Truncated Hyperbolic Smoothing Modified L-BFGS Algorithms for Finite Minimax Problems

This paper proposes a truncated hyperbolic smoothing modified L-BFGS framework (THMLBFGS) for finite minimax problems. To exploit strict negative secant curvature, three BFGS pair corrections are introduced: direct sign reflection, a Euclidean nearest-point correction, and a $B_k^{-1}$-metric nearest-point correction. Explicit admissible ranges for the curvature-classification, pair-admission, and BB2-type scaling safeguards yield positive secant curvature, a uniformly bounded SPD limited-memory metric, sufficient descent, finite Armijo backtracking, and global convergence of all three variants to Clarke-stationary points. For the corresponding modified BFGS methods, the Dennis--Moré condition is derived from the stated local assumptions and yields Q-superlinear convergence; a common numerical protocol is specified for comparing the three variants with smoothing, quasi-Newton, and Newton baselines.