Analytical formulas for calculating the extremal ranks and inertias of A + BXB∗ when X is a fixed-rank Hermitian matrix

The rank of a matrix and the inertia of a square matrix are two of the most generic concepts in matrix theory for describing the dimension of the row/column vector space and the sign distribution of the eigenvalues of the matrix. Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which 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 first establish several groups of explicit formulas for calculating the maximal and minimal ranks and inertias of matrix expression A+X subject to a Hermitian matrix X that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximal and minimal ranks and inertias of matrix expression A+BXB∗ subject to a Hermitian matrix X that satisfies a fixed-rank and semi-definiteness restrictions and use the formulas obtained to characterize behaviors of A + BXB∗.