arXiv subjects

Martin P. Neuenhofen

Publications and source records attributed to Martin P. Neuenhofen.

4 recordsLinked to original sources

arXiv2021

Dynamic Optimization with Convergence Guarantees

We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. We prove convergence of our numerical method under reasonably mild assumptions: boundedness and Lipschitz-continuity of the problem-defining functions. We do not require uniqueness, differentiability or constraint qualifications to hold and we avoid the use of Lagrange multipliers. Our approach differs fundamentally from well-known methods based on collocation; we follow a penalty-barrier approach, where we compute integral quadratic penalties on the equality path constraints and point constraints, and integral log-barriers on the inequality path constraints. The resulting penalty-barrier functional can be minimized numerically using finite elements and penalty-barrier interior-point nonlinear programming solvers. Order of convergence results are derived, even if components of the solution are discontinuous. We also present numerical results to compare our method against collocation methods. The numerical results show that for the same degree and mesh, the computational cost is similar, but that the new method can achieve a smaller error and converges in cases where collocation methods fail.

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arXiv2020

An Integral Penalty-Barrier Direct Transcription Method for Optimal Control

Some direct transcription methods can fail to converge, e.g. when there are singular arcs. We recently introduced a convergent direct transcription method for optimal control problems, called the penalty-barrier finite element method (PBF). PBF converges under very weak assumptions on the problem instance. PBF avoids the ringing between collocation points, for example, by avoiding collocation entirely. Instead, equality path constraint residuals are forced to zero everywhere by an integral quadratic penalty term. We highlight conceptual differences between collocation- and penalty-type direct transcription methods. Theoretical convergence results for both types of methods are reviewed and compared. Formulas for implementing PBF are presented, with details on the formulation as a nonlinear program (NLP), sparsity and solution. Numerical experiments compare PBF against several collocation methods with regard to robustness, accuracy, sparsity and computational cost. We show that the computational cost, sparsity and construction of the NLP functions are roughly the same as for orthogonal collocation methods of the same degree and mesh. As an advantage, PBF converges in cases where collocation methods fail. PBF also allows one to trade off computational cost, optimality and violation of differential and other equality equations against each other.

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arXiv2020

A Direct Method for Solving Integral Penalty Transcriptions of Optimal Control Problems

We present a numerical method for the minimization of objectives that are augmented with large quadratic penalties of overdetermined inconsistent equality constraints. Such objectives arise from quadratic integral penalty methods for the direct transcription of equality constrained optimal control problems. The Augmented Lagrangian Method (ALM) has a number of advantages over the Quadratic Penalty Method (QPM) for solving this class of problems. However, if the equality constraints of the discretization are inconsistent, then ALM might not converge to a point that minimizes the unconstrained bias of the objective and penalty term. Therefore, in this paper we explore a modification of ALM that fits our purpose. Numerical experiments demonstrate that the modified ALM can minimize certain quadratic penalty-augmented functions faster than QPM, whereas the unmodified ALM converges to a minimizer of a significantly different problem.

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arXiv2017

A restarted GMRES-based implementation of IDR(s)stab(L) to yield higher robustness

In this thesis we propose a novel implementation of IDRstab that avoids several unlucky breakdowns of current IDRstab implementations and is further capable of benefiting from a particular lucky breakdown scenario. IDRstab is a very efficient short-recurrence Krylov subspace method for the numerical solution of linear systems. Current IDRstab implementations suffer from slowdowns in the rate of convergence when the basis vectors of their oblique projectors become linearly dependent. We propose a novel implementation of IDRstab that is based on a successively restarted GMRES method. Whereas the collinearity of basis vectors in current IDRstab implementations would lead to an unlucky breakdown, our novel IDRstab implementation can strike a benefit from it in that it terminates with the exact solution whenever a new basis vector lives in the span of the formerly computed basis vectors. Numerical experiments demonstrate the superior robustness of our novel implementation with regards to convergence maintenance and the achievable accuracy of the numerical solution.

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