# A New Jacobian-Like Method for the Polyhedral Cone-Constrained Eigenvalue Problem

Authors: Guo Sun
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The eigenvalue problem over a polyhedral cone is studied in this paper. Based on the F-B NCP function, we reformulate this problem as a system of equations and propose a Jacobian-like method. The global convergence and local quadratic convergence of the proposed method are established under suitable assumptions. Preliminary numerical experiments for a special polyhedral cone are reported in this paper to show the validity of the proposed method.


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## Body

## 1. Introduction

In this paper, we consider the cone-constrained eigenvalue problem of finding a nonzero vectorx∈Rn and λ∈R such that
(1.1)K∋x⊥(Ax-λBx)∈K+,
with 𝒦 being a closed convex cone, 𝒦+ being its dual in the Euclidean space, and (A,B) being a pair of order n matrices such that B is symmetric and positive definite. If (x,λ) is a solution of the problem, λ is called a 𝒦-eigenvalue of (A,B) and x is called a 𝒦-eigenvector of (A,B) corresponding to λ.For this problem, we consider the polyhedral cone𝒦 in Rn as
(1.2)K={v∈Rn∣Uv⩾0},
where U∈Rs×n. It is easy to verify that its dual cone 𝒦+ assumes the following form:
(1.3)K+={u∈Rn∣u=U⊤ξ,ξ∈R+s},
where R+s stands for the nonnegative orthant of Rs.Throughout this paper, the cone-constrained eigenvalue problem (1.1) with 𝒦 being of the form (1.2) is referred, for simplicity, as the polyhedral cone-constrained eigenvalue problem in this paper.When𝒦=R+n, the cone-constrained eigenvalue problem reduces to the Pareto eigenvalue problem [1], which has the following form:
(1.4)x≥0,Ax-λBx≥0,〈x,Ax-λBx〉=0.
If A is symmetric and B is symmetric and positive definite in addition, the problem is reduced to a generalized Rayleigh quotient problem [2].The polyhedral cone-constrained eigenvalue problem is an important problem arising from the study of static equilibrium states of finite-dimensional mechanical systems with unilateral frictional contact [3]. Due to wide applications of the cone-constrained eigenvalue problem, the theoretical analysis and numerical solution methods are established [4–6]. Recently, Adly and Seeger have transformed the cone-constrained eigenvalue problem into a system of equations [1]; therefore, it is possible to apply the semismooth and smooth Newton method to solve the problem. Motivated by the work in [7, 8] for nonlinear complementarity problems, we design a smoothing Jacobian-like Method to solve the polyhedral cone-constrained eigenvalue problem. Compared with the smoothing method given in [7], the smoothing parameter is adjusted in a relaxed manner. The given numerical experiments show the efficiency of the proposed method.The remainder of this paper is organized as follows. In Section2, the reformulation of the polyhedral cone-constrained eigenvalue problem is presented, some basic well-known conclusions are reviewed, and the nonsingularity condition of the Jacobian matrix of the system of equations is established. In Section 3, we develop a new Jacobian-like method for the polyhedral cone-constrained eigenvalue problem, and the global convergence and local quadratic convergence of the proposed algorithm are discussed. Numerical experiments are presented in Section 4. Finally, the conclusions are given in Section 5.To finish this section, we give some notations used in this paper. For a continuously differentiable mappingG:Rn→Rm, let G′(x)∈Rm×n denote the Jacobian of G at a point x∈Rn, and let ∇G(x) denote its transposition. For a locally Lipschitz continuous function G:Rn→Rm, its generalized Jacobian in the sense of Clarke [9] is defined as
(1.5)∂G(x)=conv{V∈Rm×n∣V=lim⁡xk→x⁡G′(xk),G(x)is differentiable atxk},
where conv(A) is the convex hull of a set A. Proposition 2.6.2 (e) in [9] provides an overestimation of the generalized Jacobian of mapping G as follows:
(1.6)∂G(x)⊤⊆∂G1(x)×⋯×∂Gm(x),
where the right-hand side denotes the set of matrices in Rn×m whose i th column is the generalized gradient of the ith component function Gi. Traditionally, the right-hand side is denoted as
(1.7)∂CG(x)⊤=∂G1(x)×⋯×∂Gm(x)
and ∂CG(x) is called the C-subdifferential of G at x [10].The inner product of vectorsx, y∈Rn is denoted by x⊤y. We denote by ∥x∥ the Euclidean norm of x and denote by ∥·∥B the ellipsoidal norm with respect to symmetric and positive definite matrix B. Similarly, ∥A∥ denotes the spectral norm of a matrix A∈Rn×n, which is the induced matrix norm of the Euclidean vector norm.IfA∈Rn×n is any given matrix and Ξ⊆Rn×n is a nonempty set of matrices, we denote the distance from A to Ξ by dist(A,Ξ)=infB∈Ξ∥A-B∥.For a vectora∈Rn, Da=diag(a) denotes the diagonal matrix whose ith diagonal element is ai. R+n (R++n) stands for the nonnegative (positive) orthant of Rn.


