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NACO
Nonlinear equations and nonlinear least squares problems have many
applications in physics, chemistry, engineering, biology, economics,
finance and many other fields. In this paper, we will review some
recent results on numerical methods for these two special problems,
particularly on Levenberg-Marquardt type methods, quasi-Newton type
methods, and trust region algorithms. Discussions on variable
projection methods and subspace methods are also given. Some
theoretical results about local convergence results of the
Levenberg-Marquardt type methods without non-singularity assumption
are presented. A few model algorithms based on line searches and
trust regions are also given.
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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