This paper addresses the convergence analysis of a variant of the LevenbergMarquardt method (LMM) designed for nonlinear least-squares problems with non-zero residue. This variant, called LMM with Singular Scaling (LMMSS), allows the LMM scaling matrix to be singular, encompassing a broader class of regularizers, which has proven useful in certain applications. In order to establish local convergence results, a careful choice of the LMM parameter is made based on the gradient linearization error, dictated by the nonlinearity and size of the residual. Under completeness and local error bound assumptions we prove that the distance from an iterate to the set of stationary points goes to zero superlinearly and that the iterative sequence converges. Furthermore, we also study a globalized version of the method obtained using linesearch and prove that any limit point of the generated sequence is stationary. Some examples are provided to illustrate our theoretical results.

翻译：本文针对非零残差非线性最小二乘问题，研究一种Levenberg-Marquardt方法（LMM）变体的收敛性分析。该变体称为具有奇异缩放的LMM（LMMSS），它允许LMM缩放矩阵为奇异矩阵，从而涵盖了一类更广泛的正则化子，这已在某些应用中证明是有用的。为了建立局部收敛结果，我们基于梯度线性化误差（由残差的非线性和大小决定）对LMM参数进行了谨慎选择。在完备性和局部误差界假设下，我们证明了迭代点到驻点集的距离以超线性速度趋于零，并且迭代序列收敛。此外，我们还研究了通过线搜索获得的该方法的全局化版本，并证明了生成序列的任何极限点都是驻点。文中提供了一些示例以说明我们的理论结果。