| Issue |
ESAIM: COCV
Volume 26, 2020
|
|
|---|---|---|
| Article Number | 32 | |
| Number of page(s) | 16 | |
| DOI | https://doi.org/10.1051/cocv/2019016 | |
| Published online | 09 April 2020 | |
Improved regularity assumptions for partial outer convexification of mixed-integer PDE-constrained optimization problems*,**,***
Institut für Mathematische Optimierung, Technische Universität Carolo-Wilhelmina zu Braunschweig, Universitätsplatz 2,
38106
Braunschweig, Germany.
**** Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
19
April
2018
Accepted:
22
March
2019
Abstract
Partial outer convexification is a relaxation technique for MIOCPs being constrained by time-dependent differential equations. Sum-Up-Rounding algorithms allow to approximate feasible points of the relaxed, convexified continuous problem with binary ones that are feasible up to an arbitrarily small δ > 0. We show that this approximation property holds for ODEs and semilinear PDEs under mild regularity assumptions on the nonlinearity and the solution trajectory of the PDE. In particular, requirements of differentiability and uniformly bounded derivatives on the involved functions from previous work are not necessary to show convergence of the method.
Mathematics Subject Classification: 49M20 / 49N60 / 90C11 / 49J20
Key words: Mixed-integer optimal control with PDEs / relaxations of mixed-integer optimal control / regularity
P. Manns and C. Kirches acknowledge funding by Deutsche Forschungsgemeinschaft through Priority Programme 1962.
C. Kirches acknowledges financial support by the German Federal Ministry of Education and Research, program “Mathematics for Innovations in Industry and Service”, grants 05M2016-MOPhaPro, 05M17MBA-MOReNet, and program “IKT 2020: Software Engineering”, grant 61210304-ODINE.
The authors would like to thank Robert Haller-Dintelmann, TU Darmstadt, and Dirk Lorenz, TU Braunschweig, for helpful discussions on the topic.
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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