Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints

Authors:
Christian Kanzow;Nobuo Yamashita;Masao Fukushima
Affiliations:
Institute of Applied Mathematics and Statistics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany;Department of Applied Mathematics and Physics, Graduate School of Informatics Kyoto University, Kyoto 606-8501, Japan;Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
Venue:
Journal of Computational and Applied Mathematics
Year:
2005

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Abstract

We consider the problem of finding a solution of a constrained (and not necessarily square) system of equations, i.e., we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set. To this end, we present two Levenberg-Marquardt-type algorithms that differ in the way they compute their search directions. The first method solves a strictly convex minimization problem at each iteration, whereas the second one solves only one system of linear equations in each step. Both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition. Both methods can be globalized in an easy way. Some numerical results for the second method indicate that the algorithm works quite well in practice.