## 2. Preliminaries

First, using a method similar to the one of [11], we transform the polyhedral cone-constrained eigenvalue problem as a system of equations via the well-known Fischer-Burmeister (F-B) function [12] from R2 to R:
(2.1)ϕFB(a,b)=a2+b2-a-bfora,b∈R.For the polyhedral cone-constrained eigenvalue problem, if(x,λ) is a solution of the problem, then (tx,λ) is also a solution of the problem for any t>0. Hence, to solve the problem, we only need to find a solution with x being a unit vector in the B-norm. In this sense, (x,λ) is a solution of the problem if and only if the following statements hold:
(2.2)x∈K={v∈Rn∣Uv≥0},Ax-λBx∈K+={u∈Rn∣u=U⊤ξ,ξ∈R+s},x⊤(Ax-λBx)=0,12‖x‖B2-1=0.By simple reformulation, the above form is equivalent to the following system:(2.3)Ux≥0,ξ≥0,ξ⊤Ux=0,Ax-λBx=U⊤ξ,12‖x‖B2-1=0.Using the F-B merit function, we can express the problem (1.1) as the following system of
(2.4)Φ(Ux,ξ)=0,Ax-λBx-U⊤ξ=0,12‖x‖B2-1=0,
where
(2.5)Φ(Ux,ξ)=(ϕFB((Ux)1,ξ1)⋮ϕFB((Ux)s,ξs)).Denotez=(x,ξ,λ) and a vector-valued function H:Rn+s+1→Rn+s+1 as follows:
(2.6)H(z)=H(x,ξ,λ)=(Φ(Ux,ξ)Ax-λBx-U⊤ξ12x⊤Bx-1).
Then the polyhedral cone-constrained eigenvalue problem is equivalent to the equation H(z)=0.Due to the nonsmoothness of the equation above, we approximate it by the following function [13]:
(2.7)ϕFBμ(a,b)=a2+b2+μ2-a-bforμ∈R++,a,b∈R.Correspondingly, the nonsmooth functionsΦ(Ux,ξ) and H(z) can be approximated via the following smooth functions Φμ(Ux,ξ) and Hμ(z), respectively,
(2.8)Φμ(Ux,ξ)=(ϕFBμ((Ux)1,ξ1)⋮ϕFBμ((Ux)s,ξs)),(2.9)Hμ(z)=(Φμ(Ux,ξ)Ax-λBx-U⊤ξ12x⊤Bx-1).Obviously,Hμ(z)→H(z) as μ→0+. Thus, the original problem is equivalent to the following equation:
(2.10)Hμ(z)=0.In the following, we will summarize some properties of functionHμ(z), which will be used in the subsequent analysis. The first result can be proved similarly to Proposition  2.3 in [8].Proposition 2.1.
The functionΦμ defined by (2.8) satisfies the inequality
(2.11)‖Φμ1(Ux,ξ)-Φμ2(Ux,ξ)‖≤κ‖μ1-μ2‖
for all x∈Rn, ξ∈Rs, and μ1, μ2>0, where κ=s. In particular,
(2.12)‖Φμ(Ux,ξ)-Φ(Ux,ξ)‖≤κμ
for all x∈Rn, ξ∈Rs and μ>0.As an immediate consequence of Proposition2.1, we obtain the following corollary.Corollary 2.2.
The functionHμ defined by (2.9) satisfies the inequality
(2.13)‖Hμ1(z)-Hμ2(z)‖=‖(Φμ1(Ux,ξ)-Φμ2(Ux,ξ)00)‖≤κ|μ1-μ2|
for all z=(x,ξ,λ)∈Rn+s+1 and μ1, μ2>0, where κ=s. In particular,
(2.14)‖Hμ(z)-H(z)‖≤κμ
for all z=(x,ξ,λ)∈Rn+s+1 and μ>0.By direct calculation, one has the following conclusion.Proposition 2.3.
For an arbitraryz∈Rn+s+1, it holds that
(2.15)Hμ′(z)=(D¯aUD¯b0A-λB-U⊤-Bx(Bx)⊤00),
where D¯a=diag(a-1,a-2,…,a-s), and D-b=diag(b-1,b-2,…,b-s)  such that
(2.16)a-i=(Ux)i(Ux)i2+ξi2+μ2-1,b-i=ξi(Ux)i2+ξi2+μ2-1,
