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A novel gradient stepsize is derived at the motivation of equipping theBarzilai-Borwein (BB) method with two dimensional quadratic terminationproperty. A remarkable feature of the novel stepsize is that its computationonly depends on the BB stepsizes in previous iterations and does not requireany exact line search or the Hessian, and hence it can easily be extended fornonlinear optimization. By adaptively taking long BB steps and some short stepsassociated with the new stepsize, we develop an efficient gradient method forquadratic optimization and general unconstrained optimization and extend it tosolve extreme eigenvalues problems. The proposed method is further extended forbox-constrained optimization and singly linearly box-constrained optimizationby incorporating gradient projection techniques. Numerical experimentsdemonstrate that the proposed method outperforms the most successful gradientmethods in the literature.

Equipping Barzilai-Borwein method with two dimensional quadratic termination property