Necessary conditions for linear convergence of Picard iterations and application to alternating projections
D. Russell Luke, Nguyen Hieu Thao, Marc Teboulle · arXiv (Cornell University) · 2017
We present necessary conditions for monotonicity, in one form or another, of fixed point iterations of mappings that violate the usual nonexpansive property in some quantifiable way. We show that most reasonable notions of linear-type monotonicity of fixed point sequences imply metric subregularity of the fixed point mapping. This is specialized to the alternating projections iteration where metric subregularity of the fixed point mapping takes on the distinct geometric property of subtransversality of sets at points of intersection. We show the necessity of subtransversality for a number of reasonable types of sequential monotonicity, under varying degrees of assumptions on the regularity of the sets. Based on the results we obtain, we conjecture that subtransversality is necessary for R-linear convergence to fixed points of iterates of the alternating projections sequence.