for μ>0.Obviously,(2.17)-2<a-i<0,-2<b-i<0,∀i=1,2,…,s.Hence, bothD-a and D-b are negative definite diagonal matrices.Next, we will characterize differential properties of the nonsmooth mappingH(z) defined in (2.6). For the mapping H(z), we can make an overestimate of its generalized Jacobian in the sense of Clarke [9] by the definition of the C-subdifferential in the following Proposition.Proposition 2.4.
For an arbitraryz∈Rn+s+1, it holds that
(2.18)∂CH(z)=(DaUDb0A-λB-U⊤-Bx(Bx)⊤00),
where Da=diag(a1,a2,…,as), and Db=diag(b1,b2,…,bs)  such that
(2.19)ai=(Ux)i(Ux)i2+ξi2-1,bi=ξi(Ux)i2+ξi2-1
if (Ux)i2+ξi2≠0, and ai=τi-1, bi=ηi-1 for any (τi,ηi)∈R2 satisfying τi2+ηi2≤1 for the case that (Ux)i2+ξi2=0.In order to guarantee local fast convergence of our proposed algorithm, we need to consider the compatibility of the Jacobian matrixHμ′(z) in the following result.Lemma 2.5.
For anyz∈Rn+s+1, it holds that
(2.20)lim⁡μ↓0⁡dist⁡(Hμ′(z),∂CH(z))=0.Proof.
Since the difference only lies in the firsts columns between ∇Hμ(z) and ∂CH(z)⊤, it suffices to consider the distance between the first s columns of ∇Hμ(z) and ∂CH(z)⊤.
Define(2.21)β(z)={i∣[Ux]i=ξi=0,i=1,2,…,s}.
Let∇Hμ,i(z) be the ith column of matrix ∇Hμ(z) and Ui⊤ the ith column of matrix U⊤.
Then(2.22)lim⁡μ↓0∇Hμ,i(z)={(Ui⊤aieibi0),i∈{1,2,…,s}∖β(z),(-Ui⊤-ei0),i∈β(z),
whereei∈Rs is the ith unit vector.
The assertion follows from Proposition2.4 with (τi,ηi)=(0,0) for i∈β(z).To establish global convergence of our algorithm, it is necessary to study the condition under whichHμ′(z) is nonsingular for any μ>0.Theorem 2.6.
Forz=(x,ξ,λ)∈Rn×Rs×R, if x is a nonzero vector, and (A-λB) is a positive definite matrix, then Hμ′(z) is nonsingular for any μ>0.Proof.
Noticing that the nonsingularity of the matrix∇Hμ′(z) defined in Proposition 2.3 is equivalent to showing that the only solution of the following system:
(2.23)(D¯aUD¯b0A-λB-U⊤-Bx(Bx)⊤00)(pqr)=0
is the zero vector. This system can be rewritten as
(2.24)D¯aUp+D¯bq=0,(2.25)(A-λB)p-U⊤q-Bxr=0,(2.26)(Bx)⊤p=0.
Recalling thatD¯b defined in Proposition 2.3 is a negative definite matrix, we have and D-b is nonsingular. From (2.24), it holds that
(2.27)q=-D¯b-1D¯bUp.
By (2.25) and (2.27), we obtain that
(2.28)[(A-λB)+U⊤D¯b-1D¯aU]p-Bxr=0.
Premultiplying the equation above byp⊤ yields that
(2.29)p⊤[(A-λB)+U⊤D¯b-1D¯aU]p-p⊤Bxr=0.
Using (2.26), then
(2.30)pT[(A-λB)+U⊤D¯b-1D¯aU]p=0.
From the fact that bothD¯a and D¯b are negative definite matrices, it is not difficult to prove that the matrix U⊤D¯b-1D¯aU is positive semidefinite. Since (A-λB) is a positive definite matrix, (A-λB)+U⊤D¯b-1D¯aU must be a positive definite one. This implies, by (2.27) and (2.30),
(2.31)p=0,q=0.
Now, it follows from (2.25) that
(2.32)Bxr=0.
Pre-multiplying (2.25) by x⊤ yields
(2.33)x⊤Bxr=0.
From the positive definiteness ofB and the fact that x is nonzero, we assert that r=0. Hence, Hμ′(z) is nonsingular and the desired result follows.


