TY - JOUR
T1 - A generalized projective dynamic for solving extreme and interior eigenvalue problems
AU - Zhang, Lei Hong
AU - Liao, Li Zhi
N1 - Funding information:
This research was supported in part by FRG grants from Hong Kong Baptist University and grants from the Research Grant Council of Hong Kong.
PY - 2008/11
Y1 - 2008/11
N2 - In [18] (Golub and Liao), a continuous-time system which is based on the projective dynamic is proposed to solve some concave optimization problems (with the unit ball constraint) resulted from extreme and interior eigenvalue problems. The convergence inside the unit ball is established; however, neither further convergence result outside the unit ball nor the stability analysis is available. Moreover, preliminary numerical experience indicates that this method is sensitive to a parameter whose optimal value is still difficult to determine. After analyzing the stability of this dynamic, in this paper, we develop a generalized model and analyze the convergence of the new model both inside and outside the unit ball. The flow of the generalized model is proved to converge almost globally to some eigenvector corresponding to the smallest eigenvalue, and share many surprisingly analogous properties with the Rayleigh quotient gradient flow. Links of our generalized projective dynamical system with other related works are also discussed. The efficiency of our new model is both addressed in theory and verified in numerical testing.
AB - In [18] (Golub and Liao), a continuous-time system which is based on the projective dynamic is proposed to solve some concave optimization problems (with the unit ball constraint) resulted from extreme and interior eigenvalue problems. The convergence inside the unit ball is established; however, neither further convergence result outside the unit ball nor the stability analysis is available. Moreover, preliminary numerical experience indicates that this method is sensitive to a parameter whose optimal value is still difficult to determine. After analyzing the stability of this dynamic, in this paper, we develop a generalized model and analyze the convergence of the new model both inside and outside the unit ball. The flow of the generalized model is proved to converge almost globally to some eigenvector corresponding to the smallest eigenvalue, and share many surprisingly analogous properties with the Rayleigh quotient gradient flow. Links of our generalized projective dynamical system with other related works are also discussed. The efficiency of our new model is both addressed in theory and verified in numerical testing.
KW - Continuous-time system
KW - Gradient flow
KW - Projective dynamical system
KW - Rayleigh quotient gradient flow
KW - Stability analysis
KW - Symmetric eigenproblem
UR - http://www.scopus.com/inward/record.url?scp=57749195076&partnerID=8YFLogxK
U2 - 10.3934/dcdsb.2008.10.997
DO - 10.3934/dcdsb.2008.10.997
M3 - Journal article
AN - SCOPUS:57749195076
SN - 1531-3492
VL - 10
SP - 997
EP - 1019
JO - Discrete and Continuous Dynamical Systems - Series B
JF - Discrete and Continuous Dynamical Systems - Series B
IS - 4
ER -