Advanced Stability Analysis of Regularized Newton Methods for Monotone Inclusions With Bounded Variation

This paper presents a comprehensive stability analysis of regularized Newton methods for solving monotone inclusion problems of the form 0∈Ax+Fx, where A is a maximal monotone operator and F is a Lipschitz continuous operator with bounded variation. By modeling the problem as a dynamical system, we analyze its stability using a Lyapunov function approach, establishing existence, uniqueness, and exponential convergence under strong monotonicity conditions. The bounded variation property enables tighter convergence bounds compared to traditional methods. Numerical experiments validate the theoretical results, demonstrating robust stability in nonsmooth optimization contexts. The findings extend to applications in machine learning, distributed optimization, and dynamic networks, offering insights into adaptive regularization strategies and practical implementations.