ON QUADRATICAL CONVERGENCE OF INEXACT LEVENBERG-MARQUARDT METHODS UNDER LOCAL ERROR BOUND CONDITION

In the present paper, we propose a new inexact Levenberg-Marquardt method (LMM) with different residual. Under a local error bound condition which is same as given by [Optim. Methods Softw. 17 (2002), 605-626], we show that a sequence generated by the new inexact LMM converges to a solution of f(x) = 0 superlinearly and even quadratically for some special parameters, while in [Optim. Methods Softw. 17 (2002), 605-626], it's just showed that d(x(k), S) converges to 0 superlinearly or quadratically, where S is the solution set of f(x) = 0. Hence, our results improve the corresponding results in [Optim. Methods Softw. 17 (2002), 605-626]. Furthermore, we also propose the global version of the new inexact LMM by virtue of the Armijo, Wolfe or Goldstein line-search schemes, and establish its global quadratical convergence under the local error bound condition. Finally, numerical results have also been provided.