Generalized matrix nearness problems

TL;DR

This paper develops closed-form solutions and iterative methods for generalized matrix nearness problems, enabling efficient computation under various constraints like rank, symmetry, and structured matrices.

Contribution

It extends existing solutions to include a wider range of constraints and introduces a convergent iterative method for complex structured constraints.

Findings

Closed-form solutions for various constraints using SVD and GSVD.

An iterative method with proven linear and global convergence.

Applicability to structured matrices like Toeplitz, Hankel, and positive semidefinite.

Abstract

We show that the global minimum solution of $\lVert A - BXC \rVert$ can be found in closed-form with singular value decompositions and generalized singular value decompositions for a variety of constraints on $X$ involving rank, norm, symmetry, two-sided product, and prescribed eigenvalue. This extends the solution of Friedland--Torokhti for the generalized rank-constrained approximation problem to other constraints as well as provides an alternative solution for rank constraint in terms of singular value decompositions. For more complicated constraints on $X$ involving structures such as Toeplitz, Hankel, circulant, nonnegativity, stochasticity, positive semidefiniteness, prescribed eigenvector, etc, we prove that a simple iterative method is linearly and globally convergent to the global minimum solution.