Cx ∈ (Lθ)^∗, a^Tx − 1 ≥ 0, x^T(−λBx − Cx) + λ(a^Tx − 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(G₁) (11).

(b) If λ^∗ < 0, then (x^∗, −λ^∗) solvesthe CCCP(G₂) (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^Tx^∗ = 1.

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

(x^∗)^T(λ^∗Bx^∗− Cx^∗) + λ^∗(a^Tx^∗− 1) = (x^∗)^T(λ^∗Bx^∗ − Cx^∗) = 0, which implies that (x^∗, λ^∗) solves the CCCP (G₁) (11).

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

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

This indicates that (x^∗, −λ^∗) solves the CCCP(G₂) (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(G₁) (11), then (x^∗, λ^∗) solves the CCEiCP(B,C) (3).

(b) If λ^∗ 6= 0 and (x^∗, λ^∗) solves the CCCP(G₂) (12), then (x^∗, −λ^∗) solves the CCEiCP(B,C) (3).

(c) If x^∗ solves the CCLCP(−C, 0) (10) and x^∗ 6= 0, then (_aT^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(G₁) (11), then we have







x^∗ ∈ L_θ, λ^∗ > 0,

λ^∗Bx^∗− Cx^∗ ∈ (L_θ)^∗, a^Tx^∗− 1 ≥ 0, (x^∗)^T(λ^∗Bx^∗− Cx^∗) + λ^∗(a^Tx^∗− 1) = 0.

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

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

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

Case (b): if λ^∗ 6= 0 and (x^∗, λ^∗) solves the CCCP(G₂) (12), we have







x^∗ ∈ L_θ, λ^∗ > 0,

−λBx^∗− Cx^∗ ∈ (L_θ)^∗, a^Tx^∗− 1 ≥ 0, (x^∗)^T(λ^∗Bx^∗− Cx^∗) + λ^∗(a^Tx^∗− 1) = 0.

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

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

Remark 3.2. If let φ₁ : IRⁿ × IRⁿ → 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

Φ₁(z) = Φ₁(x, y, λ) :=





φ₁(x, y) λBx − Cx − y

a^Tx − 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(G₁) (11) and the CCCP(G₂) (12), let θ = ^π₃. 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, C₁), 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 w₁^∗ = (x^∗₁ = (α, 0)^T, λ^∗₁ = 1) are also the solutions of CCCP(G1) (11).

For the CCEiCP(B, C₂), 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(G₂) (12).

For the CCEiCP(B, C₃), it is easy to see that w₃^∗ := (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(−C₃, 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(−C₃, 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 (_aT^x∗3x^∗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) ∈ IRⁿ× IRⁿ× IR such that

CCQEiCP(A, B, C) :







y = λ²Ax + λBx + Cx, x ∈ L_θ, y ∈ (L_θ)^∗, x^Ty = 0, a^Tx = 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^TBx)² ≥ 4(x^TAx)(x^TCx),

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 : IRⁿ→ IRⁿ be defined as follows:

F_i(x) = λ²_i(x)Ax + λ_i(x)Bx + Cx if x 6= 0,

0 if x = 0, (15)

where λ_i(x) = ^{−(xTBx)+(−1)i+1}

√

(x^TBx)²−4(x^TAx)(x^TCx)

2(x^TAx) 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(F₁) or the CCCP(F₂).

(b) If ¯x is a nonzero solution to the CCCP(F₁) or the CCCP(F₂), then (x^∗, λ^∗) is a solution of the CCQEiCP(A, B, C) with x^∗ = _aT^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, λ²_i(¯x)A¯x + λ_i(¯x)B ¯x + C ¯x ∈ L_θ and ¯x^T(λ²_i(¯x)A¯x + λ_i(¯x)B ¯x + C ¯x) = 0.

Since a ∈ int((L_θ)^∗) and ¯x ∈ L_θ, we have _aT¹x¯ > 0. From all the above, we conclude that x^∗ := _aT¹x¯x ∈ L¯ _θ,

w^∗ := (λ^∗)²Ax^∗+ λ^∗Bx^∗+ Cx^∗ = _aT¹x¯ (λ²_i(¯x)A¯x + λ_i(¯x)B ¯x + C ¯x) ∈ (L_θ)^∗, (x^∗)^Tw^∗ = _aT¹¯x

2

¯

x^T(λ²_i(¯x)A¯x + λ_i(¯x)B ¯x + C ¯x)



= 0, a^Tx^∗ = ^a_a^TT^xx^¯¯ = 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) ... µ₁(x) µ1(Fi(x)).











