• 沒有找到結果。

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

N/A
N/A
Protected

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

Copied!
20
0
0

(1)

### 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.

(2)

### 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, xTy = 0, (1)

where B, C ∈ IRn×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, xTy = 0, aTx = 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 ∈ IRn×n, the QEiCP consists of finding (x, y, λ) ∈ IRn × IRn× IR such

that 

y = λ2Ax + λBx + Cx, x ≥ 0, y ≥ 0, xTy = 0, aTx = 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, xTy = 0, aTx = 1,

or

y = λ2Ax + λBx + Cx, x ∈ K, y ∈ K, xTy = 0, aTx = 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

(3)

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 Lθ and p-order cone Kp. For the concepts of circular cone Lθ and p-order cone Kp, 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 Lθ 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].

(4)

Let Lθ 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 Lθ can be expressed as

(Lθ) = {x = (x1, x2) ∈ IR × IRn−1| kx2k ≤ x1cot θ} = Lπ

2−θ.

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

x = µ1(x)vx(1)+ µ2(x)vx(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 = kxx2

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

As for the p-order cone. Let Kp 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 · kp 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 Kp can be expressed as

(Kp) = {x = (x1, x2) ∈ IR × IRn−1| kx2kq ≤ x1} = Kq,

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

x = α1(x)vx(1)+ α2(x)vx(2), where α1(x), α2(x), vx1, and v2x are expressed as

α1(x) = (x1+ kx2kp)

2 , α2(x) = (x1− kx2kp)

2 ,

and

vx(1) =

 1 w



, vx(2) =

 1

−w

 . with w = kxx2

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

(5)

### 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 ∈ Lθ, y ∈ (Lθ), xTy = 0, aTx = 1,

(3)

where B, C ∈ IRn×n and a is an arbitrary fixed point with a ∈ int((Lθ)). We first study the solvability of the CCEiCP that will be used in subsequent analysis.

Proposition 3.1. Suppose that x := (x1, x2) ∈ Lθ, y := (y1, y2) ∈ (Lθ). Then, the following hold.

(a) xTy ≥ 0.

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

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

xTy = 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 xTy > 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

xTy = x1y1+ xT2y2 ≥ 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

(6)

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

x1y1 = |xT2y2| ≤ 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

xT2y2 = −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 = xT2y2 = −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 ∈ Lθ, F (x) ∈ (Lθ), xTF (x) = 0, (8) where

F (x) =

 xTCx

xTBxBx − 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 = xx¯¯TTC ¯B ¯xx and x = aTx¯x¯.

(7)

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 ∈ Lθ, x¯TC ¯x

¯

xTB ¯x· B ¯x − C ¯x ∈ (Lθ), and ¯xT  ¯xTC ¯x

¯

xTB ¯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 := λBx − Cx = aT1x¯

hx¯TC ¯x

¯ xTB ¯x



B ¯x − C ¯xi

∈ (Lθ), (x)Tw = aT1x¯

2h

¯ xT

¯xTC ¯x

¯

xTB ¯xB ¯x − C ¯x

i

= 0, aTx = aaTTxx¯¯ = 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(Lθ) and F (x) ∈ bd((Lθ)). 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 ∈ Lθ, −Cx ∈ (Lθ), xT(−Cx) = 0; (10)

CCCP(G1) :

x ∈ Lθ, λ ≥ 0,

λBx − Cx ∈ (Lθ), aTx − 1 ≥ 0, xT(λBx − Cx) + λ(aTx − 1) = 0;

(11) and

CCCP(G2) :

x ∈ Lθ, λ ≥ 0,

−λBx − Cx ∈ (Lθ), aTx − 1 ≥ 0, xT(−λBx − Cx) + λ(aTx − 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.

(8)

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, λBx− Cx ∈ (Lθ), (x)TBx− Cx) = 0 and aTx = 1.

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

(x)TBx− Cx) + λ(aTx− 1) = (x)TBx − Cx) = 0, which implies that (x, λ) solves the CCCP (G1) (11).

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

(x)TBx− Cx) − λ(aTx− 1) = (x)TBx− Cx) = 0.

This indicates that (x, −λ) solves the CCCP(G2) (12).

(c) If λ = 0, then it follows that

λBx− Cx = −Cx ∈ (Lθ) and (x)T(−Cx) = (x)TBx− 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 (aTxx, 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 ∈ Lθ, λ > 0,

λBx− Cx ∈ (Lθ), aTx− 1 ≥ 0, (x)TBx− Cx) + λ(aTx− 1) = 0.

