We present necessary conditions for monotonicity of fixed point iterations of mappings that may violate the usual nonexpansive property. Notions of linear-type monotonicity of fixed point sequences—weaker than Fejér monotonicity—are shown to imply metric subregularity . This, together with the almost averaging property recently introduced by Luke et al. (Math Oper Res, 2018 . https://doi.org/10.1287/moor.2017.0898 ), guarantees linear convergence of the sequence to a fixed point. We specialize these results to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called subtransversality . Subtransversality is shown to be necessary for linear convergence of alternating projections for consistent feasibility.

中文翻译：

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迭代膨胀集值映射线性收敛的必要条件

我们提出了映射的不动点迭代的单调性的必要条件，这可能违反通常的非扩展性。不动点序列的线性类型单调性的概念——比 Fejér 单调性弱——被证明意味着度量次正则性。这一点，再加上最近由 Luke 等人提出的几乎平均的特性。(Math Oper Res, 2018. https://doi.org/10.1287/moor.2017.0898)，保证序列线性收敛到一个固定点。我们将这些结果专门用于交替投影迭代，其中度量子正则性属性在称为子横向的交叉点处对集合进行独特的几何表征。次横向性被证明对于交替投影的线性收敛是必要的，以获得一致的可行性。