## 3. Algorithm and Convergence

Based on the analysis in Section2, the polyhedral cone-constrained problem is equivalently transformed into the system of smooth equations (2.10). Motivated by the work in [7, 8] for nonlinear complementarity problem, we propose a new smoothing Jacobian-like method for the polyhedral cone-constrained eigenvalue problem. For convenience of description, we denote
(3.1)ψ(z)=12H(z)⊤H(z),ψμ(z)=12Hμ(z)⊤Hμ(z).At each iteration of the new proposed method, the searching directionΔz is computed by the Newton equation of the mapping ψμ(z). Meanwhile, μ is viewed as a smoothing parameter, which needs to be updated according to certain rule. Compared with the smoothing method given in [7], μ is adjusted in a relaxed manner. The following is the formal description of our designed method.Algorithm 3.1.
Step 0. Take tolerance ϵ≥0, choose constants z0∈Rn+s+1, η∈(0,1), α∈(0,1/2), σ∈(0,1-α), γ>0,κ=s, μ0=(α/2κ)∥H(z0)∥. Set k=0.
Step 1. Solve the linear system
(3.2)Hμk′(zk)Δzk=-H(zk)
to obtain Δzk∈Rs+n+1 and find the smallest mk∈{0,1,2,…} such that
(3.3)ψμk(zk+ηmkΔzk)-ψμk(zk)≤-2σηmkψ(zk).
Lettk=ηmk, zk+1=zk+tkΔzk.
Step 2. If H(zk+1)≤ϵ, stop; otherwise, if
(3.4)‖H(zk+1)‖≤α‖H(zk)‖+καμk,
then take μk+1 such that
(3.5)0<μk+1≤min⁡{μk2,α2κ‖H(zk+1)‖},dist⁡(Hμ′(zk+1),∂CH(zk+1))≤γ‖H(zk+1)‖,
and in other case, let μk+1=μk.Step 3. Let k=k+1 and go to Step  1.Remark 3.2.
(1) The positive sequence{μk} is nonincreasing and satisfies
(3.6)κμk≤α‖H(zk)‖,∀k=0,1,2,…
(2) Parameterμk remains unchanged except if condition (3.4) is met.The following result shows that Algorithm3.1 is well defined.Lemma 3.3.
Under the conditions in Theorem2.6, Algorithm 3.1 is well defined.Proof.
Obviously, for the sequence{μk} generated by Algorithm 3.1, it holds that μk>0. Hence, the linear system (3.2) is solvable by Theorem 2.6. To show the assertion, it suffices to show that for every k, the line search step terminates after a finite number of steps.
Since∇ψμk(zk)=∇Hμk(zk)Hμk(zk), one has
(3.7)ψμk(zk+tΔzk)-ψμk(zk)=t∇ψμk⊤(zk)Δzk+o(t)=tHμk⊤(zk)Hμk′(zk)Δzk+o(t)=-tHμk⊤(zk)H(zk)+o(t)=-tH⊤(zk)H(zk)+t(H(zk)-Hμk(zk))⊤H(zk)+o(t)≤-t‖H(zk)‖2+κμkt‖H(zk)‖+o(t)≤-(1-α)t‖H(zk)‖2+o(t)≤-2σψ(zk)+o(t),
