Continuous Models for Eigenvalue Problems

Transcription

1 Cotiuous Models for Eigevalue Problems Li-Zhi Liao ( Departmet of Mathematics Hog Kog Baptist Uiversity Dedicate this talk to the memory of Gee Golub Program for Gee aroud the world o Feb. 29, 2008, HKBU

2 hree Problems: (All matrices are symmetric) Etreme eigevalue problems Iterior eigevalue problems 1. Golub ad Liao, LAA, Zhag ad Liao, DCDS, (uder revisio) Geeralized eigevalue problems 3. Gao, Golub, ad Liao, LAA, 2008 Key idea: covert these problems ito optimizatio problems

3 Etreme eigevalue problems (I) mi ( λ, ) s. t. mi ( R s. t. mi ( R s. t. ) ) λ A = λ, c c = 1. A = 1. A c 1. (1) where c λ + 1, or c = A 1 ma + 1.

4 Etreme eigevalue problems (II) Properties of problem (1): is a local miimizer of (1) c is a global miimizer of (1) c satisfies A= λ mi, =1.

5 Iterior eigevalue problems (I) 1,.. ) )( ( mi 1... ) )( ( mi. 1,,.. 1 mi ), ( = = = s t c bi A ai A s t bi A ai A b a A s t R R c λ λ λ 1. ) )( ( or 1, ) )( ma( where = + b A a A c b a c i i i λ λ (2)

6 Iterior eigevalue problems (II) Properties of problem (2): 1) is a local miimizer of (2) is a global miimizer of (2) 2) Let * be a global miimizer of (2) ad η = (*) (A-aI )(A-bI )*, the a) If η > 0, the there eists o eigevalue of A i [a, b]. a) If η 0 the there eists at least oe eigevalue of A i [a, b]. b) If η = 0, the oe of the eigevalues of A must be a or b.

7 Iterior eigevalue problems (III) If we combie problems (1) ad (2), we have mi R H c (3) s. t. 1, where H=A for (1) ad H=(A-aI )(A-bI ) for (2) ad c λ ma (H)+1. Note: the objective fuctio is a cocave fuctio, so the solutio is always o the boudary.

8 Cotiuous models (I) he followig dyamic models are the results of cotiuous methods for optimizatio problems. For details of the cotiuous method, see L.-Z. Liao, H. D. Qi, ad L. Qi, Neurodyamical optimizatio, J. Global Optimi., 28, pp , Dyamic Model 1: Merit fuctio: Dyamical system: where Ω = { R f ( ) = H c. d( t) = { PΩ [ f ( )]}: e( ), (5) dt 1} ad P ( ) is a projectio operatio defied by P Ω ( y) = arg mi Ω Ω y 2, y R. (4)

9 Cotiuous models (II) Properties of Model 1: 1) For ay 0 R, there eists a uique solutio (t) of the dyamical system (5) with (t=t 0 )= 0 i [t 0,+ ). 2) If e( 0 )=0 (t) 0, t t 0. If e( 0 ) 0 lim t + e((t))=0. 3) e()=0 with 0 is a eigevector of H with 2 =1. 4) If 0 2 > 1, (t) 2 is mootoically decreasig to 1. If 0 2 < 1, (t) 2 is mootoically icreasig to 1. If 0 2 =1, (t) 2 1.

10 Cotiuous models (III) 5) If 0 0 * such that lim t + (t)=* ad * 2 =1. 1) For H=A, we have where k lim = mi{ i Note : ( λ, *) is what 1 t + ( t) A( t) = λ, 0, we wat! i = 1, L, }. 2) For H=(A-aI )(A-bI ), we have where k Θ = diag( θ, θ, L, θ ), ad V Note : his θ lim = mi{ i 1 k 2 t u v i ( t)( A ai i 0, V may ot be a eigevalue of k )( A bi i = 1, L, }, H = I with V ) ( t) = θ, = VΘV A. = ( v 1 k,, v 2, L, v ).

