Input-to-State Stability of Newton Methods for Generalized Equations in Nonlinear Optimization

TL;DR

The paper analyzes robustness of Newton methods for generalized equations that arise as KKT systems in nonlinear optimization. It adopts an input-to-state stability (ISS) framework to characterize how disturbances affect the iterates and to derive convergence guarantees. Its main contributions are (i) a local ISS proof for perturbed Josephy-Newton methods, (ii) a multistep Newton-type method that achieves robust local convergence under disturbances, and (iii) applications showing ISS implications for approximate sequential programming and the augmented Lagrangian method. Together, these results provide a principled approach to designing and analyzing robust optimization algorithms in interconnected or inexact computation settings.

Abstract

We show that Newton methods for generalized equations are input-to-state stable with respect to disturbances such as due to inexact computations. We then use this result to obtain convergence and robustness of a multistep Newton-type method for multivariate generalized equations. We demonstrate the usefulness of the results with other applications to nonlinear optimization. In particular, we provide a new proof for (robust) local convergence of the augmented Lagrangian method.