← 研踪 PaperTrailarXiv:2611.13916 · v1
Dual-Space Preconditioning for Variational Inequalities and Root-Finding Problems
首次提交 2026-11-10 19:06
版本更新 2026-11-10 19:06
雷达观测 2026-11-11 11:04
摘要
This paper develops a dual-space preconditioning framework for variational inequalities and root-finding problems beyond classical cocoercivity and Lipschitz continuity assumptions. To this end, we introduce both a novel cocoercivity-type condition and a relaxed Lipschitz continuity condition, and derive convergence of deterministic and stochastic methods under these generalized conditions. For isotropic preconditioners, the resulting algorithms can be interpreted as variable-stepsize variants of classical methods, extending recent analyses based on $(L_0, L_1)$-type growth conditions while allowing simpler proofs and larger stepsize ranges. We also show that the framework naturally handles variational inequalities through weak Minty-type arguments. Finally, we study the associated continuous-time dynamics and derive a variational and optimal control interpretation of the preconditioned flow. Overall, the results show that dual-space preconditioning provides a flexible mechanism for designing and analyzing operator methods beyond the standard Euclidean Lipschitz setting.
作者备注：NeurIPS 2026 poster
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v1 · 2026-11-10 19:06 · 最新观测版本
Dual-Space Preconditioning for Variational Inequalities and Root-Finding Problems
This paper develops a dual-space preconditioning framework for variational inequalities and root-finding problems beyond classical cocoercivity and Lipschitz continuity assumptions. To this end, we introduce both a novel cocoercivity-type condition and a relaxed Lipschitz continuity condition, and derive convergence of deterministic and stochastic methods under these generalized conditions. For isotropic preconditioners, the resulting algorithms can be interpreted as variable-stepsize variants of classical methods, extending recent analyses based on $(L_0, L_1)$-type growth conditions while allowing simpler proofs and larger stepsize ranges. We also show that the framework naturally handles variational inequalities through weak Minty-type arguments. Finally, we study the associated continuous-time dynamics and derive a variational and optimal control interpretation of the preconditioned flow. Overall, the results show that dual-space preconditioning provides a flexible mechanism for designing and analyzing operator methods beyond the standard Euclidean Lipschitz setting.
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