  A novel gradient stepsize is derived at the motivation of equipping the
Barzilai-Borwein (BB) method with two dimensional quadratic termination
property. A remarkable feature of the novel stepsize is that its computation
only depends on the BB stepsizes in previous iterations and does not require
any exact line search or the Hessian, and hence it can easily be extended for
nonlinear optimization. By adaptively taking long BB steps and some short steps
associated with the new stepsize, we develop an efficient gradient method for
quadratic optimization and general unconstrained optimization and extend it to
solve extreme eigenvalues problems. The proposed method is further extended for
box-constrained optimization and singly linearly box-constrained optimization
by incorporating gradient projection techniques. Numerical experiments
demonstrate that the proposed method outperforms the most successful gradient
methods in the literature.
