Application of \(m\)-weak group inverse in solving optimization problems (Q6085363)

The authors consider the \(m\)-weak group inverse of a matrix. This notion was introduced in [\textit{Y. Zhou} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 115, No. 1, Paper No. 2, 12 p. (2021; Zbl 1455.16036)]. They provide some new representations of these inverses using the full-rank factorization of \(A^k\), where \(A\) is the given matrix and \(k\) is its inverse (i.e., the least natural number satisfying \(\mathrm{rank}(A^k)=\mathrm{rank}(A^{k+1})\)). Moreover, they apply the \(m\)-weak inverse to the restricted optimization problem in the Frobenius form \[ \min\|A^{m+1}X-A^mB\|_F\text{ on condition that }\mathcal R(X)\subseteq\mathcal R(A^k). \]