Mathematical Algorithmic Optimization - Otto-von-Guericke-University Magdeburg


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

Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J. & Mahajan, A. Mixed-Integer Nonlinear Optimization 2013 Acta Numerica   incollection DOI
  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 = {},
  doi = {}
Joseph-Duran, B., Jung, M., Ocampo-Martinez, C., Sager, S. & Cambrano, G. Minimization of Sewage Network Overflow 2014 Water Resources Management   article
  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}
Jung, M., Reinelt, G. & Sager, S. The Lagrangian Relaxation for the Combinatorial Integral Approximation Problem 2015 Optimization Methods and Software   article
  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}
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
  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 = {}
Sager, S., Jung, M. & Kirches, C. Combinatorial Integral Approximation 2011 Mathematical Methods of Operations Research   article DOI
  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 = {},
  doi = {}

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

Last Modification: 2016-05-12 - Contact Person: Sebastian Sager - Impressum