  Rank-constrained optimization problems have received an increasing intensity
of interest recently, because many optimization problems in communications and
signal processing applications can be cast into a rank-constrained optimization
problem. However, due to the non-convex nature of rank constraints, a
systematic solution to general rank-constrained problems has remained open for
a long time. In this paper, we focus on a rank-constrained optimization problem
with a Schur-convex/concave objective function and multiple
trace/logdeterminant constraints. We first derive a structural result on the
optimal solution of the rank-constrained problem using majorization theory.
Based on the solution structure, we transform the rank-constrained problem into
an equivalent problem with a unitary constraint. After that, we derive an
iterative projected steepest descent algorithm which converges to a local
optimal solution. Furthermore, we shall show that under some special cases, we
can derive a closed-form global optimal solution. The numerical results show
the superior performance of our proposed technique over the baseline schemes.
