Mixedinteger optimal control problems (MIOCPs) in differential equations have gained increasing interest over the last years. This is probably due to the fact that the underlying processes have a high potential for optimization. Typical examples are the choice of gears in transport or processes in chemical engineering involving onoff valves. Also experimental design problems for continuous processes can be formulated as MIOCPs.
Although the first MIOCPs, namely the optimization of subway trains that are equipped with discrete acceleration stages, were already solved in the early eighties for the city of New York, the socalled indirect methods used there do not seem appropriate for generic largescale optimal control problems with underlying nonlinear differential algebraic equation systems. Instead direct methods, in particular allatonce approaches like the Direct Multiple Shooting Method, have become the methods of choice for most practical problems.
In direct methods infinitedimensional control functions are discretized by basis functions and corresponding finitedimensional parameters that enter into the optimization problem. The drawback of direct methods with binary control functions obviously is that they lead to highdimensional vectors of binary variables. For many practical applications a fine control discretization is required, however. Therefore, techniques from mixedinteger nonlinear programming like Branch and Bound or Outer Approximation will work only on limited and small time horizons because of the exponentially growing complexity of the problem.
We propose to use an outer convexification with respect to the binary controls. The reformulated control problem has two main advantages compared to standard formulations or convexifications. First, especially for timeoptimal control problems, the optimal solution of the relaxed problem will exhibit a bangbang structure, and is thus already integer feasible. Second, theoretical results have recently been found that show that even for pathconstrained and sensitivityseeking arcs the optimal solution of the relaxed problem yields the exact lower bound on the minimum of the integer problem. This allows to calculate precise error estimates, if a coarser control discretization grid, a simplified switching structure for the optimization of switching times, or heuristics are used.
The group actively contributes to the creation of an online benchmark library for mixedinteger optimal control problems. Several of the applications that could be solved with our algorithms are listed there.
Author  Title  Year  Journal/Proceedings  Reftype  Link 

Jung, M., Reinelt, G. & Sager, S.  The Lagrangian Relaxation for the Combinatorial Integral Approximation Problem [BibTeX] 
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 = {5480} } 

JosephDuran, B., Jung, M., OcampoMartinez, C., Sager, S. & Cambrano, G.  Minimization of Sewage Network Overflow [BibTeX] 
2014  Water Resources Management  article  
BibTeX:
@article{JosephDuran2014, author = {B. JosephDuran and M. Jung and C. OcampoMartinez 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 = {4163} } 

Jung, M.  Relaxations and Approximations for MixedInteger Optimal Control [BibTeX] 
2013  School: University Heidelberg  phdthesis  url 
BibTeX:
@phdthesis{Jung2013a, author = {M. Jung}, title = {{R}elaxations and {A}pproximations for {M}ixed{I}nteger {O}ptimal {C}ontrol}, school = {University Heidelberg}, year = {2013}, url = {http://www.ub.uniheidelberg.de/archiv/16036} } 

Sager, S.  Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin's Maximum Principle [BibTeX] 
2013  SIAM Journal on Control and Optimization  article  url 
BibTeX:
@article{Sager2013, author = {Sager, S.}, title = {{S}ampling {D}ecisions in {O}ptimum {E}xperimental {D}esign in the {L}ight of {P}ontryagin's {M}aximum {P}rinciple}, journal = {{SIAM} Journal on Control and Optimization}, year = {2013}, volume = {51}, number = {4}, pages = {31813207}, url = {https://mathopt.de/PUBLICATIONS/Sager2013.pdf} } 

Sager, S., Bock, H. & Diehl, M.  The Integer Approximation Error in MixedInteger Optimal Control [BibTeX] 
2012  Mathematical Programming A  article  url 
BibTeX:
@article{Sager2012a, author = {Sager, S. and Bock, H.G. and Diehl, M.}, title = {{T}he {I}nteger {A}pproximation {E}rror in {M}ixed{I}nteger {O}ptimal {C}ontrol}, journal = {{M}athematical {P}rogramming {A}}, year = {2012}, volume = {133}, number = {12}, pages = {123}, url = {https://mathopt.de/PUBLICATIONS/Sager2012a.pdf} } 

Sager, S.  A benchmark library of mixedinteger optimal control problems [BibTeX] 
2012  Mixed Integer Nonlinear Programming  inproceedings  url 
BibTeX:
@inproceedings{Sager2012b, author = {S. Sager}, title = {{A} benchmark library of mixedinteger optimal control problems}, booktitle = {{M}ixed {I}nteger {N}onlinear {P}rogramming}, publisher = {Springer}, year = {2012}, editor = {J. Lee and S. Leyffer}, pages = {631670}, url = {https://mathopt.de/PUBLICATIONS/Sager2012b.pdf} } 

Sager, S., Jung, M. & Kirches, C.  Combinatorial Integral Approximation [BibTeX] 
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 = {363380}, url = {https://mathopt.de/PUBLICATIONS/Sager2011a.pdf}, doi = {http://dx.doi.org/10.1007/s0018601103554} } 

Sager, S.  On the Integration of Optimization Approaches for MixedInteger Nonlinear Optimal Control [BibTeX] 
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.