A new 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 new stepsize is that its computation only depends on the BB stepsizes in previous iterations without the use of exact line searches and 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 unconstrained optimization and extend it to solve extreme eigenvalues problems. The proposed method is further extended for box-constrained 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.
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