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

Abstract

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.