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

Abstract

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 nonconvex 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/log-determinant 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.