where the second inequality follows from Remark 3.2 (1), and the last inequality uses the fact that (1-α)>σ. This shows that the inequality (3.3) holds for all m sufficiently large. Hence the desired result follows.To establish the convergence of Algorithm3.1, we need a technical lemma.Lemma 3.4.
Let sequence{zk} be generated by Algorithm 3.1, then {zk} remains in the level set L0={z∈Rs+n+1∣∥H(z)∥≤((4α+1)/(2α-1))∥H(z0)∥}.Proof.
For convenience, denoteΛ={0}∪{k∣∥H(zk)∥≤α∥H(zk-1)∥+(κ/α)μk-1}.
AssumeΛ consists of k0=0<k1<k2<⋯. For any k∈N={0,1,2,3,…}, let ki be the largest number in Λ such that ki≤k<ki+1. Then one has μk=μki by Remark 3.2 (2).
Together with the line search rule (3.2), one further has
(3.8)‖Hμk(zk)‖=‖Hμki(zk)‖≤‖Hμki(zk-1)‖≤⋯≤‖Hμki(zki)‖.
Combining this with Corollary2.2 yields
(3.9)‖H(zk)‖≤‖H(zk)-Hμk(zk)‖+‖Hμk(zk)‖=‖H(zk)-Hμk(zk)‖+‖Hμki(zk)‖≤κμk+‖Hμki(zki)‖≤κμki+‖Hμki(zki)-H(zki)‖+‖H(zki)‖≤2κμki+‖H(zki)‖.
On the other hand, it follows from condition (3.4) and Corollary 2.2 that
(3.10)‖H(zki+1)‖≤α‖H(zki+1-1)‖+κα-1μki+1-1≤α‖Hμki(zki+1-1)‖+α‖Hμki(zki+1-1)-H(zki)‖+κα-1μki≤α‖Hμki(zki)‖+κ(α+α-1)μki≤α‖H(zki)‖+κ(2α+α-1)μki.
Obviously, from the update rule of μk in (3.5), we obtain
(3.11)μki≤12μki-1=12μki-1≤⋯≤12iμ0=12iα2κ‖H(z0)‖.
Using (3.10) and (3.11), one further has
(3.12)‖H(zki+1)‖≤‖αH(zki)‖+κ(2α+α-1)μki≤α(α‖H(zki-1)‖+κ(2α+α-1)μki-1)+κ(2α+α-1)μki⋮≤αi+1‖H(z0)‖+αiκ(2α+α-1)μk0+αi-1κ(2α+α-1)μk1+⋯+κ(2α+α-1)μki≤αi+1‖H(z0)‖+αiκ(2α+α-1)μk0[1+12α+(12α)2+⋯+(12α)i]≤αi+1‖H(z0)‖+αiκ(2α+α-1)μk0(11-(1/2α))=(2α2+2α2α-1)αi+1‖H(z0)‖,
where the last equality uses the fact that μk0=μ0=(α/2κ)∥H(z0)∥.
Combining (3.9) with (3.11) and (3.12) yields
(3.13)‖H(zk)‖≤2κ12iμ0+(2α2+2α2α-1)αi‖H(z0)‖=2κ12iα2κ‖H(z0)‖+(2α2+2α2α-1)αi‖H(z0)‖≤(4α+12α-1)αi‖H(z0)‖≤(4α+12α-1)‖H(z0)‖.
This shows that {zk}⊂L0.Now, we are at the position to state our main result in this section.Theorem 3.5.
Suppose the level setL0 defined in Lemma 3.3 is bounded. Let {zk=(xk,ξk,λk)} be a sequence generated by Algorithm 3.1, and let z*=(x*,ξ*,λ*) be an accumulation point of {zk}. Under the conditions of Theorem 2.6, then ∥H(z*)∥=0, that is, (x*,λ*) is a solution of the polyhedral cone-constrained eigenvalue problem.Proof.
IfΛ is an infinite set, we conclude from (3.13) that
(3.14)‖H(z*)‖=lim⁡k→∞‖H(zk)‖≤lim⁡i→∞(4α+12α-1)αi‖H(z0)‖=0,
and the desired results follow.
In the following, we will show that the index setΛ is an infinite set by reductio ad absurdum. Suppose that Λ is a finite set. Let k^ be the largest number in Λ, then for all k-1≥k-, one has μk-1=μk- and
