Preconditioned proximal point methods and notions of partial subregularity

Abstract

Based on the needs of convergence proofs of preconditioned proximal point methods, we introduce notions of partial strong submonotonicity and partial (metric) subregularity of set-valued maps. We study relationships between these two concepts, neither of which is generally weaker or stronger than the other one. For our algorithmic purposes, the novel submonotonicity turns out to be easier to employ than more conventional error bounds obtained from subregularity. Using strong submonotonicity, we demonstrate the linear convergence of the Primal-Dual Proximal splitting method to some strictly complementary solutions of example problems from image processing and data science. This is without the conventional assumption that all the objective functions of the involved saddle point problem are strongly convex.