AI 中文总结
该研究提出一种针对有限极小极大问题的负曲率感知L-BFGS方法，通过三种最小割线修正处理负曲率，无需特征分解等操作，可退化为普通L-BFGS，便于变体比较。
AI 中文摘要
我们研究一种用于无约束有限极小极大问题的有限内存拟牛顿方法。非光滑最大值由截断双曲平滑模型处理，而L-BFGS所用的曲率对仅在检测到真实负曲率观测值时才被修改。我们提出三种修正规则：第一种是将$y_k$替换为$-y_k$；第二种来自欧氏最近点问题；第三种来自$B_k^{-1}$度量下的最近点问题。所有三种规则在严格负曲率对上强制相同的正割线曲率$s_k^T\tilde{y}_k=|s_k^Ty_k|$，之后直接使用标准BFGS更新和标准L-BFGS双环递归，无需任何改动。特别地，该方法不需要特征分解、Krylov负曲率搜索或各向同性偏移$B_k+\mu_kI$。我们证明每一对被接受的曲率对都保持正定性，在显式度量和平滑假设下建立Armijo回溯的充分下降性和有限终止性，并推导延续框架的全局Clark稳定性结果。我们还表明，在足够强凸的邻域内，所有三种修正均不再生效，因此局部方法退化为普通L-BFGS。该论文将通用算法框架与负曲率修正的选择分离，使得三种变体可在相同的平滑、线搜索和内存规则下进行比较。
英文摘要
This paper proposes a preconditioned ordinary hyperbolic smoothing modified L-BFGS framework for finite minimax problems. To mitigate the deterioration of Euclidean conditioning as the smoothing parameter decreases, a problem-derived hyperbolic majorization preconditioner is incorporated into the initial L-BFGS metric; it combines component-gradient Lipschitz curvature with the curvature induced by the hyperbolic smoothing Jacobian, is uniformly positive definite, and provably majorizes the Hessian of the smoothed objective. To exploit strict negative secant curvature, three correction strategies are introduced---direct sign reflection, a Euclidean nearest-point correction, and a $B_k^{-1}$-metric nearest-point correction---for which explicit formulas are derived and Clarke-stationary accumulation points are obtained for the idealized continuation. For a fixed smoothing parameter, Q-linear convergence of the HMLBFGS objective values is obtained under a Polyak--Lojasiewicz condition, local strong convexity further yields R-linear convergence of the iterates, and the corresponding full-memory modified BFGS methods attain local Q-superlinear convergence under the usual unit-step assumptions. Numerical experiments demonstrate the effectiveness of the proposed preconditioned methods and their numerical advantages over the selected comparison methods.