(3.15)‖H(zk)‖>α‖H(zk-1)‖+καμk-1≥μk->0.
On the other hand, from the line search condition (3.3), one has
(3.16)2σtkψ(zk)≤ψμk-(zk)-ψμk-(zk+1).
Summarizing these inequalities yields that limk→∞tkψ(zk)=0.
Since the sequence{ψ(zk)} is bounded away from zero by (3.15), we claim that tk→0 as k→∞. Hence the full step size is never accepted for all k sufficiently large, and inequality (3.3) does not hold for t-k=ηmk-1. That is, for sufficiently large k, it holds that
(3.17)-2σψ(zk)<ψμk-(zk+ηmk-1Δzk)-ψμk-(zk)ηmk-1.
Since the level setL0 is bounded and matrix Hμ′(z) is nonsingular by Theorem 2.6, then the sequence {Δzk} generated by the linear equation (3.2) is also bounded. Without loss of generality, we may assume that {zk} and {Δzk} converge to vectors z- and Δz-,  respectively. Taking limits in both sides of inequality above yields that
(3.18)-2σψ(z-)≤∇ψμk-⊤(z-)Δz-=Hμk-(z-)⊤Hμk-′(z-)Δz-=-Hμk-(z-)⊤H(z-)=-‖H(z-)‖2+(H(z-)-Hμk-(z-))⊤H(z-)≤-2(1-α)ψ(z-),
where the last inequality follows from Remark 3.2 (1).
Using the fact that the sequence{ψ(zk)} is bounded away from zero by (3.15) again, one has σ≥1-α, which is a contradiction with the choice of the parameter σ. Thus Λ is an infinite set and the desired result is proved.In order to analyze the local superlinear convergence of the Algorithm3.1, we need the concepts of semismooth and strongly semismooth [14, 15].For a locally Lipschitizian mappingG(x):Rn→Rm, it is said to be semismooth at x∈Rn if the limit
(3.19)lim⁡V∈∂G(x+th′)h′→h,t↓0Vh′
exists for any h∈Rn [15].G
(
x
)
:
R
n
→
R
m is called strongly semismooth at x∈Rn if G(x) is semismooth at x, and for any V∈∂G(x+h) with h>0 being sufficiently small, it holds that
(3.20)G(x+h)-G(x)-Vh=O(‖h‖2).Obviously, the vector-value functionH defined in (2.6) is semismooth and strongly semismooth according to Lemma 2.4(a) in [16]. In the following, we can obtain the following convergence rate result in a similar way to the proof of Theorem 3 in [7].Theorem 3.6.
Suppose the conditions in Theorem2.6 hold, and assume the level set L0 defined in Lemma 3.3 is bounded. z*=(x*,ξ*,λ*) is an accumulation of the sequence {zk} generated by Algorithm 3.1. If every element in ∂cH(z*) is nonsingular, then {zk} converges quadratically to z*.


