On the connection between the conjugate gradient method and quasi-Newton methods on quadratic problems

TL;DR

This paper explores the precise conditions under which quasi-Newton methods, specifically the one-parameter Broyden family, produce the same iterates as the conjugate gradient method on quadratic problems with positive-definite Hessians, revealing their deep connection.

Contribution

It precisely characterizes the conditions for equivalence between conjugate gradient and quasi-Newton methods using Broyden updates, and establishes a complete framework for their relationship.

Findings

The Broyden family is complete under certain conjugacy conditions.

A one-to-one correspondence exists between Broyden parameters and search direction scaling.

Update matrices are almost always well-defined, except for a specific symmetric rank-one case.

Abstract

It is well known that the conjugate gradient method and a quasi-Newton method, using any well-defined update matrix from the one-parameter Broyden family of updates, produce identical iterates on a quadratic problem with positive-definite Hessian. This equivalence does not hold for any quasi-Newton method. We define precisely the conditions on the update matrix in the quasi-Newton method that give rise to this behavior. We show that the crucial facts are, that the range of each update matrix lies in the last two dimensions of the Krylov subspaces defined by the conjugate gradient method and that the quasi-Newton condition is satisfied. In the framework based on a sufficient condition to obtain mutually conjugate search directions, we show that the one-parameter Broyden family is complete. A one-to-one correspondence between the Broyden parameter and the non-zero scaling of the search direction obtained from the corresponding quasi-Newton method compared to the one obtained in the conjugate gradient method is derived. In addition, we show that the update matrices from the one-parameter Broyden family are almost always well-defined on a quadratic problem with positive-definite Hessian. The only exception is when the symmetric rank-one update is used and the unit steplength is taken in the same iteration. In this case it is the Broyden parameter that becomes undefined.