Mixed-Integer Nonlinear Programming

We are particularly interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). We focus, e.g., on combinatorial constraints such as restrictions on the number of switches on a fixed time grid.

We propose a novel approach that is based on a decomposition of the MINLP into a NLP and a MILP. We discuss the relation of the MILP solution to the MINLP solution and formulate bounds for the gap between the two, depending on Lipschitz constants and the control discretization grid size. The MILP solution can also be used for an efficient initialization of the MINLP solution process.

The speedup of the solution of the MILP compared to the MINLP solution is considerable already for general purpose MILP solvers. We analyze the structure of the MILP that takes switching constraints into account and propose a tailored Branch and Bound strategy that outperforms state-of-the-art solvers on a numerical case study.

Selected publications

2020
incollection
Garmatter, D., Maggi, A., Wenzel, M., Monem, S., Hahn, M., Stoll, M., Sager, S., Benner, P., Sundmacher, K.
Power-to-Chemicals: A Superstructure Problem for Sustainable Syngas Production
Mathematical Modeling, Simulation and Optimization for Power Engineering and Management
@incollection{Garmatter2020,
    author = {Garmatter, D. and Maggi, A. and Wenzel, M. and Monem, S. and Hahn, M. and Stoll, M. and Sager, S. and Benner, P. and Sundmacher, K.},
    title = {Power-to-Chemicals: A Superstructure Problem for Sustainable Syngas Production},
    booktitle = {Mathematical Modeling, Simulation and Optimization for Power Engineering and Management},
    publisher = {Springer},
    year = {2020},
    pages = {145--168}
}
2020
article
Liers, F., Martin, A., Merkert, M., Mertens, N., Michaels, D.
Solving Mixed-Integer Nonlinear Optimization Problems using Simultaneous Convexification - a Case Study for Gas Networks
accepted in Journal of Global Optimization
@article{Liers2020,
    author = {Liers, F. and Martin, A. and Merkert, M. and Mertens, N. and Michaels, D.},
    title = {Solving Mixed-Integer Nonlinear Optimization Problems using Simultaneous Convexification - a Case Study for Gas Networks},
    journal = {accepted in {Journal of Global Optimization}},
    year = {2020},
    note = {accepted},
    url = {http://www.optimization-online.org/DB_HTML/2020/02/7628.html}
}
2020
article
Merkert, M., Orlinskaya, G., Weninger, D.
An Exact Projection-Based Algorithm for Bilevel Mixed-Integer Problems with Nonlinearities
Preprint (Optimization Online), submitted to Journal of Global Optimization
@article{Merkert2021,
    author = {Merkert, Maximilian and Orlinskaya, Galina and Weninger, Dieter},
    title = {An Exact Projection-Based Algorithm for Bilevel Mixed-Integer Problems with Nonlinearities},
    journal = {Preprint (Optimization Online), submitted to {Journal of Global Optimization}},
    year = {2020},
    note = {submitted},
    url = {http://www.optimization-online.org/DB_HTML/2020/12/8153.html}
}
2020
article
Uebbing, J., Rihko-Struckmann, L., Sager, S., Sundmacher, K.
CO2 methanation process synthesis by superstructure optimization
Journal of CO2 Utilization
@article{Uebbing2020,
    author = {Uebbing, Jennifer and Rihko-Struckmann, Liisa and Sager, Sebastian and Sundmacher, Kai},
    title = {CO2 methanation process synthesis by superstructure optimization},
    journal = {Journal of CO2 Utilization},
    publisher = {Elsevier},
    year = {2020},
    volume = {40},
    pages = {101228}
}
2018
article
Diedam, H., Sager, S.
Global optimal control with the direct multiple shooting method
Optimal Control Applications and Methods
@article{Diedam2018,
    author = {Diedam, H. and Sager, S.},
    title = {Global optimal control with the direct multiple shooting method},
    journal = {Optimal Control Applications and Methods},
    year = {2018},
    volume = {39},
    number = {2},
    pages = {449--470},
    doi = {10.1002/oca.2324}
}
2017
inproceedings
Matke, C., Bienstock, D., Munoz, G., Yang, S., Kleinhans, D., Sager, S.
Robust optimization of power network operation: storage devices and the role of forecast errors in renewable energies
Studies in Computational Intelligence: Complex Networks and Their Applications V
@inproceedings{Matke2017,
    author = {Matke, C. and Bienstock, D. and Munoz, G. and Yang, S. and Kleinhans, D. and Sager, S.},
    title = {Robust optimization of power network operation: storage devices and the role of forecast errors in renewable energies},
    booktitle = {Studies in Computational Intelligence: Complex Networks and Their Applications V},
