Fast Sparse Nonlinear Optimization and Its Application to Optimal Control

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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.