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Kim, Yunho
Mathematical Imaging Lab
Research Interests
  • Optimization, inverse problems, convex analysis, calculus of variations, partial differential equations, computational mathematics, medical/biomedical imaging

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An unconstrained global optimization framework for real symmetric eigenvalue problems

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Title
An unconstrained global optimization framework for real symmetric eigenvalue problems
Author
Kim, Yunho
Issue Date
2019-10
Publisher
Elsevier BV
Citation
APPLIED NUMERICAL MATHEMATICS, v.144, pp.253 - 275
Abstract
In this work, we interpret real symmetric eigenvalue problems in an unconstrained global optimization framework. More precisely, given two N×N matrices, a symmetric matrix A, and a symmetric positive definite matrix B, we propose and analyze a nonconvex functional F whose local minimizers are, indeed, global minimizers. These minimizers correspond to eigenvectors of the generalized eigenvalue problem Ax=λBx associated with its smallest eigenvalue. To minimize the proposed functional F, we consider the gradient descent method and show its global convergence. Furthermore, we provide explicit error estimates for eigenvalues and eigenvectors at the k th iteration of the method in terms of the gradient of F at the k th iterate x k . At the end, we provide a few numerical experiments to confirm our analysis and to compare with other methods, which reveals interesting numerical aspects of our proposed model.
URI
https://scholarworks.unist.ac.kr/handle/201301/26620
URL
https://www.sciencedirect.com/science/article/pii/S0168927419301138
DOI
10.1016/j.apnum.2019.05.006
ISSN
0168-9274
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