  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.
