Global Complexity Bound Analysis of the Levenberg-Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem

We investigate a global complexity bound of the Levenberg-Marquardt Method (LMM) for nonsmooth equations F (x) = 0. The global complexity bound is an upper bound to the number of iterations required to get an approximate solution such that ∥∇f(x)∥ ≤ ε, where f is a least square merit function and ε is a given positive constant. We show that the bound of the LMM is O(ε−2). We also show that it is reduced to O(log ε−1) under some regularity assumption on the generalized Jacobian of F . Furthermore, by applying these results to nonsmooth equations equivalent to the nonlinear complementarity problem (NCP), we get global complexity bounds for the NCP. In particular, we show that the bound is O(log ε−1) when the mapping involved in the NCP is a uniformly P-function.