Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A+BXB∗A+BXB∗

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. If we take the inertia and rank of a Hermitian matrix as objective functions, then they are neither differentiable nor smooth. In this case, maximizing and minimizing the inertia and rank of a Hermitian matrix function could be regarded as a continuous-integer optimization problem. In this paper, we use some pure algebraic operations of matrices and their generalized inverses to derive explicit expansion formulas for calculating the global maximum and minimum ranks and inertias of the linear Hermitian matrix function A+BXB∗A+BXB∗ subject to some rank and definiteness restrictions on the variable matrix XX. Various direct consequences of the formulas in characterizing algebraic properties of A+BXB∗A+BXB∗ are also presented. In particular, solutions to a group of constrained optimization problems on the rank and inertia of a partially specified block Hermitian matrix are given.