A generalized projective dynamic for solving extreme and interior eigenvalue problems

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.