Mathematical Sciences: Projected Newton Methods for Optimal Control Problems and Other Large-Scale Structured Nonlinear Programs

这里提出的研究的主要目标是设计，分析和测试所谓的计算优化中的投影牛顿方法，特别强调这些方法在最优控制问题中的应用。考虑了两类问题：离散时间动态优化问题和连续时间问题。本研究的结果似乎也可能适用于其他大规模结构非线性优化问题。投影牛顿法算法比标准的一阶方法（即基于梯度的方法）收敛速度更快，但相关的每次迭代计算成本仍然仅与n成正比，其中n为所考虑的时间区间的子区间数。每次迭代工作的相当大一部分可以组织为并行计算。本研究属于优化与控制方面的计算方法。这种类型的问题出现在许多应用中，特别是在空间飞行器的轨迹优化和各种工业过程控制计算机的最佳程序计算中。

The investigation proposed here has as its main objective the design, analysis, and testing of the so called projected Newton methods in computational optimization, with a special emphasis on the application of these methods to optimal control problems. Two kinds of problems are considered: disrete-time dynamical optimization problems and continuous time problems. It seems likely that the results of this research may be also applicable to other large-scale structured nonlinear optimization problems. The projected Newton method algorithms converge more rapidly than the standard first order, i.e. gradient based methods, yet the associated per iteration computational costs are still only proportional to n, where n is the number of subintervals of the time interval under consideration. A sizeable portion of the per iteration work can be organized for parallel computation. This research belongs to computational methods in optimization and control. Problems of this type arise in many applications, typically in optimization of trajectories for space vehicles and in calculations of optimal programs for control computers for various industrial processes.