  We propose a new stepsize for the gradient method. It is shown that this new
stepsize will converge to the reciprocal of the largest eigenvalue of the
Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational
Optimization and Applications, 2006, 33(1): 73-88) is applied for minimizing
quadratic objective functions. Based on this spectral property, we develop a
monotone gradient method that takes a certain number of steps using the
asymptotically optimal stepsize by Dai and Yang, and then follows by some short
steps associated with this new stepsize. By employing one step retard of the
asymptotic optimal stepsize, a nonmonotone variant of this method is also
proposed. Under mild conditions, $R$-linear convergence of the proposed methods
is established for minimizing quadratic functions. In addition, by combining
gradient projection techniques and adaptive nonmonotone line search, we further
extend those methods for general bound constrained optimization. Two variants
of gradient projection methods combining with the Barzilai-Borwein stepsizes
are also proposed. Our numerical experiments on both quadratic and bound
constrained optimization indicate that the new proposed strategies and methods
are very effective.
