Necessary conditions for linear convergence of iterated expansive, set-valued mappings - Luke, D. Russell; Teboulle, Marc; Thao, Nguyen H. (Mathematical programming, 2020)
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.