Analysis of monotone gradient methods

The gradient method is one simple method in nonlinear optimization. In this paper, we give a brief review on monotone gradient methods and study their numerical properties by introducing a new technique of long-term observation. We find that, one monotone gradient algorithm which is proposed by Yuan recently shares with the Barzilai-Borwein (BB) method the property that the gradient components with respect to the eigenvectors of the function Hessian are decreasing together. This might partly explain why this algorithm by Yuan is comparable to the BB method in practice. Some examples are also provided showing that the alternate minimization algorithm and the other algorithm by Yuan may fall into cycles. Some more efficient gradient algorithms are provided. Particularly, one of them is monotone and performs better than the BB method in the quadratic case.