Generalized Monotonicity and the Proximal Point Algorithm

We study the proximal point algorithm when the operator of interest is metrically subregular and satisfies a submonotonicity property. The latter property can be viewed as a quantified weakening of the standard definition of a monotone operator. Our main result gives a condition under which locally, the proximal point algorithm generates sequences that are linearly convergent to a zero of the underlying operator. General properties of our notion of submonotonicity are also explored as well as connections to other concepts in the literature. Funding: D. R. Luke was supported in part by the Deutsche Forschungsgemeinschaft [Grants 541767835 and SFB1456/1-432680300]. M. K. Tam was supported in part by the Australian Research Council [Grant DP230101749].