Generalization of Quasi-Newton Methods: Application to Robust Symmetric Multisecant Updates

TL;DR

This paper introduces a new quasi-Newton symmetric update method that uses multiple secant equations in a least-squares framework, unifying existing approaches and providing robustness guarantees.

Contribution

A novel quasi-Newton update scheme that generalizes and unifies previous methods while ensuring symmetry and robustness through multiple secant equations.

Findings

Provides a unified framework for quasi-Newton updates

Ensures symmetry and robustness in Hessian approximation

Generalizes existing secant-based methods

Abstract

Quasi-Newton techniques approximate the Newton step by estimating the Hessian using the so-called secant equations. Some of these methods compute the Hessian using several secant equations but produce non-symmetric updates. Other quasi-Newton schemes, such as BFGS, enforce symmetry but cannot satisfy more than one secant equation. We propose a new type of quasi-Newton symmetric update using several secant equations in a least-squares sense. Our approach generalizes and unifies the design of quasi-Newton updates and satisfies provable robustness guarantees.