## 4. Numerical Experiments

In this section, we give preliminary numerical experiments for a special polyhedral cone-constrained eigenvalue problem, that is, the Pareto eigenvalue problem. We implement the algorithm in MATLAB 7.0.In our numerical experiments, we consider the eigenvalue complementarity problem withB=I. As for the parameters used in our numerical experiments, we take η=0.5, α=0.25, σ=0.3, γ=20, and let the algorithm terminates when ∥∇H(z)∥≤10-6.To generate the initial point for these examples, we first generate a random vectorς∈ℝn with uniform distribution on [-1,1]n, and then set
(4.1)x0=ς〈1n,ς〉,λ0=〈x0,Ax0〉〈x0,Bx0〉,y0=Ax0-λ0Bx0,μ0=α2κ‖H(z0)‖
to generate the initial point z0=(μ0,x0,y0,λ0).Example 4.1.
Consider the pareto eigenvalue problem [1] for which
(4.2)A=[8-1434122-126].
This problem has 9 Pareto eigenvalues [5], which can all be detected using our algorithm via distinct initial points as seen from Table 1. In this table, SP denotes the starting point of z0, Iter denotes the number of iterations, and HV denotes the final value of H(zk) when the algorithm terminates.Table 1
Numerical result of Example4.1.
SP
Iter
Pareto
Pareto eigenvectorx
HV
λ
x
eigenvalue
x
1
x
2
x
3
4.4737
−0.03
0.8305
0.5563
3
λ
1 = 4.133975
0
0.9659
0.2588
6.51 ×10-14
5.6473
−0.15
0.6575
−0.7384
5
λ
2 = 4.602084
0.1814
0.9792
0.0907
2.7873 ×10-20
4.9735
0.3011
0.8987
0.0112
2
λ
3 = 5
0.3162
0.9487
0
3.269 ×10-20
4.179
−0.1735
0.3621
0.9158
4
λ
4 = 5.866025
0
0.2588
0.9659
5.142 ×10-16
4.5108
−0.075
−0.83
0.5536
5
λ
5 = 6
0
0
1
6.537 ×10-18
7.7682
0.7014
0.689
0.1823
3
λ
6 = 7
0.7071
0.7017
0
4.9206 ×10-16
6.672
0.98
−0.077
−0.1846
3
λ
7 = 8
1
0
0
4.0538 ×10-18
8.9328
0.5986
0.5013
0.2996
3
λ
8 = 9.397916
0.7875
0.4741
0.3937
2.7843 ×10-22
10.1813
0.7681
0.038
0.6391
4
λ
9 = 10
0.8944
0
0.4472
1.6339 ×10-18Example 4.2.
Consider the Pareto eigenvalue problem withA being randomly generated from [-1,1] such that(4.3)
[
-
0.02418
-
0.3211
0.8033
-
0.5086
-
0.1005
-
0.5833
0.3562
0.8881
0.692
-
0.2711
0.783
-
0.981
-
0.3515
0.7952
0.2862
-
0.4964
0.0175
0.8242
-
0.6552
0.9514
0.5246
-
0.3324
0.1454
0.2082
-
0.5095
-
0.207
-
0.4462
0.63
-
0.926
0.4338
0.3106
-
0.5063
-
0.63
-
0.7205
-
0.064
-
0.0387
0.1577
0.3793
-
0.3749
0.2866
0.943
0.9881
0.4758
-
0.6774
-
0.0051
0.0186
0.6457
-
0.3827
0.6346
0.5651
-
0.6577
0.8111
-
0.7765
0.7663
-
0.85
0.25
0.883
0.1164
-
0.5307
-
0.067
-
0.7281
-
0.3001
-
0.2325
0.9572
0.5332
0.2511
-
0.1114
0.2735
0.1749
-
0.5354
0.5104
-
0.4438
0.7247
0.2026
-
0.9093
0.9823
-
0.1535
0.5382
0.8484
-
0.3642
-
0.3736
-
0.8142
0.7142
-
0.2764
-
0.6698
-
0.2816
0.9924
-
0.892
-
0.34
0.5776
-
0.0735
-
0.5189
0.6387
-
0.5371
0.5544
-
0.448
0.2282
-
0.7703
-
0.5889
0.2689
]
.For this matrix, if we take the initial point(4.4)z0=(0.03572,0.3626,-0.4312,0.1223,0.4,-0.4128,-0.1927,0.1593,0.0775,-0.3911,0.3409,0.1971,1.3022,1.2591,0.58,-0.5704,0.1238,-0.1847,0.3483,0.6675,0.4573,-0.7037)T
and apply the JL method, we can obtain one Pareto eigenvalue 0.0592 and a corresponding eigenvector of (0,0,0.1204,0.315,0,0,0.8402,0.227,0,0.3591)T after 6 iteration steps within computer 1.016 seconds on a PIV 2.0 GHz personal computer.


## 5. Conclusions

In this paper, we first reformulate the the polyhedral cone-constrained eigenvalue problem as a system of equations based on the F-B function and then establish a smooth version by parameterizing the F-B function. A smoothing Jacobian-like method is developed to solve the problem. Some preliminary numerical results reported in the paper suggest efficiency of the proposed algorithm.



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*Source: 849249-2012-09-12.xml*