Imperial College London
Tuesday, December 11, 2018, 11:00 - 12:30
Room 01-012, Georges-Köhler-Allee 102, Freiburg 79110, Germany
We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. These problems arise in a large class of optimal control, estimation and system design applications. In order to provide numerical convergence guarantees, we show that it is sufficient for the functions that define the problem to satisfy boundedness and Lipschitz conditions. Our assumptions are the most general to date; we do not require uniqueness, differentiability or constraint qualifications to hold and we avoid the use of Lagrange multipliers. Our approach differs fundamentally from state-of-the-art methods based on collocation. We follow a least-squares approach to finding approximate solutions to the differential equations. The objective is augmented with the integral of a quadratic penalty on the differential equation residual and a logarithmic barrier for the inequality constraints, as well as a quadratic penalty on the point constraint residual. The resulting unconstrained infinite-dimensional optimization problem is discretized using finite elements, while integrals are replaced by quadrature approximations if they cannot be evaluated analytically. We will present order of convergence results, even if components of solutions are allowed to be discontinuous. We demonstrate via numerical experiments that our method is able to solve problems where certain collocation methods fail. Our method is also able to achieve orders of magnitude better accuracy than collocation methods, while only needing to solve optimization problems of similar or smaller size than those resulting from collocation.
Eric Kerrigan is a Reader in Control Engineering and Optimization at Imperial College London. He received a BSc(Eng) from the University of Cape Town and a PhD from the University of Cambridge. His research is in the design of efficient numerical methods and computing architectures for solving optimal control problems in real-time, with applications in the design of aerospace, renewable energy and information systems. He is the chair of the IFAC Technical Committee on Optimal Control, a Senior Editor for IEEE Transactions on Control Systems Technology and an Associate Editor of IEEE Transactions on Automatic Control.