    year = {2017},
    number = {693},
    pages = {809--820},
    doi = {10.1007/978-3-319-50901-3}
}
2017
inproceedings
Matke, C., Medjroubi, W., Kleinhans, D., Sager, S.
Structure Analysis of the German Transmission Network Using the Open Source Model SciGRID
Advances in Energy System Optimization
@inproceedings{Matke2017a,
    author = {Matke, Carsten and Medjroubi, Wided and Kleinhans, David and Sager, Sebastian},
    title = {Structure Analysis of the German Transmission Network Using the Open Source Model SciGRID},
    booktitle = {Advances in Energy System Optimization},
    publisher = {Springer International Publishing},
    year = {2017},
    editor = {Bertsch, Valentin and Fichtner, Wolf and Heuveline, Vincent and Leibfried, Thomas},
    pages = {177--188},
    address = {Cham}
}
2017
phdthesis
Merkert, M.
Solving Mixed-Integer Linear and Nonlinear Network Optimization Problems by Local Reformulations and Relaxations
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
@phdthesis{Merkert2017,
    author = {Merkert, Maximilian},
    title = {Solving Mixed-Integer Linear and Nonlinear Network Optimization Problems by Local Reformulations and Relaxations},
    school = {Friedrich-Alexander-Universit{\"a}t Erlangen-N{\"u}rnberg (FAU)},
    year = {2017},
    url = {https://opus4.kobv.de/opus4-fau/frontdoor/index/index/docId/9403}
}
2015
article
Jung, M., Reinelt, G., Sager, S.
The Lagrangian Relaxation for the Combinatorial Integral Approximation Problem
Optimization Methods and Software
@article{Jung2015,
    author = {Jung, M. and Reinelt, G. and Sager, S.},
    title = {{T}he {L}agrangian {R}elaxation for the {C}ombinatorial {I}ntegral {A}pproximation {P}roblem},
    journal = {{O}ptimization {M}ethods and {S}oftware},
    year = {2015},
    volume = {30},
    number = {1},
    pages = {54--80}
}
2014
article
Joseph-Duran, B., Jung, M., Ocampo-Martinez, C., Sager, S., Cambrano, G.
Minimization of Sewage Network Overflow
Water Resources Management
@article{Joseph-Duran2014,
    author = {Joseph-Duran, B. and Jung, M. and Ocampo-Martinez, C. and Sager, S. and Cambrano, G.},
    title = {{M}inimization of {S}ewage {N}etwork {O}verflow},
    journal = {{W}ater {R}esources {M}anagement},
    year = {2014},
    volume = {28},
    number = {1},
    pages = {41--63}
}
2013
incollection
Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.
Mixed-Integer Nonlinear Optimization
Acta Numerica
@incollection{Belotti2013,
    author = {Belotti, P. and Kirches, C. and Leyffer, S. and Linderoth, J.T. and Luedtke, J. and Mahajan, A.},
    title = {{M}ixed-{I}nteger {N}onlinear {O}ptimization},
    booktitle = {{A}cta {N}umerica},
    publisher = {Cambridge University Press},
    year = {2013},
    editor = {Iserles, Arieh},
    volume = {22},
    pages = {1--131},
    url = {https://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8877390&fulltextType=RA&fileId=S0962492913000032},
    doi = {10.1017/S0962492913000032}
}
2011
article
Sager, S., Jung, M., Kirches, C.
Combinatorial Integral Approximation
Mathematical Methods of Operations Research
@article{Sager2011a,
    author = {Sager, S. and Jung, M. and Kirches, C.},
    title = {{C}ombinatorial {I}ntegral {A}pproximation},
    journal = {{M}athematical {M}ethods of {O}perations {R}esearch},
    year = {2011},
    volume = {73},
    number = {3},
    pages = {363--380},
    url = {https://mathopt.de/publications/Sager2011a.pdf},
    doi = {10.1007/s00186-011-0355-4}
}
2011
misc
Sager, S.
On the Integration of Optimization Approaches for Mixed-Integer Nonlinear Optimal Control
@misc{Sager2011d,
    author = {Sager, S.},
    title = {{O}n the {I}ntegration of {O}ptimization {A}pproaches for {M}ixed-{I}nteger {N}onlinear {O}ptimal {C}ontrol},
    year = {2011},
    note = {Habilitation},
    url = {https://mathopt.de/publications/Sager2011d.pdf}
}

Prof. Dr. Sebastian Sager
Head of MathOpt group
at the Institute of Mathematical Optimization
at the Faculty of Mathematics
at the Otto von Guericke University Magdeburg

Universitätsplatz 2, 02-224
39106 Magdeburg, Germany

: +49 391 67 58745
:

Susanne Heß

Universitätsplatz 2, 02-201
39106 Magdeburg, Germany

: +49 391 67 58756
:

Prof. Dr. Sebastian Sager
Head of MathOpt group
at the Institute of Mathematical Optimization
at the Faculty of Mathematics
at the Otto von Guericke University Magdeburg

Universitätsplatz 2, 02-224
39106 Magdeburg, Germany

: +49 391 67 58745
:

Susanne Heß

Universitätsplatz 2, 02-201
39106 Magdeburg, Germany

: +49 391 67 58756
: