  This paper presents a group of analytical formulas for calculating the global
maximal and minimal ranks and inertias of the quadratic matrix-valued function
$\phi(X) = (\, AXB + C\,)M(\, AXB + C)^{*} + D$ and use them to derive
necessary and sufficient conditions for the two types of multiple quadratic
matrix-valued function {align*} (\, \sum_{i = 1}^{k}A_iX_iB_i + C
\,)M(\,\sum_{i = 1}^{k}A_iX_iB_i + C \,)^{*} +D, \ \ \ \sum_{i =
1}^{k}(\,A_iX_iB_i + C_i\,)M_i(\,A_iX_iB_i + C_i \,)^{*} +D {align*} to be
semi-definite, respectively, where $A_i,\ B_i,\ C_i,\ C,\ D,\ M_i$ and $M$ are
given matrices with $M_i$, $M$ and $D$ Hermitian, $i =1,..., k$. L\"owner
partial ordering optimizations of the two matrix-valued functions are studied
and their solutions are characterized.
