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


AuthorTitleYearJournal/ProceedingsReftypeLink
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 2020 Mathematical Modeling, Simulation and Optimization for Power Engineering and Management   incollection
 
BibTeX:
@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}
}
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 2020 accepted in Journal of Global Optimization   article
url  
BibTeX:
@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}
}
Merkert, M., Orlinskaya, G. & Weninger, D. An Exact Projection-Based Algorithm for Bilevel Mixed-Integer Problems with Nonlinearities 2020 Preprint (Optimization Online), submitted to Journal of Global Optimization   article
url  
BibTeX:
@article{Merkert2021,
  author = {Maximilian Merkert and Galina Orlinskaya and Dieter Weninger},
  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}
}
Uebbing, J., Rihko-Struckmann, L., Sager, S. & Sundmacher, K. CO2 methanation process synthesis by superstructure optimization 2020 Journal of CO2 Utilization   article
 
BibTeX:
@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}
}
Diedam, H. & Sager, S. Global optimal control with the direct multiple shooting method 2018 Optimal Control Applications and Methods   article DOI
 
BibTeX:
@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 = {http://dx.doi.org/10.1002/oca.2324}
}
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 2017 Studies in Computational Intelligence: Complex Networks and Their Applications V   inproceedings DOI
 
BibTeX:
@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 = {http://dx.doi.org/10.1007/978-3-319-50901-3}
}
Matke, C., Medjroubi, W., Kleinhans, D. & Sager, S. Structure Analysis of the German Transmission Network Using the Open Source Model SciGRID 2017 Advances in Energy System Optimization   inproceedings
 
BibTeX:
@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}
}
Merkert, M. Solving Mixed-Integer Linear and Nonlinear Network Optimization Problems by Local Reformulations and Relaxations 2017 School: Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)   phdthesis
url  
BibTeX:
@phdthesis{Merkert2017,
  author = {Maximilian Merkert},
  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}
}
Jung, M., Reinelt, G. & Sager, S. The Lagrangian Relaxation for the Combinatorial Integral Approximation Problem 2015 Optimization Methods and Software   article
 
BibTeX:
@article{Jung2015,
  author = {M. Jung and G. Reinelt and S. Sager},
  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}
}
Joseph-Duran, B., Jung, M., Ocampo-Martinez, C., Sager, S. & Cambrano, G. Minimization of Sewage Network Overflow 2014 Water Resources Management   article
 
BibTeX:
@article{Joseph-Duran2014,
  author = {B. Joseph-Duran and M. Jung and C. Ocampo-Martinez and S. Sager and G. Cambrano},
  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}
}
Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J. & Mahajan, A. Mixed-Integer Nonlinear Optimization 2013 Acta Numerica   incollection DOI
url  
BibTeX:
@incollection{Belotti2013,
  author = {P. Belotti and C. Kirches and S. Leyffer and J.T. Linderoth and J. Luedtke and A. Mahajan},
  title = {{M}ixed-{I}nteger {N}onlinear {O}ptimization},
  booktitle = {{A}cta {N}umerica},
  publisher = {Cambridge University Press},
  year = {2013},
  editor = {Arieh Iserles},
  volume = {22},
  pages = {1--131},
  url = {https://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8877390&fulltextType=RA&fileId=S0962492913000032},
  doi = {http://dx.doi.org/10.1017/S0962492913000032}
}
Sager, S., Jung, M. & Kirches, C. Combinatorial Integral Approximation 2011 Mathematical Methods of Operations Research   article DOI
url  
BibTeX:
@article{Sager2011a,
  author = {S. Sager and M. Jung and C. Kirches},
  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 = {http://dx.doi.org/10.1007/s00186-011-0355-4}
}
Sager, S. On the Integration of Optimization Approaches for Mixed-Integer Nonlinear Optimal Control 2011   misc
url  
BibTeX:
@misc{Sager2011d,
  author = {S. Sager},
  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}
}

Further references of the MathOpt group can be found on this page.

  • Jun 22nd 2021, Andreas Himmel succesfully defended his PhD thesis on "Optimization-based Operation Strategy and Storage Design for Coupled Processes".

...more

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
:

  • Jun 22nd 2021, Andreas Himmel succesfully defended his PhD thesis on "Optimization-based Operation Strategy and Storage Design for Coupled Processes".

...more

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
: