Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty of the problem, and motivates the need for efficient algorithms amenable to large-scale implementations. In this paper, we propose and analyze a Levenberg-Marquardt algorithm for nonlinear least squares subject to nonlinear equality constraints. Our algorithm is based on inexact solves of linear least-squares problems, that only require Jacobian-vector products. Global convergence is guaranteed by the combination of a composite step approach and a nonmonotone step acceptance rule. We illustrate the performance of our method on several test cases from data assimilation and inverse problems: our algorithm is able to reach the vicinity of a solution from an arbitrary starting point, and can outperform the most natural alternatives for these classes of problems.

翻译：最起码的方形构成最突出的优化问题类别之一, 科学计算和数据安装方面有许多应用。 当这种配方旨在建模复杂的系统时, 优化过程必须通过纳入限制来说明非线性动态。 此外, 这些系统往往包含大量变量, 增加问题的难度, 并促使需要高效算法, 以便大规模实施。 在本文中, 我们提议并分析非线性最低方格的利文伯格- 马尔夸特算法, 受非线性平等制约。 我们的算法基于线性最小方格问题的不精确解决方案, 只需要雅各布- 矢量产品。 全球趋同得到综合步骤方法和非单步接受规则的结合的保证。 我们用数据同化和反问题等数个测试案例来说明我们的方法的性能: 我们的算法能够从任意的起点接近解决方案的近处, 并且能够超越这些问题中最自然的替代方法。