  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.
