  We consider the following constrained Rayleigh quotient optimization problem
(CRQopt) $$ \min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\,
x^{T}x=1\,\mbox{and}\,C^{T}x=b, $$ where $A$ is an $n\times n$ real symmetric
matrix and $C$ is an $n\times m$ real matrix.
  Usually, $m\ll n$. The problem is also known as the constrained eigenvalue
problem in the literature because it becomes an eigenvalue problem if the
linear constraint $C^{T}x=b$ is removed. We start by equivalently transforming
CRQopt into an optimization problem, called LGopt, of minimizing the Lagrangian
multiplier of CRQopt, and then an problem, called QEPmin, of finding the
smallest eigenvalue of a quadratic eigenvalue problem. Although such
equivalences has been discussed in the literature, it appears to be the first
time that these equivalences are rigorously justified. Then we propose to
numerically solve LGopt and QEPmin by the Krylov subspace projection method via
the Lanczos process. The basic idea, as the Lanczos method for the symmetric
eigenvalue problem, is to first reduce LGopt and QEPmin by projecting them onto
Krylov subspaces to yield problems of the same types but of much smaller sizes,
and then solve the reduced problems by some direct methods, which is either a
secular equation solver (in the case of LGopt) or an eigensolver (in the case
of QEPmin). The resulting algorithm is called the Lanczos algorithm. We perform
convergence analysis for the proposed method and obtain error bounds. The
sharpness of the error bound is demonstrated by artificial examples, although
in applications the method often converges much faster than the bounds suggest.
Finally, we apply the Lanczos algorithm to semi-supervised learning in the
context of constrained clustering.
