Decomposition methods for mixed-integer optimal control

Description

The objective of this project A05 inside of the TRR 154 is the development of mathematical algorithms to find an optimal control for mixed-integer problems on transport networks with the help of decomposition methods. For the sake of synergy inside of the TRR 154 the focus is on gas networks, but the methods should also be useful for water networks or other energy networks. The optimization problems are planned to be decomposed with respect to variables but also with respect to subsystems, with the result that we are getting a time-expansive MINLP with a hierarchic structure. On the upper level, there are integer decisions, while on the lower level the focus is on continuous variables. Eventually the continuous variables are discretized for the numerical realization. This approach investigates the whole range from totally discrete MINLPs to PDE-based MINLPs in the Banach space. While we use well-known finite-volume methods to simulate the gas equations at the beginning, we want to include methods from the sub-project C02 inside of TRR 154 during the progress. The same holds for the inclusion of MINLP-Solvers from the sub-project B07. So the focus of this project is on the mathematical analysis of structured MINLPs in the light of hierarchic models. The methods of many classical decomposition approaches like Benders, Outer Approximation or Dantzig-Wolfe focus on a generation of cutting planes in the subproblem, which tighten the relaxed set in the masterproblem to achieve a convergence between the values of the objective functions of the masterproblem (dual bound) and the subproblem (primal bounds). In this sub-project we want the subproblem to provide disjunctions for the masterproblem as well, because such an approach enables the algorithm to find global optima for non-convex problems as well.

People involved

Alexander Martin
Martin Schmidt
Mathias Sirvent

Contact

For further details about this project please contact Mathias Sirvent (mathias.sirvent[at]fau.de).

Supported by

Deutsche Forschungsgemeinschaft, Sonderforschungsbereich/Transregio 154

Partners

This project is part of the Sonderforschungsbereich/Transregio 154. Collaborative researchers in this project are

Prof. Dr. Günter Leugering (FAU, Applied Mathematics 2)
Prof. Dr. Martin Gugat (FAU, Applied Mathematics 2)
David Wintergerst (FAU, Applied Mathematics 2)

Related Talks

May 24th, 2017 by Mathias Sirvent
SIOPT: SIAM Conference on Optimization, Vancouver, Canada
MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems
http://www.siam.org/meetings/op17/
September 2nd, 2015 by Mathias Sirvent
OR: International Conference on Operations Research, Vienna, Austria
A Decomposition Method for Mixed-Integer Programs with Differential Equations (Version 2015)
http://or2015.univie.ac.at/
July 5th, 2016 by Mathias Sirvent
EURO: European Conference on Operational Research, Poznań, Poland
A Decomposition Method for Mixed-Integer Programs with Differential Equations (Version 2016)
http://www.euro2016.poznan.pl/
July 19th, 2016 by Mathias Sirvent
7ECM: 7th European Congress of Mathematics, Berlin, Germany
A Decomposition Method for Mixed-Integer Programs with Differential Equations (Version 2016)
http://www.7ecm.de/

Publications

Submitted work / Preprints

The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities. Martin Schmidt, Mathias Sirvent, and Winnifried Wollner. Submitted. Preprint (4/2018): Optimization Online, TRR154 Preprint Server.
A Decomposition Method for MINLPs with Lipschitz Continuous Nonlinearities. Martin Schmidt, Mathias Sirvent, and Winnifried Wollner. Submitted. Updated Preprint (3/2018): Optimization Online, TRR154 Preprint Server.

Journal Articles

MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems. Martin Gugat, Günter Leugering, Alexander Martin, Martin Schmidt, Mathias Sirvent, and David Wintergerst. In Computational Optimization and Applications, Volume 70, Issue 1, pp. 267-294, May 2018. DOI: 10.1007/s10589-017-9970-1.
Towards Simulation Based Mixed-Integer Optimization with Differential Equations. Martin Gugat, Günter Leugering, Alexander Martin, Martin Schmidt, Mathias Sirvent, and David Wintergerst. In Networks, 2018. DOI: 10.1002/net.21812.
GasLib – A Library of Gas Network Instances. Jointly with Martin Schmidt, Denis Aßmann, Robert Burlacu, Jesco Humpola, Imke Joormann, Nikolaos Kanelakis, Djamal Oucherif, Marc E. Pfetsch, Lars Schewe, Robert Schwarz, and Mathias Sirvent. In Data, Volume 2, Issue 4, December 2017. DOI: 10.3390/data2040040.
Nonoverlapping Domain Decomposition for Optimal Control Problems governed by Semi-Linear Models for Gas Flow in Networks. Jointly with Günter Leugering, Alexander Martin, Martin Schmidt, and Mathias Sirvent. In Control and Cybernetics, Volume 46, Issue 3, pp. 191-225, 2017.
A Linearized Model for the Optimization of the Coupled Electricity and Natural Gas System. Mathias Sirvent, Nikolaos Kanelakis, Björn Geißler, and Pandelis Biskas. In Journal of Modern Power Systems and Clean Energy, Volume 5, Issue 3, pp. 364-374, May 2017. DOI: 10.1007/s40565-017-0275-2.