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An efficient preconditioning method for state box-constrained optimal control problems

Owe Axelsson, Maya Neytcheva and Anders Ström

Abstract

An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrix blocks is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau–Yosida penalization method. It is shown that there occur very few nonlinear iteration steps and also few iterations to solve the arising linearized equations on the fine mesh. This holds for a wide range of the penalization and discretization parameters. The arising nonlinearity can be handled with a hybrid nonlinear-linear procedure that raises the computational efficiency of the overall solution method.

MSC 2010: 65F08; 49J20; 65N30

Acknowledgements

Comments by the anonymous reviewers on the original version of this paper enabled improvements of the presentation and are highly appreciated.

We are indebted to Johanna Brodin who performed anew all numerous numerical tests with the software that enabled more detailed analysis and deeper understanding of the numerical behaviour of the coupled nonlinear-linear solver used in the experiments.

The work of the first author was supported by The Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II) project IT4Innovations excellence in scienceLQ1602.

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Received: 2017-04-02
Revised: 2018-10-02
Accepted: 2018-10-25
Published Online: 2018-10-31
Published in Print: 2018-12-19

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