- Analytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions Tian Yongge(yongge.tiangmail.com) Abstract: group of analytical formulas formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function $$\phi(X) = \left(\, AXB + C\,\right)\!M\!\left(\, AXB + C \right)^{*} + D$$ are established and their consequences are presented, where $A$, $B$, $C$ and $D$ are given complex matrices with $A$ and $C$ Hermitian. As applications, necessary and sufficient conditions for the two general quadratic matrix-valued functions \begin{align*} \left(\, \sum_{i = 1}^{k}A_iX_iB_i + C \,\right)\!M\!\left(\,\sum_{i = 1}^{k}A_iX_iB_i + C \, \right)^{\!*} +D, \ \ \ \sum_{i = 1}^{k}\left(\,A_iX_iB_i + C_i\,\right)\!M_i\!\left(\,A_iX_iB_i + C_i \,\right)^{*} +D \end{align*} to be semi-definite are derived, respectively, where $A_i, \ B_i, \ C_i, \ C, \ D, \ M_i$ and $M$ are given matrices with $M_i$, $M$ and $D$ Hermitian, $i =1, \ldots, k$. L\"owner partial ordering optimizations of the two matrix-valued functions are studied and their solutions are characterized. Keywords: quadratic matrix-valued function; quadratic matrix equation; quadratic matrix inequality; Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Category 2: Global Optimization Category 3: Nonlinear Optimization Citation: Download: [PDF]Entry Submitted: 05/27/2012Entry Accepted: 05/28/2012Entry Last Modified: 05/27/2012Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.