Preprint C130/2011
The method and the trajectory of Levenberg-Marquardt

Roger Behling

**Keywords: **
calmness | upper-Lipschitz continuity | nonlinear equations | error bound | Levenberg-Marquardt | Gauss-Newton | central path | interiors points | constrained equation | convergence rate | inexactness | non-isolated solution.

In this thesis we study both the method and the trajectory of Levenberg-Marquardt,
which stem from the forties and sixties. Recently, the method turned out to be a
valuable tool for solving systems of nonlinear equations subject to convex constraints,
even in the presence of non-isolated solutions. We consider basically a projected and
an inexact constrained version of a Levenberg-Marquardt type method in order to solve
such systems of equations. Our results unify and extend several recent ones on the local
convergence of Levenberg-Marquardt and Gauss-Newton methods. They were carried
out under a regularity condition called calmness, which is also called upper-Lipschitz
continuity and is described by an error bound. This hypothesis became quite popular
in the last decade, since it generalizes the classical regularity condition that implies
that solutions are isolated. In this direction, one of the most interesting results in this
work states that the solution set of a calm problem must be locally a dierentiable
manifold.
We have also obtained primal-dual relations between the central path and the
Levenberg-Marquardt trajectory. These results are directly applicable in convex programming
and path following methods.