On the acceleration of the Barzilai-Borwein method

TL;DR

This paper introduces a new stepsize for the Barzilai-Borwein method to accelerate convergence, especially for strongly convex quadratic problems, and demonstrates significant performance improvements through extensive numerical experiments.

Contribution

A novel stepsize strategy that accelerates the BB method and its extension to constrained problems, outperforming recent gradient methods.

Findings

Accelerated convergence for strongly convex quadratic problems.

Enhanced performance over existing gradient descent methods.

Effective extension to constrained optimization problems.

Abstract

The Barzilai-Borwein (BB) gradient method is efficient for solving large-scale unconstrained problems to the modest accuracy and has a great advantage of being easily extended to solve a wide class of constrained optimization problems. In this paper, we propose a new stepsize to accelerate the BB method by requiring finite termination for minimizing two-dimensional strongly convex quadratic function. Combing with this new stepsize, we develop gradient methods which adaptively take the nonmonotone BB stepsizes and certain monotone stepsizes for minimizing general strongly convex quadratic function. Furthermore, by incorporating nonmonotone line searches and gradient projection techniques, we extend these new gradient methods to solve general smooth unconstrained and bound constrained optimization. Extensive numerical experiments show that our strategies of properly inserting monotone gradient steps into the nonmonotone BB method could significantly improve its performance and the new resulted methods can outperform the most successful gradient decent methods developed in the recent literature.