Augmented Newton Method for Optimization: Global Linear Rate and Momentum Interpretation

Abstract

We propose two variants of Newton method for solving unconstrained minimization problem. Our method leverages optimization techniques such as penalty and augmented Lagrangian method to generate novel variants of the Newton method namely the Penalty Newton method and the Augmented Newton method. In doing so, we recover several well-known existing Newton method variants such as Damped Newton, Levenberg, and Levenberg-Marquardt methods as special cases. Moreover, the proposed Augmented Newton method can be interpreted as Newton method with adaptive heavy ball momentum. We provide global convergence results for the proposed methods under mild assumptions that hold for a wide variety of problems. The proposed methods can be sought as the penalty and augmented extensions of the results obtained by Karimireddy et. al [24].