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Riemannian optimization methods for some eigenvalue problems

English Abstract

In recent years, Riemannian optimization has become a popular research area. It has important applications in various fields, such as eigenvalue optimization problems, singular value problems, signal processing, machine learning, computer vision, principal component analysis, and so on. The applications of Riemannian optimization methods require some typical knowledge in Riemannian geometry. In the first two parts of this thesis, we just make a review of some basic concepts and introduce certain significant theoretical results in Riemannian geometry and Riemannian optimization. The general ideas of some Riemannian optimization methods are also discussed. The main research topic of this thesis is the applications of some Riemannian optimization methods to two special kinds of eigenvalue problems. In the third part, we consider a class of nonlinear eigenvalue problems (NEP) which arose from electronic structure calculations. A Riemannian Newton-Type method for minimizing a total energy function subject to orthogonality constraint is formulated. Under some mild assumptions, we establish the global and local quadratic convergence of the proposed method. In addition, a kind of positive definiteness condition for Riemannian Hessian of the total energy function at a solution point is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems. The fourth part is devoted to stochastic inverse eigenvalue problem (StIEP). We focus on the stochastic inverse eigenvalue problem of reconstructing a real stochastic matrix from prescribed spectrum. The StIEP is reformulated as an equivalent constrained optimization problem over several matrix manifolds. Then we propose a geometric Polak-Ribiere-Polyak-based nonlinear conjugate gradient method to solve the ` constrained optimization problem. The global convergence of the proposed method is established. Moreover, this method is further extended to the stochastic inverse eigenvalue problem with prescribed entries. Numerical results show the efficiency of the proposed method for large-scale StIEP. Finally, we make a conclusion of this thesis, and list some possible research topics in our further study.

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Zhao, Zhi,


Faculty of Science and Technology


Department of Mathematics




Riemannian manifolds



Jin, Xiao Qing

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