AI 中文总结
研究指定特征值的逆结构化对称矩阵问题，开发无导数优化模型，应用确定性DFO方案及启发式策略求解，通过计算对称矩阵特征值评估目标函数，给出数值结果比较不同方法在各场景下的性能。
AI 中文摘要
本文开发并分析了一种无导数优化（DFO）模型，用于解决指定特征值的逆结构化对称矩阵问题。部分（零和非零）元素预先指定且不可更改，其他元素应为非零但值未给定，其余元素完全自由。所得矩阵须满足这些条件并具有指定特征值。此特殊逆特征值问题与多种应用相关，与确定矩阵稀疏模式的无向边加权图有关。我们的新型优化模型需计算对称矩阵的特征值以评估不可微目标函数。应用确定性DFO方案，特别是著名的方向直接搜索（DDS）方法的全局变体GLODS。讨论了基于模型目标函数是Lipschitz连续的收敛性质。此外，探索了使用几种成熟启发式策略解决所提出优化模型的潜在益处。给出了初步数值结果，以说明和比较在各种可能情况下所考虑的确定性和启发式DFO选项的性能。
英文摘要
A Derivative-Free Optimization (DFO) model is developed and analyzed for solving inverse structured symmetric matrix problems for which the eigenvalues are specified. Some (zero and nonzero) entries are preassigned and cannot be changed, while others should be nonzero but their values are not given. The rest of the entries are completely free. The obtained matrix must meet these criteria and have the specified eigenvalues. This specialized inverse eigenvalue problem is relevant to various applications and is linked to determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our novel optimization model requires computing the eigenvalues of a symmetric matrix to evaluate the non-differentiable objective function. We apply deterministic DFO schemes, specifically the global variant GLODS of the well-known family of directional direct search (DDS) methods. We discuss its convergence properties which are based on the fact that the objective function of our model is Lipschitz continuous. Additionally, we explore the potential benefits of using several well-established heuristic strategies to solve the proposed optimization model. We present preliminary numerical results to illustrate and compare the performance of the considered deterministic and heuristic DFO options in various possible scenarios.
