TY - JOUR
T1 - A Nonlinear Eigenvalue Problem Associated with the Sum-of-Rayleigh-Quotients Maximization
AU - Zhang , Lei-Hong
AU - Chang , Rui
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 - <p style="text-align: justify;">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.</p>