Necessary conditions for linear convergence of iterated expansive, set-valued mappings

• Published in: Mathematical programming Vol. 180; no. 1-2; pp. 1 - 31
• Main Authors: Luke, D. Russell, Teboulle, Marc, Thao, Nguyen H.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2020; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Euclidean space; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Projectors; Theoretical; Metric subregularity; 65K05; Fixed point iteration; Fejér monotone; 90C26; Elemental regularity; Nonexpansive; Transversality; Secondary 49K40; Cyclic projections; Averaged operators; Almost averaged mappings; Fixed points; 65K10; Feasibility; 49M05; 49M27; Metric regularity; Calmness; Primary 49J53; Nonconvex; Subtransversality
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-018-1343-8

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.