We introduce LMLS and LMQR, two globally convergent Levenberg–Marquardt methods for finding zeros of Hölder metrically subregular mappings that may have non-isolated zeros. The first method unifies the Levenberg–Marquardt direction and an Armijo-type line search, while the second incorporates this direction with a non-monotone quadratic regularization technique. For both methods, we prove the global convergence to a first-order stationary point of the associated merit function. Furthermore, the worst-case global complexity of these methods are provided, indicating that an approximate stationary point can be computed in at most $\mathcal{O}\left({\epsilon }^{-2}\right)$ function and gradient evaluations, for an accuracy parameter $\epsilon >0$. We also study the conditions for the proposed methods to converge to a zero of the associated mappings. Computing a moiety conserved steady state for biochemical reaction networks can be cast as the problem of finding a zero of a Hölder metrically subregular mapping. We report encouraging numerical results for finding a zero of such mappings derived from real-world biological data, which supports our theoretical foundations.

我们介绍了 LMLS 和 LMQR，这两种全局收敛的 Levenberg-Marquardt 方法用于寻找可能具有非孤立零点的 Hölder 度量次正则映射的零点。第一种方法统一了 Levenberg-Marquardt 方向和 Armijo 型线搜索，而第二种方法将此方向与非单调二次正则化技术相结合。对于这两种方法，我们证明了全局收敛到相关评价函数的一阶驻点。此外，提供了这些方法的最坏情况全局复杂度，表明最多可以计算出一个近似静止点$\mathcal{○}\left({\epsilon }^{-2}\right)$函数和梯度评估，用于精度参数$\epsilon >0$. 我们还研究了所提出的方法收敛到相关映射的零的条件。计算生化反应网络的部分守恒稳态可以看作是找到 Hölder 度量次正则映射的零点的问题。我们报告了令人鼓舞的数值结果，从现实世界的生物数据中找到了零这样的映射，这支持了我们的理论基础。