Fast Sparse Nonlinear Optimization and Its Application to Optimal Control

本计画将发展新的计算演算法来解决稀疏最佳化问题。这些都是科学、工程和工业中出现的大型复杂问题，其目标是以有效的方式操作大型互连系统。应用范围从电网到空中交通管制系统，再到计算机芯片的制造。在该项目中开发的一个具体应用是最优控制，它具有广泛的用途，包括空间飞行机动，空气动力学形状的优化设计和制造工艺的优化设计。该项目中为稀疏优化问题开发的算法提供了比以前更快、更准确的解决方案。 在该项目中开发的计算技术将建立在一个新的误差估计，这产生了一个严格的边界的错误在一个解决方案中的优化问题的违反一阶最优性条件，和一个新的双为基础的方法多面体投影。 在估计解中的误差的同时，生成了对偶乘子在平稳点处的近似。新的多面体投影算法将被纳入一个新的算法多面体约束非线性优化。结合线性化和全局化技术，新算法将用于实现一般稀疏约束非线性优化问题的快速精确求解。新的优化框架将被用来开发解决最优控制问题的hp-正交配置技术。 研究将集中在网格细化技术，快速准确的优化求解器将有助于推进技术。

New computational algorithms will be developed in this project for solving sparse optimization problems. These are large, complex problems that arise in science, engineering, and industry where the goal is to operate a large interconnected system in an efficient way. Applications range from power grids to air traffic control systems to the fabrication of computer chips. A specific application developed in the project is to optimal control which has a wide range of uses including space flight maneuvers, the optimal design of aerodynamic shapes, and the optimal design of manufacturing processes. The algorithms developed in the project for the sparse optimization problem provide solutions much faster and with much greater accuracy than was previously possible. The computational techniques developed in the project will be built around a new error estimator, which yields a tight bound for the error in a solution to an optimization problem in terms of the violation in the first-order optimality conditions, and a new dual-based approach to polyhedral projection. At the same time that the error in the solution is estimated, approximations to the dual multipliers at a stationary point are generated. The new polyhedral projection algorithm will be incorporated into a new algorithm for polyhedral constrained nonlinear optimization. With linearization and globalization techniques, the new algorithms will be used to achieve a fast and accurate solver for the general sparse constrained nonlinear optimization problem. The new optimization framework will be used to develop hp-orthogonal collocation techniques for solving optimal control problems. The research will focus on mesh refinement techniques for which the fast and accurate optimization solver will be instrumental in advancing the technology.