Generalized Metric Subregularity with Applications to High-Order Regularized Newton Methods

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 Kurdyka-Łojasiewicz (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) Hadamard reparameterized compressed sensing model, we achieve global convergence with a quadratic local convergence rate toward a global minimizer under a strict complementarity condition.

Funding: The research of G. Li and B. Mordukhovich was supported by the Australian Research Council Discovery Projects [Grants DP190100555 and DP250101112]. The research of B. Mordukhovich was also supported by the U.S. National Science Foundation [Grant DMS-2204519] and by Project 111 of China [Grant D21024]. The research of J. Zhu was supported by the Basic Research Program of Yunnan Province [Grant 202201AT070066], the Project for Young-notch Talents in the Ten Thousand Talent Program of Yunnan Province [Grant YNWR-QNBJ-2020-080], and the National Natural Science Foundation of People’s Republic of China [Grants 12171419 and 12261109].