矩阵束的半正定区间及其在广义置信域子问题中的应用
Positive semidefinite interval of matrix pencil and its applications for the generalized trust region subproblems
摘要：
我们关心的是找到实值集$I_{\succeq}(A,B)$ $μ$使得矩阵束$A+μB$是半正定的。如果$A, B$不能通过同余（SDC）同时对角化，则$I_{\succeq}(A,B)$为空或只有一个值$μ.$当$A, B$为SDC时，$I_{\succeq}(A,B),$如果不为空，则可以是单例或区间。特别地，如果$I_{\succeq}(A,B)$是一个区间，并且至少有一个矩阵是非奇异的，那么它的内部是正定区间$I_{\succ}(A,B).$。如果$A, B$都是奇异的，那么甚至$I_{\succeq}(A,B)$也是一个区间，它的内部可能不是$I_{\succ}(A,B),$，但$A, B$然后被分解为具有$B_1$非奇异的子矩阵$A_1, B_1$的块对角，使得$I_{\succeq}(A,B)=I_{\succeq}(A_1,B_1).$应用$I_{\succeq}(A,B),$广义信赖域子问题的硬情况（GTRS）可以通过仅求解线性方程组来处理，或者简化为较小尺寸的GTRS的简单情况。
英文摘要：
We are concerned with finding the set $I_{\succeq}(A,B)$ of real values $μ$ such that the matrix pencil $A+μB$ is positive semidefinite. If $A, B$ are not simultaneously diagonalizable via congruence (SDC), $I_{\succeq}(A,B)$ either is empty or has only one value $μ.$ When $A, B$ are SDC, $I_{\succeq}(A,B),$ if not empty, can be a singleton or an interval. Especially, if $I_{\succeq}(A,B)$ is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval $I_{\succ}(A,B).$ If $A, B$ are both singular, then even $I_{\succeq}(A,B)$ is an interval, its interior may not be $I_{\succ}(A,B),$ but $A, B$ are then decomposed to block diagonals of submatrices $A_1, B_1$ with $B_1$ nonsingular such that $I_{\succeq}(A,B)=I_{\succeq}(A_1,B_1).$ Applying $I_{\succeq}(A,B),$ the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.
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