Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems
Abstract
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 sk−1 fixed and a variable curvature response bk, an enhanced three-term direction is introduced together with an adaptive parameter μk⋆ 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.
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BibTeX
@misc{zhao2026preconditioned,
title = {{Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems}},
author = {Wenzhe Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13754},
eprint = {2609.13754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13754},
}
RIS
TY - UNPB
TI - Preconditioned Three-Term Conjugate Gradient Framework for Nonconvex Finite Minimax Problems
AU - Wenzhe Zhao
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13754
UR - https://arxiv.org/abs/2609.13754
AB - 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 μ_k^⋆ 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.
ER -