Minimal Rank Completions for Overlapping Blocks

TL;DR

This paper addresses a multi-objective optimization problem involving the minimization of ranks in block lower triangular matrices, providing a comprehensive set of all optimal solutions.

Contribution

It proves the existence of a simultaneous rank-minimizer and constructs the complete set of all such minimizers for the problem.

Findings

Existence of a simultaneous rank-minimizer proven.

Complete set of all minimizers constructed.

Framework applicable to block matrix rank optimization.

Abstract

We consider the multi-objective optimization problem of choosing the bottom left block-entry of a block lower triangular matrix to minimize the ranks of all block sub-matrices. We provide a proof that there exists a simultaneous rank-minimizer by constructing the complete set of all minimizers.