This paper presents a group of analytical formulas for calculating the globalmaximal and minimal ranks and inertias of the quadratic matrix-valued function$\phi(X) = (\, AXB + C\,)M(\, AXB + C)^{*} + D$ and use them to derivenecessary and sufficient conditions for the two types of multiple quadraticmatrix-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 besemi-definite, respectively, where $A_i,\ B_i,\ C_i,\ C,\ D,\ M_i$ and $M$ aregiven matrices with $M_i$, $M$ and $D$ Hermitian, $i =1,..., k$. L\"ownerpartial ordering optimizations of the two matrix-valued functions are studiedand their solutions are characterized.
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机译：本文提出了一组用于计算二次矩阵值函数$ \ phi（X）=（\，AXB + C \，）M（\，AXB + C）^ { *} + D $，并使用它们为两种类型的多个二次矩阵值函数{align *}（\，\ sum_ {i = 1} ^ {k} A_iX_iB_i + C \，）M（\ ，\ sum_ {i = 1} ^ {k} A_iX_iB_i + C \，）^ {*} + D，\ \ \ \ sum_ {i = 1} ^ {k}（\，A_iX_iB_i + C_i \，）M_i（ \，A_iX_iB_i + C_i \，）^ {*} + D {align *}分别为半定数，其中$ A_i，\ B_i，\ C_i，\ C，\ D，\ M_i $和$ M $是给定矩阵与$ M_i $，$ M $和$ D $ Hermitian，$ i = 1，...，k $。研究了两个矩阵值函数的L \“偏方排序优化方法，并描述了它们的解。
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