Necessary conditions for linear convergence of iterated expansive, set-valued mappings with application to alternating projections

We present necessary conditions for monotonicity, in one form or another, of fixed point iterations of mappings that violate the usual nonexpansive property. We show that most reasonable notions of linear-type monotonicity of fixed point sequences imply em metric subregularity. This is specialized to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called em subtransversality. Our more general results for fixed point iterations are specialized to establish the necessity of subtransversality for consistent feasibility with a number of reasonable types of sequential monotonicity, under varying degrees of assumptions on the regularity of the sets.