反二次特征值问题在结构动力学、地质工程、医学、军事、环境、遥感、控制、通讯、气象、经济等领域有着重要的应用背景。本项目主要探讨几类结构化反二次特征值问题的可解性及有效的优化算法（如广义牛顿法、非光滑最优化以及正则化方法等）及其在动力系统中的应用。我们旨在使所求物理矩阵满足部分测量得到的特征信息（即特征值和特征向量），同时保持原始模型的特定结构性质（如对称性、非负性、正定性、稀疏性或带状结构及内部连接性等），并考虑到特征信息的不完备性和噪声影响以及应用中的规模化问题。这些问题的研究不仅对反特征值问题的发展具有重要的理论和算法方面的意义，而且对结构动力学、地质工程、医学、遥感、控制及气象等领域的研究和发展都具有较高的应用参考价值。

Inverse quadratic eigenvalue problems arise in the fields of structural dynamics, geological engineering, medicine, military, environment, remote sensing, control systems, communications, meteorology, economics, etc. This project is mainly concerned with the solvability and effective numerical optimization methods (e.g., generalized Newton-type methods, nonsmooth optimization methods，and regularization methods) of several structured inverse quadratic eigenvalue problems and their applications in dynamical systems. We aim to find the mass, damping and stiffness matrices such that the corresponding quadratic pencil satisfies the measdured partial eigeninformation (i.e., eigenvalues and eigenvectors), preserve the special structural properties of the original model (e.g., symmetry, nonnegativity, definiteness, sparsity or bandedness, and internal connectivity, etc.), and we also consider the influence of incomplete and noisy eigeninformation and the large-scale problem in applications. The research on these problems is not only significant to the development of inverse quadratic eigenvalue problems in theoretical and algorithmic aspects but very valuable to various applications such as structural dynamics, geological engineering, medicine, remote sensing, control systems, meteorology, etc.