论文标题
非线性基质算法，用于非线性等效限制的最小二乘问题
A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Least-Squares Problems
论文作者
论文摘要
最小二乘构成了最突出的优化问题类别之一，在科学计算和数据拟合中有许多应用。当这种配方旨在建模复杂系统时，优化过程必须通过合并约束来解释非线性动力学。此外，这些系统通常会包含大量变量，从而增加了问题的难度，并激发了对大规模实施的有效算法的需求。 在本文中，我们提出和分析了一种受非线性平等约束的非线性最小二乘正方形的Levenberg-Marquardt算法。我们的算法基于线性最小二乘问题的不精确溶液，仅需要Jacobian-Vector产品。通过复合步骤方法和非单调步骤接受规则的结合来保证全球收敛。我们说明了方法在数据同化和反面问题中的几个测试用例上的性能：我们的算法能够从任意起点从任意起点到达解决方案的附近，并且可以优于这些类别问题的最自然的选择。
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.
