Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems

This paper develops a hyperbolic-majorization preconditioned three-term nonlinear conjugate-gradient framework for nonconvex finite minimax optimization. An analytic symmetric positive definite metric is derived from a global quadratic majorization of the hyperbolic smoothing model and is used simultaneously as a curvature absorber, a preconditioner, and the line-search energy metric. With the displacement $s_{k-1}$ fixed and a variable curvature response $b_k$, an enhanced three-term direction is introduced together with an adaptive parameter ${\mu }_k^{\star }$ that is maximal under the requirement that the baseline worst-case metric-energy constant be preserved. The resulting framework yields a Dai–Liao-type conjugacy relation, enhanced sufficient descent, a smoothing-parameter-uniform Armijo lower bound, fixed-smoothing global first-order convergence and complexity, and Clarke-stationary accumulation points under continuation.