(9)

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

(x)TBx− Cx) = 0 and λ(aTx− 1) = 0.

It follows from λ > 0 that aTx− 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 ∈ Lθ, λ > 0,

−λBx− Cx ∈ (Lθ), aTx− 1 ≥ 0, (x)TBx− Cx) + λ(aTx− 1) = 0.

Then, it follows that (x)T(−λBx − Cx) = 0 and aTx − 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 (aTxx, 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

aTx − 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 w1 := (x1 = (α, 0)T, λ1 = 1) for any α > 0 are the solutions of this problems. At this moment, λ1 = 1 > 0, by Theorem 3.2, it follows that w1 = (x1 = (α, 0)T, λ1 = 1) are also the solutions of CCCP(G1) (11).

For the CCEiCP(B, C2), we know that w2 := (x2 = (α, 0)T, λ2 = −1) for any α > 0 are the solutions of this problems. Here, λ2 = −1 < 0, by Theorem 3.2, it follows that (x2 = (α, 0)T, −λ2 = 1) are the solutions of the CCCP(G2) (12).

(10)

For the CCEiCP(B, C3), it is easy to see that w3 := (x3 = (1, −√

3)T, λ3 = 0) is a solution of this problems. Now, λ3 = 0, by Theorem 3.2 again, we have x3 = (1, −√

3)T is a solution of the CCCP(−C3, 0) (10).

Conversely, we know that w1, w2 and x3 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 λi for i = 1, 2, 3, it follows that w1, (x2, −λ2) and (aTx3x3, 0) are the solutions of CCEiCP(B, Ci) 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 ∈ IRn×n, the CCQE- iCP seeks to find (x, y, λ) ∈ IRn× IRn× IR such that

CCQEiCP(A, B, C) :

y = λ2Ax + λBx + Cx, x ∈ Lθ, y ∈ (Lθ), xTy = 0, aTx = 1,

(14)

where a is an arbitrary fixed point with a ∈ int((Lθ)).

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

Definition 3.1. A triple (A, B, C), with A, B, C ∈ IRn×n is called Lθ-hyperbolic if (xTBx)2 ≥ 4(xTAx)(xTCx),

for all nonzero x ∈ Lθ.

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 Lθ-hyperbolic. Let Fi : IRn→ IRn be defined as follows:

Fi(x) = λ2i(x)Ax + λi(x)Bx + Cx if x 6= 0,

0 if x = 0, (15)

where λi(x) = −(xTBx)+(−1)i+1

(xTBx)2−4(xTAx)(xTCx)

2(xTAx) 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).

(11)

(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 = aTx¯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(Fi) with Fi as in (15) for i = 1, 2. Then, for each i, we know that

¯

x ∈ Lθ, λ2i(¯x)A¯x + λi(¯x)B ¯x + C ¯x ∈ Lθ and ¯xT2i(¯x)A¯x + λi(¯x)B ¯x + C ¯x) = 0.

Since a ∈ int((Lθ)) and ¯x ∈ Lθ, we have aT1x¯ > 0. From all the above, we conclude that x := aT1x¯x ∈ L¯ θ,

w := (λ)2Ax+ λBx+ Cx = aT1x¯2i(¯x)A¯x + λi(¯x)B ¯x + C ¯x) ∈ (Lθ), (x)Tw = aT1¯x

2

¯

xT2i(¯x)A¯x + λi(¯x)B ¯x + C ¯x)



= 0, aTx = aaTTxx¯¯ = 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(Fi), where Fi is defined as in (15) (i = 1, 2), we use the same skills to define

Hi1(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 Hi1(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 ∈ Lθ, λ ≥ 0,

λ2Ax + λBx + Cx ∈ (Lθ), aTx − 1 ≥ 0, xT2Ax + λBx + Cx) + λ(aTx − 1) = 0,

(16)

and

CCCP(G5) :

x ∈ Lθ, λ ≥ 0,

λ2Ax − λBx + Cx ∈ (Lθ), aTx − 1 ≥ 0, xT2Ax − λBx + Cx) + λ(aTx − 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(Gi) for i = 4, 5.

(12)

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+ λBx+ Cx ∈ (Lθ), (x)T(λ)2Ax+ λBx+ Cx = 0 and aTx = 1. To proceed, we discuss three cases.

