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Séminaire du département Automatique du 16/05/2019 à 15h00

 

A Novel Numerical Method for Solving Dynamic Optimization Problems with Convergence Guarantees

Intervenant : Eric KERRIGAN, Imperial College London

Lieu : B208, GIPSA-lab

 

Résumé : 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, where the differential equation is required to be satisfied at a finite number of points. We instead 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


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