∈ IRⁿ

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(G₄) :







x ∈ L_θ, λ ≥ 0,

λ²Ax + λBx + Cx ∈ (L_θ)^∗, a^Tx − 1 ≥ 0, x^T(λ²Ax + λBx + Cx) + λ(a^Tx − 1) = 0,

(16)

and

CCCP(G₅) :







x ∈ L_θ, λ ≥ 0,

λ²Ax − λBx + Cx ∈ (L_θ)^∗, a^Tx − 1 ≥ 0, x^T(λ²Ax − λBx + Cx) + λ(a^Tx − 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(G₄) (16).

(b) If λ^∗ < 0, then (x^∗, −λ^∗) solves the CCCP(G₅) (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^∗)²Ax^∗+ λ^∗Bx^∗+ Cx^∗ ∈ (Lθ)^∗, (x^∗)^T(λ^∗)²Ax^∗+ λ^∗Bx^∗+ Cx^∗ = 0 and a^Tx^∗ = 1. To proceed, we discuss three cases.

Case (a): if λ^∗ > 0, then we have

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

This implies that (x^∗, λ^∗) solves the CCCP(G₄) (16).

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

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

This indicates that (x^∗, −λ^∗) solves the CCCP(G₅) (17).

Case (c): if λ^∗ = 0, then we have

(λ^∗)²Ax^∗+ λ^∗Bx^∗+ Cx^∗ = Cx^∗ ∈ L_θ and (x^∗)^TCx^∗ = 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(G₄) (16), then (x^∗, λ^∗) solves the CCQEiCP(A,B,C) (14).

(b) If (x^∗, λ^∗) solves the CCCP(G₅) (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 (_aT^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(G₄) (16), then we have







x^∗ ∈ L_θ, λ^∗ > 0,

(λ^∗)²Ax^∗+ λ^∗Bx^∗+ Cx^∗ ∈ (L_θ)^∗, a^Tx^∗− 1 ≥ 0, (x^∗)^T(λ^∗)²Ax^∗+ λ^∗Bx^∗ + Cx^∗ + λ^∗(a^Tx^∗ − 1) = 0.

By the definition of the dual cone, this implies (x^∗)^T[(λ^∗)²Ax^∗ + λ^∗Bx^∗+ Cx^∗] = 0 and λ^∗(a^Tx^∗ − 1) = 0. In addition, it follows from λ^∗ > 0 that a^Tx^∗ − 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^∗)²Ax^∗+ (−λ^∗)Bx^∗+ Cx^∗ ∈ (L_θ)^∗, a^Tx^∗ − 1 ≥ 0, (x^∗)^T(−λ^∗)²Ax^∗+ (−λ^∗)Bx^∗+ Cx^∗ + λ^∗(a^Tx^∗− 1) = 0.

This says that (x^∗)^T[(−λ^∗)²Ax^∗ + (−λ^∗)Bx^∗ + Cx^∗] = 0 and a^Tx^∗ − 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 (_aT^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

Ψ₁(z) = Ψ₁(x, y, λ) :=





φ1(x, y)

λ²Ax + λBx + Cx − y a^Tx − 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) ∈ IRⁿ× IRⁿ× IR such that

POCEiCP(B, C) :







y = λBx − Cx,

x ∈ K_p, y ∈ (K_p)^∗, x^Ty = 0, a^Tx = 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 × IRⁿ⁻¹ and x2 = (x⁽¹⁾₂ , ..., x⁽ⁿ⁻¹⁾₂ )^T, then we denote



|x⁽¹⁾₂ |^p, · · · , |x⁽ⁿ⁻¹⁾₂ |^pT

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 := (x₁, x₂) ∈ K_p, y := (y₁, y₂) ∈ (K_p)^∗. Then, the following hold.

(a) x^Ty ≥ 0.

(b) If y ∈ int(K_p), then x^Ty > 0 if and only if x 6= 0.

