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AIMS Mathematics
JIMO
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.
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