# Analytical solutions to some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix inequalities

arXiv: https://arxiv.org/abs/1301.0986 · HTML: https://mlsanity.com/paper/1301.0986/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2013.

## Built on by (most cited first, 1 of 1)

- [Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity](https://mlsanity.com/paper/2312.06540/) (2023)

## Abstract (arXiv)

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which the decision variables are matrices running over certain feasible matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank- and inertia-objective functions of the Hermitian matrix expression A1 - B1XB1* subject to the linear matrix inequality B2XB2* succcurlyeq A2 (B2XB2* preccurlyeq A2) in the L"owner partial ordering, and give applications of these formulas in characterizing behaviors of some constrained matrix-valued functions.