In this paper, we propose two complementary variants of the projected Levenberg–Marquardt (LM) algorithm for solving convex constrained nonlinear equations. Since the orthogonal projection onto the feasible set may be computationally expensive, we first propose a local LM algorithm in which inexact projections are allowed. The feasible inexact projections used in our algorithm can be easily obtained by means of iterative methods, such as conditional gradient. Local convergence of the proposed algorithm is established by using an error bound condition which is weaker than the standard full-rank assumption. We further present and analyze a global version of this algorithm by means of a nonmonotone line search technique. Numerical experiments are reported to showcase the effectiveness of the proposed algorithms, especially when the projection onto the feasible set is difficult to compute.

在本文中，我们提出了Levenberg-Marquardt（LM）投影算法的两个互补变体，用于求解凸约束非线性方程。由于到可行集上的正交投影可能在计算上昂贵，因此我们首先提出一种允许不精确投影的局部LM算法。我们的算法中使用的可行的不精确投影可以通过迭代方法（例如条件梯度）轻松获得。该算法的局部收敛是通过使用一个误差约束条件建立的，该误差约束条件比标准的全秩假设弱。我们进一步通过非单调线搜索技术介绍并分析了该算法的全局版本。据报道，数值实验证明了所提出算法的有效性，