Given two matrices $X,B\in \mathbb{R}^{n\times m}$ and a set $\mathcal{A}\subseteq \mathbb{R}^{n\times n}$, a Procrustes problem consists in finding a matrix $A \in \mathcal{A}$ such that the Frobenius norm of $AX-B$ is minimized. When $\mathcal{A}$ is the set of the matrices whose symmetric part is positive semidefinite, we obtain the so-called non-symmetric positive semidefinite Procrustes (NSPDSP) problem. The NSPDSP problem arises in the estimation of compliance or stiffness matrix in solid and elastic structures. If $X$ has rank $r$, Baghel et al. (Lin. Alg. Appl., 2022) proposed a three-step semi-analytical approach: (1) construct a reduced NSPDSP problem in dimension $r\times r$, (2) solve the reduced problem by means of a fast gradient method with a linear rate of convergence, and (3) post-process the solution of the reduced problem to construct a solution of the larger original NSPDSP problem. In this paper, we revisit this approach of Baghel et al. and identify an unnecessary assumption used by the authors leading to cases where their algorithm cannot attain a minimum and produces solutions with unbounded norm. In fact, revising the post-processing phase of their semi-analytical approach, we show that the infimum of the NSPDSP problem is always attained, and we show how to compute a minimum-norm solution. We also prove that the symmetric part of the computed solution has minimum rank bounded by $r$, and that the skew-symmetric part has rank bounded by $2r$. Several numerical examples show the efficiency of this algorithm, both in terms of computational speed and of finding optimal minimum-norm solutions.

翻译：给定矩阵 $X,B\in \mathbb{R}^{n\times m}$ 与集合 $\mathcal{A}\subseteq \mathbb{R}^{n\times n}$，Procrustes问题旨在寻找矩阵 $A \in \mathcal{A}$ 使得 $AX-B$ 的Frobenius范数最小化。当 $\mathcal{A}$ 为对称部分半正定的矩阵集合时，即得到所谓的非对称半正定Procrustes（NSPDSP）问题。该问题出现于固体弹性结构中的柔度矩阵或刚度矩阵估计。若 $X$ 的秩为 $r$，Baghel等人（Lin. Alg. Appl., 2022）提出了一种三步半解析方法：（1）构建维度为 $r\times r$ 的约化NSPDSP问题；（2）采用具有线性收敛速率的快速梯度法求解约化问题；（3）对约化解进行后处理，以构造原始大规模NSPDSP问题的解。本文重新审视Baghel等人的方法，发现作者使用了一个不必要的假设，该假设会导致其算法无法达到最小值并产生无界范数的解。事实上，通过修正其半解析方法的后续处理阶段，我们证明NSPDSP问题的下确界总是可达的，并阐述了如何计算最小范数解。我们还证明所求解的对称部分具有不超过 $r$ 的最小秩，且反对称部分的秩不超过 $2r$。多个数值算例表明该算法在计算速度和寻找最优最小范数解方面均具有高效性。