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

x^Ty = 0 ⇒ x₁ = kx₂k_p and |y₂|^q = α|x₂|^p, where α is a positive constant. Similarly, if x 6= 0 and y 6= 0, then

x^Ty = 0 ⇒ y₁ = ky₂k_q and |x₂|^p = β|y₂|^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^Ty > 0 implies x 6= 0. We only prove the other direction as follows. Since x 6= 0, we know that x₁ > 0. Otherwise, we have that kx₂k_p ≤ x₁ = 0 implies x = 0. Thus, it yields

x^Ty = x₁y₁+ x^T₂y₂ ≥ x₁y₁− kx₂k_pky₂k_q > x₁y₁− x₁y₁ = 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 x₁ > 0. Similarly, we know that y₁ > 0. Using the condition of x^Ty = 0, the H¨older’s inequality and the definition of circular cones, we have

x₁y₁ = |x^T₂y₂| ≤ kx₂k_pky₂k_q≤ x₁y₁ (20) which leads to

x₁y₁ = kx₂k_pky₂k_q. (21) From x₁ > 0, y₁ > 0, x₁ ≥ kx₂k_p, y₁ ≥ ky₂k_q and the equality (21), it follows that

x₁ = kx₂k_p and y₁ = ky₂k_q. (22) Moreover, by (20), (21) and (22) again, we have

x^T₂y₂ = −x₁y₁ = −kx₂kky₂k.

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

|y₂|^q = α|x₂|^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^TF (x) = 0, (23) where F : IRⁿ→ IRⁿ be defined as in (9), i.e.,

F (x) =

 _x^T_Cx

x^TBxBx − 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^¯_¯^TT^{C ¯}B ¯^xx and x^∗ = _aT^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(G₁) :







x ∈ Kp, λ ≥ 0,

λBx − Cx ∈ (K_p)^∗, a^Tx − 1 ≥ 0, x^T(λBx − Cx) + λ(a^Tx − 1) = 0,

(25)

and

POCCP(G₂) :







x ∈ K_p, λ ≥ 0,

−λBx − Cx ∈ (K_p)^∗, a^Tx − 1 ≥ 0, x^T(−λBx − Cx) + λ(a^Tx − 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(G₂) (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(G₁) (25), then (x^∗, λ^∗) solves the POCEiCP(B,C) (19).

(b) If λ^∗ 6= 0 and (x^∗, λ^∗) solves the POCCP(G₂) (26), then (x^∗, −λ^∗) solves the POCEiCP(B,C) (19).

(c) If x^∗ solves the POCLCP(−C, 0)(24) and x^∗ 6= 0, then (_aT^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 φ₂ : IRⁿ×IRⁿ → 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:

Φ₂(z) = Φ₂(x, y, λ) :=





φ2(x, y) λBx − Cx − y

a^Tx − 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) ∈ IRⁿ× IRⁿ× IR such that

POCQEiCP(A, B, C) :







y = λ²Ax + λBx + Cx, x ∈ Kp, y ∈ (Kp)^∗, x^Ty = 0, a^Tx = 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^TBx)² ≥ 4(x^TAx)(x^TCx),

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 : IRⁿ → IRⁿ 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(F₁) or the POCCP(F₂).

(b) If ¯x is a nonzero solution to the POCCP(F₁) or the POCCP(F₂), then (x^∗, λ^∗) solves the POCQEiCP(A, B, C) with x^∗ = _aT^x¯x¯ and λ^∗ = λ₁(¯x) or λ₂(¯x), where

λi(x) = −(x^TBx) + (−1)ⁱ⁺¹p(x^TBx)²− 4(x^TAx)(x^TCx) 2(x^TAx)

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,

λ²Ax + λBx + Cx ∈ (K_p)^∗, a^Tx − 1 ≥ 0, x^T(λ²Ax + λBx + Cx) + λ(a^Tx − 1) = 0,

(29)

and

POCCP(G₅) :







x ∈ K_p, λ ≥ 0,

λ²Ax − λBx + Cx ∈ (K_p)^∗, a^Tx − 1 ≥ 0, x^T(λ²Ax − λBx + Cx) + λ(a^Tx − 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 (G₄) (29).

(b) If λ^∗ < 0, then (x^∗, −λ^∗) solves the POCCP(G₅) (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(G₄) (29), then (x^∗, λ^∗) solves the POCQEiCP(A,B,C) (28).

(b) If λ^∗ 6= 0 and (x^∗, λ^∗) solves the POCCP(G₅) (30), then (x^∗, −λ^∗) solves POCQEiCP(A,B,C) (28).

(c) If x^∗ solves POCLCP(−C, 0) and x^∗ 6= 0, then (_aT^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:

Ψ₂(z) = Ψ₂(x, y, λ) :=





φ2(x, y)

λ²Ax + λBx + Cx − y a^Tx − 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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