AI 中文总结
该研究针对基于模型的无导数优化中模型维护成本高的问题，利用KKT矩阵结构导出\(\mathcal{O}(n^2)\)的更新公式，集成到并行算法中，经测试在减少模型维护开销和提高成功率方面优于标准求解器。
AI 中文摘要
基于模型的无导数优化依赖二次插值，维持这些模型通常需\(\mathcal{O}(m^3)\)次线性系统求解。研究表明，对于最小Frobenius范数更新模型，相关KKT矩阵有固定内积结构。单点替换和坐标轴翻转操作会对该矩阵产生精确的秩2扰动。据此得出\(\mathcal{O}(n^2)\)的KKT逆更新公式，消除每次迭代的昂贵重构。将更新集成到并行信赖域算法中，测试表明该方法减少了模型维护开销，在严格函数评估预算下成功率更高。
英文摘要
Model-based derivative-free optimization relies on quadratic interpolation, but maintaining these models typically requires $\mathcal{O}(m^3)$ linear system solves. We show that for the least Frobenius norm updating model, the associated KKT matrix possesses a fixed inner-product structure. Both single-point replacements and a proposed coordinate-axis flipping operation induce exact Rank-2 perturbations to this matrix. Using this structure, we derive an $\mathcal{O}(n^2)$ update formula for the KKT inverse, eliminating costly refactorizations at each iteration. We integrate the update into a parallel trust-region algorithm where workers independently flip interpolation axes, refresh local models, and synchronize the best configuration. Tests on 530 benchmark problems show the method reduces model-maintenance overhead and achieves higher success rates under tight function-evaluation budgets compared to standard solvers.
