Computing the nearest low-rank correlation matrix by a simplified SQP algorithm

In this paper, we propose a numerical method for computing the nearest low-rank correlation matrix (LRCM). Motivated by the fact that the nearest LRCM problem can be reformulated as a standard nonlinear equality constrained optimization problem with matrix variables via the Gramian representation, we propose a new algorithm based on the sequential quadratic programming (SQP) method. On each iteration, we do not solve the quadratic program (QP) corresponding to the exact Hessian, but a modified QP with a simpler Hessian. This QP subproblem can be solved efficiently by equivalently transforming it to a sparse linear system. Global convergence is established and preliminary numerical results are presented to demonstrate the proposed method is potentially useful.