Hyperbolic Smoothing Modified L-BFGS Algorithms for Finite Minimax Problems

Abstract

This paper proposes an ordinary hyperbolic smoothing modified L-BFGS method for finite minimax problems. 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. Based on these strategies, explicit formulas for the three corrections are derived, and the global convergence of the resulting methods is established. For a fixed smoothing parameter, Q-linear convergence of the HMLBFGS objective values is obtained under a Polyak--\L ojasiewicz condition, and local strong convexity further yields R-linear convergence of the iterates. For the corresponding modified BFGS methods, local Q-superlinear convergence is proved under local strong convexity. Numerical experiments demonstrate the effectiveness of the proposed methods and their numerical advantages over the selected comparison methods.