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<p>For monotone linear variational inequality (LVI) problems, projection-type methods play an important role due to their low computational cost when the projection onto the constraint set is easy. Although the global convergence of projection-type methods is well understood, their iteration complexity has only been established recently in the work [<CitationRef CitationID="CR6">6</CitationRef>]. In [<CitationRef CitationID="CR6">6</CitationRef>], the ergodic convergence was established through a restricted measure that acts as the lower bound for a gap function, while the non-ergodic convergence was demonstrated by using the natural residual under the best-case scenario (i.e., the minimal iteration), instead of the final iteration. In this paper, we introduce some gap functions to discuss the complexity of projection-type methods for monotone linear variational inequality problems. We present both the ergodic complexity and the non-ergodic complexity for the projection-type methods, measured by the natural residual. Under some additional conditions, we also present their linear rate of convergence.</p>

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Revisit Iterative Complexity of Some Projection-type Methods for Monotone Linear Variational Inequality Problems

  • 期刊论文

  • 发表日期：

    2026年2月11日

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  • Yongxin Chen,
  • Xingju Cai,
  • Deren Han

摘要

For monotone linear variational inequality (LVI) problems, projection-type methods play an important role due to their low computational cost when the projection onto the constraint set is easy. Although the global convergence of projection-type methods is well understood, their iteration complexity has only been established recently in the work [6]. In [6], the ergodic convergence was established through a restricted measure that acts as the lower bound for a gap function, while the non-ergodic convergence was demonstrated by using the natural residual under the best-case scenario (i.e., the minimal iteration), instead of the final iteration. In this paper, we introduce some gap functions to discuss the complexity of projection-type methods for monotone linear variational inequality problems. We present both the ergodic complexity and the non-ergodic complexity for the projection-type methods, measured by the natural residual. Under some additional conditions, we also present their linear rate of convergence.