Rank Constrained Schur-Convex Optimization with Multiple Trace/Log-Det Constraints

In this paper, we focus on a rank constrained optimization problem with general Schur-convex/concave objective function and multiple trace/log-determinant constraints. We first derive a structural result on the optimal solution of the rank constrained problem without relaxation 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 could even derive closed form global optimal solution. The numerical results show the superior performance of the our proposed technique over the baseline schemes, the rank relaxation based randomization technique.