  The low-rank matrix optimization with affine set (rank-MOA) is to minimize a
continuously differentiable function over a low-rank set intersecting with an
affine set. Under some suitable assumptions, the intersection rule of the
Fr\'{e}chet normal cone to the feasible set of the rank-MOA is established.
This allows us to propose the first-order optimality conditions for the
rank-MOA in terms of the so-called F-stationary point and the
$\alpha$-stationary point. Furthermore, the second-order optimality analysis,
including the necessary condition and the sufficient one, is proposed as well.
All these results will enrich the theory of low-rank matrix optimization and
give potential clues to designing efficient numerical algorithms for seeking
low-rank solutions. Meanwhile, we illustrate our proposed optimality analysis
for several specific applications of the rank-MOA including the Hankel matrix
approximation problem in system identification and the low-rank representation
on linear manifold in signal processing.
