Abstract:
We propose and study the Levenberg–Marquardt method globalized by means of linesearch
for unconstrained optimization problems with possibly nonisolated solutions.
It is well-recognized that this method is an efficient tool for solving systems
of nonlinear equations, especially in the presence of singular and even nonisolated
solutions. Customary globalization strategies for the Levenberg–Marquardt method
rely on linesearch for the squared Euclidean residual of the equation being solved.
In case of unconstrained optimization problem, this equation is formed by putting
the gradient of the objective function equal to zero, according to the Fermat principle.
However, these globalization strategies are not very adequate in the context of
optimization problems, as the corresponding algorithms do not have “preferences” for
convergence to minimizers, maximizers, or any other stationary points. To that end,
in this work we considers a different technique for globalizing convergence of
the Levenberg–Marquardt method, employing linesearch for the objective function of the
original problem. We demonstrate that the proposed algorithm possesses reasonable
global convergence properties, and preserves high convergence rate of the
Levenberg–Marquardt method under weak assumptions.
Keywords:unconstrained optimization problem; nonisolated solutions;
Levenberg–Marquardt method; globalization of convergence.
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