On the non-symmetric semidefinite Procrustes problem

In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices $X,Y \in \mathbb{R}^{n,m}$, find the matrix $A \in \mathbb{R}^{n,n}$ that minimizes the Frobenius norm of $AX-Y$ and which is such that $A+A^T$ is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where $A$ is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As for the symmetric case, we first show that the NSPSDP problem can be reduced to a smaller NSPSDP problem that always has a unique solution and where the matrix $X$ is diagonal and has full rank. Then, an efficient semi-analytical algorithm to solve the NSPSDP problem is proposed, solving the smaller and well-posed problem with a fast gradient method which guarantees a linear rate of convergence. This algorithm is also applicable to solve the complex NSPSDP problem, where $X,Y \in \mathbb{C}^{n,m}$, as we show the complex NSPSDP problem can be written as an overparametrized real NSPSDP problem. The efficiency of the proposed algorithm is illustrated on several numerical examples.