Large Scale Nonlinear Optimization

本研究项目是关于极小化多变量光滑非线性函数的算法。这项研究包括实践和理论两个部分，其最终目标是开发新的通用软件。该项目有三个组成部分。(1)将考虑在大型约束问题的算法中加入可变度量信息。最初的工作将集中于使用有限记忆方法来解决有界约束问题，但也将开发出适用于更一般约束的新技术。(2)给出了非线性共轭梯度法的完全收敛理论。新的结果将填补我们目前对非线性梯度法理解的重要空白。该理论提出的新方法将在大尺度问题上进行检验，特别是在气象资料的变分同化方面。(3)将开发用于优化方法的新的满阶自动缩放技术。这个想法是为了扩大黑森近似的因式分解的列。标度技术将极大地提高拟牛顿方法的性能，特别是对于病态问题。

This research project is concerned with algorithms for minimizing smooth nonlinear functions of many variables. The research has both practical and theoretical components, and its ultimate goal is the development of new general-purpose software. The project has three components. (1) The incorporation of variable metric information in algorithms for large constrained problems will be considered. The initial effort will concentrate on the use of limited memory methods for bound constrained problems, but novel techniques useful for more general constraints will also be developed. (2) A complete convergence theory for nonlinear conjugate gradient methods will be derived. The new results will fill important gaps in our present understanding of nonlinear gradient methods. The new methods suggested by the theory will be tested on large scale problems, in particular, in the variational assimilation of metereological data. (3) New full-rank automatic scaling techniques for optimization methods will be developed. The idea is to scale the columns of a factorization of the Hessian approximation. The scaling techniques will result in dramatic improvement in the performance of quasi- Newton methods, especially for ill-conditions problems.