# Category:Implementation

This category includes all subcategories that implement the MIOCP in a specific format of a solver. These may of course be very different, depending on the particular approach to algorithmically solve the control problem.

MIOCPs include features related to different mathematical disciplines. Hence, it is not surprising that very different approaches have been proposed to analyze and solve them. There are three generic approaches to solve model-based optimal control problems, compare [Binder2001]**Author: ** *T. Binder; L. Blank; H.G. Bock; R. Bulirsch; W. Dahmen; M. Diehl; T. Kronseder; W. Marquardt; J.P. Schl\"oder; O.v. Stryk***Booktitle: ** *Online Optimization of Large Scale Systems: State of the Art***Editor: ** *M. Gr\"otschel and S.O. Krumke and J. Rambau***Pages: ** *295--340***Publisher: ** *Springer***Title: ** *Introduction to Model Based Optimization of Chemical Processes on Moving Horizons***Url: ** *http://www.zib.de/dfg-echtzeit/Publikationen/Preprints/Preprint-01-15.html***Year: ** *2001*

: first, solution of the Hamilton-Jacobi-Bellman equation and in a discrete setting Dynamic Programming, second indirect methods, also known as the first optimize, then discretize approach, and third direct methods (first optimize, then discretize) and in particular all-at-once approaches that solve the simulation and the optimization task simultaneously. The combination with the additional combinatorial restrictions on control functions comes at different levels: for free in dynamic programming, as the control space is evaluated anyhow, by means of an enumeration in the inner optimization problem of the necessary conditions of optimality in Pontryagin's maximum principle, or by various methods from integer programming in the direct methods.

Even in the case of direct methods, there are multiple alternatives to proceed. Various approaches have been proposed to discretize the differential equations by means of shooting methods or collocation, e.g., [Bock1984]**Address: ** *Budapest***Author: ** *H.G. Bock; K.J. Plitt***Booktitle: ** *Proceedings of the 9th IFAC World Congress***Pages: ** *242--247***Publisher: ** *Pergamon Press***Title: ** *A Multiple Shooting algorithm for direct solution of optimal control problems***Url: ** *http://www.iwr.uni-heidelberg.de/groups/agbock/FILES/Bock1984.pdf***Year: ** *1984*

,[Biegler1984]**Author: ** *Biegler, L.T.***Journal: ** *Computers \& Chemical Engineering***Pages: ** *243--248***Title: ** *Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation***Volume: ** *8***Year: ** *1984*

, to (re)formulate the control problem by outer convexification [Sager2009]**Author: ** *Sager, S.; Reinelt, G.; Bock, H.G.***Journal: ** *Mathematical Programming***Number: ** *1***Pages: ** *109--149***Title: ** *Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2009.pdf***Volume: ** *118***Year: ** *2009*

, to use global optimization methods by under- and overestimators, e.g., [Esposito2000]**Author: ** *W.R. Esposito; C.A. Floudas***Journal: ** *Journal of Global Optimization***Number: ** *1***Pages: ** *97--126***Title: ** *Deterministic Global Optimization in Nonlinear Optimal Control Problems***Url: ** *http://titan.princeton.edu/research.htm***Volume: ** *17***Year: ** *2000*

,[Papamichail2004]**Author: ** *Papamichail, I.; Adjiman, C.S.***Journal: ** *Computers \& Chemical Engineering***Pages: ** *403--415***Title: ** *Global optimization of dynamic systems***Volume: ** *28***Year: ** *2004*

,[Chachuat2006]**Author: ** *B. Chachuat; A.B. Singer; P.I. Barton***Journal: ** *Industrial and Engineering Chemistry Research***Number: ** *25***Pages: ** *8573--8392***Title: ** *Global methods for dynamic optimization and mixed-integer dynamic optimization***Volume: ** *45***Year: ** *2006*

, to optimize the time-points for a given switching structure, e.g., [Kaya2003]**Author: ** *C.Y. Kaya; J.L. Noakes***Journal: ** *Journal of Optimization Theory and Applications***Pages: ** *69--92***Title: ** *A Computational Method for Time-Optimal Control***Volume: ** *117***Year: ** *2003*

,[Gerdts2006]**Author: ** *M. Gerdts***Journal: ** *Optimal Control Applications and Methods***Number: ** *3***Pages: ** *169--182***Title: ** *A variable time transformation method for mixed-integer optimal control problems***Volume: ** *27***Year: ** *2006*

,[Sager2009]**Author: ** *Sager, S.; Reinelt, G.; Bock, H.G.***Journal: ** *Mathematical Programming***Number: ** *1***Pages: ** *109--149***Title: ** *Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2009.pdf***Volume: ** *118***Year: ** *2009*

