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.

我们为梯度方法提出了一个新的步骤大小。结果表明，当戴阳的渐近最优梯度方法（计算优化与应用，2006，33（1）：73-88）用于最小化二次方时，这种新的步长将收敛到黑森州最大特征值的倒数。目标函数。基于此光谱特性，我们开发了一种单调梯度方法，该方法使用Dai和Yang的渐近最优步长大小采取一定数量的步长，然后遵循与此新步长相关的一些短步长。通过采用渐近最优步长的一阶延迟，还提出了该方法的非单调变体。在温和的条件下，R建立了所提出方法的线性收敛以最小化二次函数。此外，通过将梯度投影技术和自适应非单调线搜索相结合，我们进一步扩展了这些方法用于一般约束优化。还提出了结合Barzilai-Borwein步长大小的梯度投影方法的两个变体。我们对二次和有界约束优化的数值实验表明，新提出的策略和方法非常有效。