Wenzhe Zhao
We study a limited-memory quasi-Newton method for unconstrained finite minimax problems. The nonsmooth maximum is handled by a truncated hyperbolic smoothing model, while the curvature pairs used by L-BFGS are modified only when a genuine negative-curvature observation is detected. Three correction rules are developed. The first replaces $y_k$ by $-y_k$; the second is obtained from the Euclidean nearest-point problem; and the third is obtained from the nearest-point problem in the $B_k^{-1}$ metric. All three rules enforce the same positive secant curvature $s_k^T\ytilde_k=|s_k^Ty_k|$ on a strict negative-curvature pair. The standard BFGS update and the standard L-BFGS two-loop recursion are then used without any change. In particular, the method does not require an eigendecomposition, a Krylov negative-curvature search, or an isotropic shift $B_k+μ_kI$. We prove that every accepted pair preserves positive definiteness, establish the sufficient-descent property and finite termination of the Armijo backtracking under explicit metric and smoothing assumptions, and derive a global Clarke-stationarity result for the continuation framework. We also show that all three corrections become inactive in a sufficiently strongly convex neighborhood, so the local method reduces to ordinary L-BFGS. The paper separates the common algorithmic framework from the choice of the negative-curvature correction, which allows the three variants to be compared under identical smoothing, line-search, and memory rules.