Preprint A697/2011
The effect of calmness on the solution set of systems of nonlienar equaltions

Alfredo Iusem | Behling, Roger

**Keywords: **
calmness | upper-Lipschitz continuity | nonlinear equations | error bound | Levenberg Marquardt

We address the issue of solving a system of nonlinear equations under the condition of calmness. This property, also called upper-Lipschitz continuity in the literature, can be described by a local error bound, and is being widely used as a regularity condition in Optimization.Indeed, it is known to be significantly weaker than classical regularity assumptions which
imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian matrix of the function that describes the system must be locally constant on the solution set. As a consequence, we prove that locally the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance
and algorithmic implications.