Optimization Problems of Hermitian Quadratic Matrix-Valued Functions and Applications

In this paper, we investigate optimization problems for a Hermitian quadratic matrix-valued function involving two variable matrices. We derive algebraic formulas for the maximal and minimal ranks and partial inertias of this function based on a linearization method and specific block matrix rank and inertia transformations. Further, we establish necessary and sufficient conditions for the existence of solutions satisfying the associated matrix equation and various Hermitian inequalities. As applications, we obtain extremal ranks and inertias for the Hermitian generalized Schur complement with respect to Hermitian reflexive generalized inverses, and provide conditions for Hermitian reflexive generalized inverses to satisfy specific Hermitian properties. In addition, we apply these results to analyze optimization, stability, and congestion in a traffic flow network modeled by this quadratic function.