Preprint A729/2013
A Levenberg-Marquardt method with approximate projections

Yinyu Ye | Behling, Roger | Fischer, Andreas | Herrich, Markus | Iusem, Alfredo

**Keywords: **
constrained equation | Levenberg Marquardt | approximate projection | error bound

The projected Levenberg-Marquardt method for the solution of a system of equations with convex constraints is known to converge locally quadratically to a possibly non-isolated solution if a certain error bound condition holds. This condition turns out to be quite strong, since it implies that the solution sets of the constrained and the unconstrained systems are locally the same. Under a pair of more reasonable error bound conditions, this paper proves R-linear convergence of a Levenberg-Marquardt method with approximate projections. In this way, computationally expensive projections can be avoided. The new method is also applicable if there are nonsmooth constraints having subgradients. Moreover, the projected Levenberg-Marquardt method is a special case of the new methods and shares its R-linear convergence.