具有相应特征向量约束的次随机逆特征值问题
On Substochastic Inverse Eigenvalue Problems with the Corresponding Eigenvector Constraints
AI总结：
本文研究带特征向量约束的次随机矩阵逆特征值问题，将其转化为非凸优化问题，提出交替最小化算法并证明收敛性，数值实验验证了方法效率。
摘要：
我们考虑从给定的谱参数并结合相应特征向量约束来构造次随机矩阵的逆特征值问题。这一带有特定特征向量约束的次随机逆特征值问题（SstIEP）被表述为一个非凸优化问题（NcOP）。带有特定特征向量约束的 SstIEP 的可解性，等价于判定所表述 NcOP 的零最优值是否可达。当最优目标值为零时，所表述 NcOP 的相应最优解恰好就是期望构造的次随机矩阵。我们开发了交替最小化算法来求解所表述的 NcOP，并通过提出一种获得最优解有界性的新方法建立了其收敛性。此外，还进行了若干数值实验，以证明所提方法的效率。
英文摘要：
We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identify the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix desired to be constructed. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.
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