Volume 2, Issue 2
A Nonlinear Eigenvalue Problem Associated with the Sum-of-Rayleigh-Quotients Maximization

Lei-Hong Zhang & Rui Chang

CSIAM Trans. Appl. Math., 2 (2021), pp. 313-335.

Published online: 2021-05

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  • Abstract

Recent applications in the data science and wireless communications give rise to a particular Rayleigh-quotient maximization, namely, maximizing the sum-of-Rayleigh-quotients over a sphere constraint. Previously, it is shown that maximizing the sum of two Rayleigh quotients is related with a certain eigenvector-dependent nonlinear eigenvalue problem (NEPv), and any global maximizer must be an eigenvector associated with the largest eigenvalue of this NEPv. Based on such a principle for the global maximizer, the self-consistent field (SCF) iteration turns out to be an efficient numerical method. However, generalization of sum of two Rayleigh-quotients to the sum of an arbitrary number of Rayleigh-quotients maximization is not a trivial task. In this paper, we shall develop a new treatment based on the S-Lemma. The new argument, on one hand, handles the sum of two and three Rayleigh-quotients maximizations in a simple way, and also deals with certain general cases, on the other hand. Our result gives a characterization for the solution of this sum-of-Rayleigh-quotients maximization and provides theoretical foundation for an associated SCF iteration. Preliminary numerical results are reported to demonstrate the performance of the SCF iteration.

  • Keywords

Eigenvector-dependent nonlinear eigenvalue problem, self-consistent-field iteration, Rayleigh quotient maximization, S-Lemma.

  • AMS Subject Headings

65L15, 90C26, 65H17, 15A18

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-313, author = {Lei-Hong Zhang , and Rui Chang , }, title = {A Nonlinear Eigenvalue Problem Associated with the Sum-of-Rayleigh-Quotients Maximization}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {2}, pages = {313--335}, abstract = {

Recent applications in the data science and wireless communications give rise to a particular Rayleigh-quotient maximization, namely, maximizing the sum-of-Rayleigh-quotients over a sphere constraint. Previously, it is shown that maximizing the sum of two Rayleigh quotients is related with a certain eigenvector-dependent nonlinear eigenvalue problem (NEPv), and any global maximizer must be an eigenvector associated with the largest eigenvalue of this NEPv. Based on such a principle for the global maximizer, the self-consistent field (SCF) iteration turns out to be an efficient numerical method. However, generalization of sum of two Rayleigh-quotients to the sum of an arbitrary number of Rayleigh-quotients maximization is not a trivial task. In this paper, we shall develop a new treatment based on the S-Lemma. The new argument, on one hand, handles the sum of two and three Rayleigh-quotients maximizations in a simple way, and also deals with certain general cases, on the other hand. Our result gives a characterization for the solution of this sum-of-Rayleigh-quotients maximization and provides theoretical foundation for an associated SCF iteration. Preliminary numerical results are reported to demonstrate the performance of the SCF iteration.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2021.nla.04}, url = {http://global-sci.org/intro/article_detail/csiam-am/18887.html} }
TY - JOUR T1 - A Nonlinear Eigenvalue Problem Associated with the Sum-of-Rayleigh-Quotients Maximization AU - Lei-Hong Zhang , AU - Rui Chang , JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 313 EP - 335 PY - 2021 DA - 2021/05 SN - 2 DO - http://doi.org/10.4208/csiam-am.2021.nla.04 UR - https://global-sci.org/intro/article_detail/csiam-am/18887.html KW - Eigenvector-dependent nonlinear eigenvalue problem, self-consistent-field iteration, Rayleigh quotient maximization, S-Lemma. AB -

Recent applications in the data science and wireless communications give rise to a particular Rayleigh-quotient maximization, namely, maximizing the sum-of-Rayleigh-quotients over a sphere constraint. Previously, it is shown that maximizing the sum of two Rayleigh quotients is related with a certain eigenvector-dependent nonlinear eigenvalue problem (NEPv), and any global maximizer must be an eigenvector associated with the largest eigenvalue of this NEPv. Based on such a principle for the global maximizer, the self-consistent field (SCF) iteration turns out to be an efficient numerical method. However, generalization of sum of two Rayleigh-quotients to the sum of an arbitrary number of Rayleigh-quotients maximization is not a trivial task. In this paper, we shall develop a new treatment based on the S-Lemma. The new argument, on one hand, handles the sum of two and three Rayleigh-quotients maximizations in a simple way, and also deals with certain general cases, on the other hand. Our result gives a characterization for the solution of this sum-of-Rayleigh-quotients maximization and provides theoretical foundation for an associated SCF iteration. Preliminary numerical results are reported to demonstrate the performance of the SCF iteration.

Lei-Hong Zhang & Rui Chang. (2021). A Nonlinear Eigenvalue Problem Associated with the Sum-of-Rayleigh-Quotients Maximization. CSIAM Transactions on Applied Mathematics. 2 (2). 313-335. doi:10.4208/csiam-am.2021.nla.04
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