AI 中文总结
研究无导数优化问题，提出基于秩二KKT更新的方法，通过反射插值集改变KKT矩阵，降低更新成本，支持全局收敛，嵌入主从并行算法，经530个基准问题数值结果与现有求解器对比性能。
AI 中文摘要
无导数优化（DFO）解决无约束问题$\min_{\x\in\RR^n} f(\x)$，其中$f$仅通过零阶预言机访问。基于模型的信赖域方法从$\mathcal{O}(n)$个点构建欠定二次插值模型并求解KKT系统以确定模型参数，成本为$\mathcal{O}(m^3)$操作且限制了并行扩展性。研究表明，最小Frobenius范数更新模型的KKT矩阵完全取决于平移坐标的内积。在单个坐标轴上反射插值集可保留这些内积并仅改变KKT矩阵的一行和一列，诱导至多秩二扰动，当$m=\mathcal{O}(n)$时通过Sherman-Morrison-Woodbury公式进行逆更新成本为$\mathcal{O}(n^2)$。反射在中心欧几里得信赖域中是等距变换并保留插值集的平衡常数；与标准全线性模型管理假设一起支持一阶全局收敛。该机制嵌入具有截断共轭梯度子问题求解器的主从并行算法中。在530个基准问题上的数值结果与已建立的DFO求解器比较性能。
英文摘要
Derivative-free optimization (DFO) addresses unconstrained problems $\min_{\x\in\RR^n} f(\x)$ where $f$ is accessed only through a zeroth-order oracle. Model-based trust-region methods construct underdetermined quadratic interpolation models from $\mathcal{O}(n)$ points and solve a KKT system to determine model parameters, costing $\mathcal{O}(m^3)$ operations and limiting parallel scalability. It is shown that the KKT matrix for the minimum Frobenius norm updating model depends entirely on inner products of shifted coordinates. Reflecting the interpolation set across a single coordinate axis preserves these inner products and changes only one row and column of the KKT matrix, inducing a rank-at-most-two perturbation whose inverse update via the Sherman-Morrison-Woodbury formula costs $\mathcal{O}(n^2)$ when $m=\mathcal{O}(n)$. The reflection is an isometry in centered Euclidean trust regions and preserves the poisedness constant of the interpolation set; together with standard fully linear model-management assumptions this supports first-order global convergence. The mechanism is embedded in a master-worker parallel algorithm with a Truncated Conjugate Gradient subproblem solver. Numerical results on 530 benchmark problems compare performance against an established DFO solver.
Comments23 pages
