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

TL;DR

This paper introduces a novel approach for solving rank-constrained optimization problems with Schur-convex/concave objectives and multiple trace/logdet constraints, transforming the problem into a more tractable form and providing an iterative solution with proven convergence.

Contribution

It derives a structural solution for rank-constrained problems, transforms them into problems with unitary constraints, and proposes an iterative algorithm with convergence guarantees.

Findings

Proposed an iterative projected steepest descent algorithm.

Achieved convergence to a local optimum.

Numerical results demonstrate superior performance over baseline schemes.

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 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.