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.

为节省Jacobian的计算量，Fan（2013）提出了多步Levenberg-Marquardt方法，称为Shamanskii-like LM方法，用于非线性方程组。通过在局部错误约束条件下使用信任区域技术证明了其收敛性。但是，作者想知道，对于标准线搜索，相似的收敛特性是否仍然正确，因为该方向可能不是下降方向。为此，作者提出了一种新的非单调m阶Armijo型线搜索，以确保全局收敛。在与信任区域相同的条件下，收敛速度也显示为m通过使用此行搜索技术+1。数值实验表明，该算法可为大规模问题节省大量运行时间，是一种有效的方法。