The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst-case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under suitable assumptions. We introduce a novel Levenberg-Marquardt method that matches, simultaneously, the state of the art in all of these convergence properties with a single seamless algorithm. Numerical experiments confirm the theoretical behavior of our proposed algorithm.

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求解逆问题的 Levenberg-Marquardt 算法的收敛性和复杂度分析

Levenberg-Marquardt 算法是用于寻找非线性最小二乘问题解的最流行的算法之一。在基本过程的不同修改变体中，该算法在适当的假设下享有全局收敛性、具有竞争力的最坏情况迭代复杂度率以及零和非零小残差问题的局部收敛保证率。我们引入了一种新颖的 Levenberg-Marquardt 方法，该方法同时与单个无缝算法在所有这些收敛特性中的现有技术相匹配。数值实验证实了我们提出的算法的理论行为。