Levenberg-Marquardt methods (LMMs) are the most typical algorithms for
solving nonlinear equations $F(x) = 0$, where $F : \mathbb{R}^n \to \mathbb{R}^m$ is a continuously
differentiable function. They sequentially solve subproblems represented as
squared residual of the Newton equations with the $L_2$ regularization to determine
the search direction. However, since the subproblems of the LMMs
are usually reduced to linear equations with $n$ variables, it takes much time
to solve them when $m \ll n$.
In this paper, we propose a new LMM which generalizes the $L_2$ regularization
of the subproblems of the ordinary LMMs. By virtue of the generalization,
we can choose a suitable regularization term for each given problem.
Moreover, we show that a sequence generated by the proposed method
converges globally and quadratically under some reasonable assumptions.
Finally, we conduct numerical experiments to confirm that the proposed
method performs better than the existing LMMs for some problems that
satisfy $m \ll n$.