Case (a): if λ > 0, then we have

(x)T[(λ)2Ax+ λBx+ Cx] + λ(aTx− 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− λBx+ Cx − λ(aTx− 1) = 0.

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

Case (c): if λ = 0, then we have

)2Ax+ λBx+ Cx = Cx ∈ Lθ 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 (aTxx, 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 ∈ Lθ, λ > 0,

)2Ax+ λBx+ Cx ∈ (Lθ), aTx− 1 ≥ 0, (x)T(λ)2Ax+ λBx + Cx + λ(aTx − 1) = 0.

(13)

By the definition of the dual cone, this implies (x)T[(λ)2Ax + λBx+ Cx] = 0 and λ(aTx − 1) = 0. In addition, it follows from λ > 0 that aTx − 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 ∈ Lθ, λ > 0,

(−λ)2Ax+ (−λ)Bx+ Cx ∈ (Lθ), aTx − 1 ≥ 0, (x)T(−λ)2Ax+ (−λ)Bx+ Cx + λ(aTx− 1) = 0.

This says that (x)T[(−λ)2Ax + (−λ)Bx + Cx] = 0 and aTx − 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 (aTxx, 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 aTx − 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 ∈ Kp, y ∈ (Kp), xTy = 0, aTx = 1,

(19)

(14)

where B, C ∈ IRn×n and a is an arbitrary fixed point with a ∈ int((Kp)). 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 |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 := (x1, x2) ∈ Kp, y := (y1, y2) ∈ (Kp). Then, the following hold.

(a) xTy ≥ 0.

(b) If y ∈ int(Kp), then xTy > 0 if and only if x 6= 0.

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

xTy = 0 ⇒ x1 = kx2kp and |y2|q = α|x2|p, where α is a positive constant. Similarly, if x 6= 0 and y 6= 0, then

xTy = 0 ⇒ y1 = ky2kq 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 xTy > 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 kx2kp ≤ x1 = 0 implies x = 0. Thus, it yields

xTy = x1y1+ xT2y2 ≥ x1y1− kx2kpky2kq > 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 xTy = 0, the H¨older’s inequality and the definition of circular cones, we have

x1y1 = |xT2y2| ≤ kx2kpky2kq≤ x1y1 (20) which leads to

x1y1 = kx2kpky2kq. (21) From x1 > 0, y1 > 0, x1 ≥ kx2kp, y1 ≥ ky2kq and the equality (21), it follows that

x1 = kx2kp and y1 = ky2kq. (22) Moreover, by (20), (21) and (22) again, we have

xT2y2 = −x1y1 = −kx2kky2k.

(15)

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 ∈ Kp, F (x) ∈ (Kp), xTF (x) = 0, (23) where F : IRn→ IRn be defined as in (9), i.e.,

F (x) =

 xTCx

xTBxBx − 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 = xx¯¯TTC ¯B ¯xx and x = aTx¯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), xT(−Cx) = 0, (24)

POCCP(G1) :

x ∈ Kp, λ ≥ 0,

λBx − Cx ∈ (Kp), aTx − 1 ≥ 0, xT(λBx − Cx) + λ(aTx − 1) = 0,

(25)

and

POCCP(G2) :

x ∈ Kp, λ ≥ 0,

−λBx − Cx ∈ (Kp), aTx − 1 ≥ 0, xT(−λBx − Cx) + λ(aTx − 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.

(16)

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 (aTxx, 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

aTx − 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 ∈ IRn×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), xTy = 0, aTx = 1,

(28)

where a is an arbitrary fixed point with a ∈ int((Kp)). We first introduce the definition of Kp-hyperbolic, which will be used later.

(17)

Definition 4.1. A triple (A, B, C), with A, B, C ∈ IRn×n is called Kp-hyperbolic if (xTBx)2 ≥ 4(xTAx)(xTCx),

for all nonzero x ∈ Kp.

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 = aTx¯x¯ and λ = λ1(¯x) or λ2(¯x), where

λi(x) = −(xTBx) + (−1)i+1p(xTBx)2− 4(xTAx)(xTCx) 2(xTAx)

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 ∈ Kp, λ ≥ 0,

λ2Ax + λBx + Cx ∈ (Kp), aTx − 1 ≥ 0, xT2Ax + λBx + Cx) + λ(aTx − 1) = 0,

(29)

and

POCCP(G5) :

x ∈ Kp, λ ≥ 0,

λ2Ax − λBx + Cx ∈ (Kp), aTx − 1 ≥ 0, xT2Ax − λBx + Cx) + λ(aTx − 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).

