Formulas for calculating the extremal ranks and inertias of a matrix-valued function subject to matrix equation restrictions

Yongge Tian

paper

Formulas for calculating the extremal ranks and inertias of a matrix-valued function subject to matrix equation restrictions

arXiv:1301.2850

Abstract

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain 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 $A_{1} - B_{1} X B_{1}^{*}$ subject to the common Hermitian solution of a pair of consistent matrix equations $B_{2} X B_{2}^{*} = A_{2}$ and $B_{3} X B_{3}^{*} = A_{3}$ , and Hermitian solution of the consistent matrix equation $B_{4} X = A_{4}$ , respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations $B_{1} X B_{1}^{*} = A_{1}$ , $B_{2} X B_{2}^{*} = A_{2}$ and $B_{3} X B_{3}^{*} = A_{3}$ to have a common Hermitian solution, as necessary and sufficient conditions for the two matrix equations $B_{1} X B_{1}^{*} = A_{1}$ and $B_{4} X = A_{4}$ to have a common Hermitian solution.

arXiv comment21 page. arXiv admin note: text overlap with arXiv:1301.0986