SIAM Journal on Scientific Computing, Ahead of Print.
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.

中文翻译：

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非线性等式约束最小二乘问题的非单调无矩阵算法

SIAM 科学计算杂志，提前印刷。
最小二乘法是最突出的一类优化问题，在科学计算和数据拟合中有许多应用。当此类公式旨在为复杂系统建模时，优化过程必须通过结合约束来考虑非线性动力学。此外，这些系统通常包含大量变量，这增加了问题的难度，并激发了对适用于大规模实现的高效算法的需求。在本文中，我们提出并分析了一种用于非线性等式约束的非线性最小二乘法的 Levenberg--Marquardt 算法。我们的算法基于对线性最小二乘问题的不精确求解，这些问题只需要雅可比向量乘积。综合步骤方法和非单调步骤接受规则的组合保证了全局收敛。我们从数据同化和逆问题中说明了我们的方法在几个测试用例上的性能；我们的算法能够从任意起点到达解决方案的附近，并且可以胜过这些类别问题的最自然的替代方案。