(18)

(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 (aTxx, 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 aTx − 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).

### References

[1] S. Adly and H. Rammal, A new method for solving second-order cone eigenvalue complementarity problems, Journal of Optimization Theory and Applications, vol.

165, issue 1, pp. 563–585, 2015.

[2] J.F. Bonnans and H. Ramerez, Perturbation analysis of second-order cone pro- gramming problems, Mathematical Programming, vol. 104, no. 2-3, pp. 205–227, 2005.

[3] C. Br´as, M. Fukushima, A. Iusem, and J. J´udice, On the quadratic eigenvalue complementarity problem over a general convex cone, Applied Mathematics and Com- putation, vol. 271, pp. 391–403, 2015.

[4] C. Br´as, A. Iusem, and J. J´udice, On the quadratic eigenvalue complementarity problem, Journal of Global Optimization, vol. 66, no. 2, pp. 153–171, 2016.

(19)

[5] J.-S. Chen and S.-H. Pan, Semismooth Newton Methods for the Cone spectrum of Linear Transformations Relative to Lorentz Cones, Linear and Nonlinear Analysis, vol. 1, no. 1, pp. 13–36, 2015.

[6] J.-S. Chen and S.-H. Pan, A survey on SOC complementarity functions and so- lution methods for SOCPs and SOCCPs, Pacific Journal of Optimization, vol. 8, no.

1, pp. 33–74, 2012.

[7] J.-S. Chen, Conditions for error bounds and bounded level sets of some merit func- tions for SOCCP, Journal of Optimization Theory and Applications, vol. 135, no. 3, pp. 459–473, 2007.

[8] J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of second-order cone complementarity problem, Mathematical Programming, vol. 104, no. 2-3, pp. 293–327, 2005.

[9] J.-S. Chen, X. Chen, and P. Tseng, Analysis of nonsmooth vector-valued func- tions associated with second order cones, Mathematical Programming, vol. 101, no.

1, pp. 95–117, 2004.

[10] J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo Publishing, California, 2005.

[11] L. Fernandes, M. Fukushima, J. J´udice, and H. Sherali, The second-order cone eigenvalue complementarity problem, Optimization Methods and Software, vol.

31, no. 1, pp. 24–52, 2016.

[12] J. Faraut and A. Kor´anyi, Analysis on Symmetric Cones, Oxford Mathematical Monographs (New York: Oxford University Press), 1994.

[13] F. Facchinei and J. Pang, Finite-dimensional Variational Inequalities and Com- plementarity Problems, Springer, New York, 2003.

[14] M.S. Gowda, R. Sznajder, and J. Tao, P-properties for linear transformations on Euclidean Jordan algebras, Linear Algebra and its Applications, vol. 393, pp. 203–

232, 2004

[15] A. Iusem, J. J´udice, V. Sessa, and H. Sherali, On the numerical solution of the quadratic eigenvalue complementarity problem, Numerical Algorithms, vol. 72, pp. 721–747, 2016.

[16] X.-H. Miao, S.J. Guo, N. Qi, J.-S. Chen, Constructions of complementarity functions and merit functions for circular cone complementarity problem, Computa- tional Optimization and Applications, vol. 63, pp. 495–522, 2016.

(20)

[17] X.-H. Miao, W.-M. Hsu, C.T. Nguyen, and J.-S. Chen, The solvabilities of three optimization problems associated with second-order cone, Journal of Nonlinear and Convex Analysis, vol. 22, no. 5, pp. 937-967, 2021.

[18] X.-H. Miao, Y.-C. Lin, and J.-S. Chen, An alternative approach for a distance inequality associated with the second-order cone and the circular cone, Journal of Inequalities and Applications, vol. 2016, Article ID 291, 10 pages, 2016.

[19] X.-H. Miao, Y. Lu, and J.-S. Chen, From symmetric cone optimization to nonsymmetric cone optimization: Spectral decomposition, nonsmooth analysis, and projections onto nonsymmetric cones, Pacific Journal of Optimization, vol. 14, no. 3, pp. 399–419, 2018.

[20] X.-H. Miao, N. Qi, and J.-S. Chen, Projection formula and one type of spectral factorization associted with p-order cone, Journal of Nonlinear and Convex Analysis, vol. 18, no. 9, pp. 1699–1705, 2017.

