Generic linear convergence through metric subregularity in a variable-metric extension of the proximal point algorithm

The proximal point algorithm finds a zero of a maximal monotone mapping by iterations in which the mapping is made strongly monotone by the addition of a proximal term. Here it is articulated with the norm behind the proximal term possibly shifting from one iteration to the next, but under conditions that eventually make the metric settle down. Despite the varying geometry, the sequence generated by the algorithm is shown to converge to a particular solution. Although this is not the first variable-metric extension of proximal point algorithm, it is the first to retain the flexibility needed for applications to augmented Lagrangian methodology and progressive decoupling. Moreover, in a generic sense, the convergence it generates is Q-linear at a rate that depends in a simple way on the modulus of metric subregularity of the mapping at that solution. This is a tighter rate than previously identified and reveals for the first time the definitive role of metric subregularity in how the proximal point algorithm performs, even in fixed-metric mode.