Modified inexact Levenberg–Marquardt methods for solving nonlinear least squares problems

Abstract

In the present paper, we propose a modified inexact Levenberg–Marquardt method (LMM) and its global version by virtue of Armijo, Wolfe or Goldstein line-search schemes to solve nonlinear least squares problems (NLSP), especially for the underdetermined case. Under a local error bound condition, we show that a sequence generated by the modified inexact LMM converges to a solution superlinearly and even quadratically for some special parameters, which improves the corresponding results of the classical inexact LMM in Dan et al. (Optim Methods Softw 17:605–626, 2002). Furthermore, the quadratical convergence of the global version of the modified inexact LMM is also established. Finally, preliminary numerical experiments on some medium/large scale underdetermined NLSP show that our proposed algorithm outperforms the classical inexact LMM.