[21] A. Pinto da Costa and A. Seeger, Cone-constrained eigenvalue prob- lems:theory and algorithms, Computational Optimization and Applications, vol. 45, issue 1, pp. 25–47, 2010.

[22] M. Queiroz, J. J´udice, and C. Humes, The symmetric eigenvalue complemen- tarity problem, Mathmatics of Computation, vol. 73, no.248 , pp. 1849–1863, 2003.

[23] A. Seeger, Eigenvalue analysis of equilibrium processes defined by linear comple- mentarity conditions, Linear Algebra and its Applications, vol. 292, pp. 1–14, 1999.

[24] A. Seeger, Quadratic eigenvalue problems under conic constraints, SIAM Journal on Matrix Analysis and Applications, vol. 32, no.3, pp. 700–721, 2011.

[25] A. Seeger and M. Torki, On eigenvalues induced by a cone constraint, Linear Algebra and its Applications,vol. 372, pp. 181–206, 2003.

[26] J. Wu and J.-S. Chen, A proximal point algorithm for the monotone second-order cone complementarity problem, Computational Optimization and Applications, vol.

51, no. 3, pp. 1037–1063, 2012.

[27] G.-L. Xue and Y.-Y. Ye, An efficient algorithm for minimizing a sum of p-norm, SIAM Journal on Optimization, vol. 10, pp. 551–579, 2000.

[28] J.C. Zhou and J.-S. Chen, Properties of circular cone and spectral factorization associated with circular cone, Journal of Nonlinear and Convex Analysis, vol. 14, no.

4, pp. 807–816, 2013.

Error bounds for the analytic center pOCP soln when distances have small errors... Decrease µ by a factor

Interior-point methods not amenable to warm start.. Chen,T

Numerical results are reported for some convex second-order cone programs (SOCPs) by solving the unconstrained minimization reformulation of the KKT optimality conditions,

Based on a class of smoothing approximations to projection function onto second-order cone, an approximate lower order penalty approach for solving second-order cone

In this paper, we have shown that how to construct complementarity functions for the circular cone complementarity problem, and have proposed four classes of merit func- tions for

According to the authors’ earlier experience on symmetric cone optimization, we believe that spectral decomposition associated with cones, nonsmooth analysis regarding

[1] F. Goldfarb, Second-order cone programming, Math. Alzalg, M.Pirhaji, Elliptic cone optimization and prime-dual path-following algorithms, Optimization. Terlaky, Notes on duality

In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases.. In addition,

Chen, Conditions for error bounds and bounded level sets of some merit func- tions for the second-order cone complementarity problem, Journal of Optimization Theory and

Chen, Properties of circular cone and spectral factorization associated with circular cone, to appear in Journal of Nonlinear and Convex Analysis, 2013.

It is well-known that, to deal with symmetric cone optimization problems, such as second-order cone optimization problems and positive semi-definite optimization prob- lems, this

According to the authors’ earlier experience on symmetric cone optimization, we believe that spectral decomposition associated with cones, nonsmooth analysis regarding cone-

For circular cone, a special non-symmetric cone, and circular cone optimization, like when dealing with SOCP and SOCCP, the following studies are cru- cial: (i) spectral

Abstract We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate

Taking second-order cone optimization and complementarity problems for example, there have proposed many ef- fective solution methods, including the interior point methods [1, 2, 3,

For finite-dimensional second-order cone optimization and complementarity problems, there have proposed various methods, including the interior point methods [1, 15, 18], the

Fukushima, On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity, SIAM Journal on Optimization,

This kind of algorithm has also been a powerful tool for solving many other optimization problems, including symmetric cone complementarity problems [15, 16, 20–22], symmetric

Theorem 5.6.1 The qd-algorithm converges for irreducible, symmetric positive definite tridiagonal matrices.. It is necessary to show that q i are in

Optim. Humes, The symmetric eigenvalue complementarity problem, Math. Rohn, An algorithm for solving the absolute value equation, Eletron. Seeger and Torki, On eigenvalues induced by

Due to the important role played by σ -term in the analysis of second-order cone, before developing the sufficient conditions for the existence of local saddle points, we shall

Chen, Conditions for error bounds and bounded level sets of some merit func- tions for the second-order cone complementarity problem, Journal of Optimization Theory and

Abstract In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone.. The