A gradient method exploiting the two dimensional quadratic termination property

TL;DR

This paper introduces a new gradient method that exploits the two-dimensional quadratic termination property, achieving asymptotic eigenvector alignment and $R$-linear convergence for quadratic optimization.

Contribution

It develops a gradient method with a novel stepsize that guarantees quadratic termination and asymptotic eigenvector alignment, improving convergence efficiency.

Findings

Method converges $R$-linearly with rate $1-1/\kappa$

Numerical experiments demonstrate high efficiency

Stepsize converges to reciprocal of largest eigenvalue

Abstract

The quadratic termination property is important to the efficiency of gradient methods. We consider equipping a family of gradient methods, where the stepsize is given by the ratio of two norms, with two dimensional quadratic termination. Such a desired property is achieved by cooperating with a new stepsize which is derived by maximizing the stepsize of the considered family in the next iteration. It is proved that each method in the family will asymptotically alternate in a two dimensional subspace spanned by the eigenvectors corresponding to the largest and smallest eigenvalues. Based on this asymptotic behavior, we show that the new stepsize converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, by adaptively taking the long Barzilai--Borwein stepsize and reusing the new stepsize with retard, we propose an efficient gradient method for unconstrained quadratic optimization. We prove that the new method is $R$-linearly convergent with a rate of $1-1/\kappa$, where $\kappa$ is the condition number of Hessian. Numerical experiments show the efficiency of our proposed method.