, to consider a static optimization problem instead of the transient behavior, e.g., [Grossmann2005]**Author: ** *I.E. Grossmann; P.A. Aguirre; M. Barttfeld***Journal: ** *Computers \& Chemical Engineering***Pages: ** *1203--1215***Title: ** *Optimal synthesis of complex distillation columns using rigorous models***Volume: ** *29***Year: ** *2005*

, to approximate nonlinearities by piecewise-linear functions, e.g., [Martin2006]**Author: ** *A. Martin; M. M\"oller; S. Moritz***Journal: ** *Mathematical Programming***Pages: ** *563--582***Title: ** *Mixed integer models for the stationary case of gas network optimization***Volume: ** *105***Year: ** *2006*

, or by approximating the combinatorial decisions by continuous formulations, as in [Burgschweiger2009]**Author: ** *J. Burgschweiger; B. Gn\"adig; M.C. Steinbach***Journal: ** *The Open Applied Mathematics Journal***Pages: ** *1--16***Title: ** *Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks***Volume: ** *3***Year: ** *2009*

for drinking water networks.

We do not want to discuss these methods here, but rather refer to [Sager2009]**Author: ** *Sager, S.; Reinelt, G.; Bock, H.G.***Journal: ** *Mathematical Programming***Number: ** *1***Pages: ** *109--149***Title: ** *Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2009.pdf***Volume: ** *118***Year: ** *2009*

,[Sager2009b]**Author: ** *S. Sager***Journal: ** *Journal of Process Control***Number: ** *8***Pages: ** *1238--1247***Title: ** *Reformulations and Algorithms for the Optimization of Switching Decisions in Nonlinear Optimal Control***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2009b.pdf***Volume: ** *19***Year: ** *2009*

for more comprehensive surveys. The main purpose of mentioning them is to point out that they all discretize the optimization problem in function space in a different manner, and hence result in different mathematical problems that are actually solved on a computer.

Powerful commercial MILP solvers and advances in MINLP solvers make the usage of general purpose MILP/MINLP solvers more and more attractive. *Please be aware however that the MINLP formulations we provide in some of the categories are only one out of many possible ways to formulate the underlying MIOCP problems.*

## References

[Biegler1984] | Biegler, L.T. (1984): Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Computers \& Chemical Engineering, 8, 243--248 | |

[Binder2001] | T. Binder; L. Blank; H.G. Bock; R. Bulirsch; W. Dahmen; M. Diehl; T. Kronseder; W. Marquardt; J.P. Schl\"oder; O.v. Stryk (2001): Introduction to Model Based Optimization of Chemical Processes on Moving Horizons. Online Optimization of Large Scale Systems: State of the Art | |

[Bock1984] | H.G. Bock; K.J. Plitt (1984): A Multiple Shooting algorithm for direct solution of optimal control problems. Pergamon Press, Proceedings of the 9th IFAC World Congress | |

[Burgschweiger2009] | J. Burgschweiger; B. Gn\"adig; M.C. Steinbach (2009): Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks. The Open Applied Mathematics Journal, 3, 1--16 | |

[Chachuat2006] | B. Chachuat; A.B. Singer; P.I. Barton (2006): Global methods for dynamic optimization and mixed-integer dynamic optimization. Industrial and Engineering Chemistry Research, 45, 8573--8392 | |

[Esposito2000] | W.R. Esposito; C.A. Floudas (2000): Deterministic Global Optimization in Nonlinear Optimal Control Problems. Journal of Global Optimization, 17, 97--126 | |

[Gerdts2006] | M. Gerdts (2006): A variable time transformation method for mixed-integer optimal control problems. Optimal Control Applications and Methods, 27, 169--182 | |

[Grossmann2005] | I.E. Grossmann; P.A. Aguirre; M. Barttfeld (2005): Optimal synthesis of complex distillation columns using rigorous models. Computers \& Chemical Engineering, 29, 1203--1215 | |

[Kaya2003] | C.Y. Kaya; J.L. Noakes (2003): A Computational Method for Time-Optimal Control. Journal of Optimization Theory and Applications, 117, 69--92 | |

[Martin2006] | A. Martin; M. M\"oller; S. Moritz (2006): Mixed integer models for the stationary case of gas network optimization. Mathematical Programming, 105, 563--582 | |

[Papamichail2004] | Papamichail, I.; Adjiman, C.S. (2004): Global optimization of dynamic systems. Computers \& Chemical Engineering, 28, 403--415 | |

[Sager2009] | Sager, S.; Reinelt, G.; Bock, H.G. (2009): Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems. Mathematical Programming, 118, 109--149 | |

[Sager2009b] | S. Sager (2009): Reformulations and Algorithms for the Optimization of Switching Decisions in Nonlinear Optimal Control. Journal of Process Control, 19, 1238--1247 |