  This paper pursues a twofold goal. First, we introduce and study in detail a
new notion of variational analysis called generalized metric subregularity,
which is a far-going extension of the conventional metric subregularity
conditions. Our primary focus is on examining this concept concerning
first-order and second-order stationary points. We develop an extended
convergence framework that enables us to derive superlinear and quadratic
convergence under the generalized metric subregularity condition, broadening
the widely used KL convergence analysis framework. We present verifiable
sufficient conditions to ensure the proposed generalized metric subregularity
condition and provide examples demonstrating that the derived convergence rates
are sharp. Second, we design a new high-order regularized Newton method with
momentum steps, and apply the generalized metric subregularity to establish its
superlinear convergence. Quadratic convergence is obtained under additional
assumptions. Specifically, when applying the proposed method to solve the
(nonconvex) over-parameterized compressed sensing model, we achieve global
convergence with a quadratic local convergence rate towards a global minimizer
under a strict complementarity condition.
