Shamanskii-Like Levenberg-Marquardt Method with a New Line Search for Systems of Nonlinear Equations

Abstract

To save the calculations of Jacobian, a multi-step Levenberg-Marquardt method named Shamanskii-like LM method for systems of nonlinear equations was proposed by Fan (2013). Its convergence properties have been proved by using a trust region technique under the local error bound condition. However, the authors wonder whether the similar convergence properties are still true with standard line searches since the direction may not be a descent direction. For this purpose, the authors present a new nonmonotone m-th order Armijo type line search to guarantee the global convergence. Under the same condition as trust region case, the convergence rate also has been shown to be m + 1 by using this line search technique. Numerical experiments show the new algorithm can save much running time for the large scale problems, so it is efficient and promising.