11 Cotiuous models (IV) Steps to obtai a eigevalue of A i [a, b]: 1) If θ k =0, oe of a or b is a eigevalue of A. his ca be verified by checkig the values of A*-a* 2 or A*- b* 2. 2) If θ k <0, solve (λ k - a)(λ k - b) = θ k. wo λ k values ca be obtaied. Pick the oe such that A*- λ k * 2 is very small. A eigevalue of A i [a, b] is foud. 3) If θ k >0, a ew startig poit has to be selected to start over. If after several tries, all θ k 's are positive, we may coclude that there is o eigevalue of A i [a, b].

12 Cotiuous models (V) Numerical results idicate that the performace (or CPU) is sesitive to the value of c. o improve this, we develop a ew model.

13 Cotiuous models (VI) Dyamic Model 2: Remember e() = P Ω [- f()]. Now, we defie e γ = P Ω [-γ f()], 0 γ 1. Merit fuctio: (o chage) f ( ) = H c. Dyamical system: (6) where d( t) = α ( ) f ( ) e dt ( ), eγ ( ) η 0, α( ) = f ( ) γη = 0. γ with 0 η 1. It ca be show that α() is locally Lipschitz cotiuous.

14 Cotiuous models (VI) All the theoretical results of Model 1 are also held for Model 2. I additio, Model 2 ejoys df ( ) α ( ) f ( ) e ( ). 2 γ dt γ 2

15 Numerical results (I) Eample 1: We costruct Eample 1 i the followig steps: 1. Select Λ=diag(-1e-4, -1e-4, 0, 0, 1,, 1) R. 2. Let B=rad(,) ad [Q,R]=qr(B). 3. Defie A=Q ΛQ. Stoppig criterio: d( t) dt Dyamical system solver: Matlab ODE45 with Relol=10-6 ad Absol=10-9

16 Numerical results (II) We select 0 =(1,,1), η = γ =1, ad 1) [a, b] (1) =[-310-4, ] (o eigevalue); 2) [a, b] (2) =[0.9, 1.1] (oe eigevalue); ad 3) [a, b] (3) =[0, 2] (two eigevalues). We fi =5,000, the iitial value at P Ω ( 0 ) ad select c i = A i, i=0,1,2. c able 1 CPUs for Model 1 (Model 2) of Eample 1 [a, b] [a, b] (1) [a, b] (2) [a, b](3) c (230.6) (34.08) (30.52) c (293.8) (32.30) (28.86) c (577.6) (26.34) (22.92)

17 Geeralized eigevalue problems (I) Problem: A =A, B =B R ad B is positive semidefiite, fid a scalar λ ad a ozero vector such that A= λb. (7) Assumptios: (i) here eists a costat c such that cb-a is positive semi-defiite. (ii) he miimum eigevalue of (A,B) is fiite. Result: For system (7), all its eigevalues are real. Furthermore, uder the above assumptios, system (7) has r real eigepairs, where r=rak(b).

18 Geeralized eigevalue problems (II) mi A s. t. B = 1, (8) Results: (i) is a local miimizer of problem (8) is a global miimizer of (8). (ii) is a global miimizer of (8) is a eigevector of (7) correspodig to the miimum geeralized eigevalue of (A,B).

19 Geeralized eigevalue problems (III) mi ( A cb) s. t. B 1, (9) where c is chose such that cb-a is positive semidefiite. Results: (i) (ii) is a local miimizer of problem (9) is a global miimizer of (9); is a global miimizer of (9) is a eigevector of (7) correspodig to the miimum geeralized eigevalue of (A,B).

20 Geeralized eigevalue problems (IV) mi ( A cb) s. t. mi y c B 1, ( Aˆ cp) y s. t. y Py 1, (10) where Â=L -1 AL -, y=l, P = I r 0 0, 0 ad L is a product of a permutatio matri ad a lower triagular matri.

21 Ed haks for your attetio