  The NEPv approach has been increasingly used lately for optimization on the
Stiefel manifold arising from machine learning. General speaking, the approach
first turns the first order optimality condition, also known as the KKT
condition, into a nonlinear eigenvalue problem with eigenvector dependency
(NEPv) or a nonlinear polar decomposition with orthogonal factor dependency
(NPDo) and then solve the nonlinear problem via some variations of the
self-consistent-field (SCF) iteration. The difficulty, however, lies in
designing a proper SCF iteration so that a maximizer is found at the end.
Currently, each use of the approach is very much individualized, especially in
its convergence analysis to show that the approach does work or otherwise. In
this paper, a unifying framework is established. The framework is built upon
some basic assumptions. If the basic assumptions are satisfied, globally
convergence is guaranteed to a stationary point and during the SCF iterative
process that leads to the stationary point, the objective function increases
monotonically. Also a notion of atomic functions is proposed, which include
commonly used matrix traces of linear and quadratic forms as special ones. It
is shown that the basic assumptions are satisfied by atomic functions and by
convex compositions of atomic functions. Together they provide a large
collection of objectives for which the NEPv/NPDo approach is guaranteed to
work.
