Category:Implementation

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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
Link to Google Scholar
: 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
Link to Google Scholar
,[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
Link to Google Scholar
, 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
Link to Google Scholar
, 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
Link to Google Scholar
,[Papamichail2004]Author: Papamichail, I.; Adjiman, C.S.
Journal: Computers \& Chemical Engineering
Pages: 403--415
Title: Global optimization of dynamic systems
Volume: 28
Year: 2004
Link to Google Scholar
,[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
Link to Google Scholar
, 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
Link to Google Scholar
,[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
Link to Google Scholar
,[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
Link to Google Scholar
, 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
Link to Google Scholar
, 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
Link to Google Scholar
, 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
Link to Google Scholar
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
Link to Google Scholar
,[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
Link to Google Scholar
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--248Link to Google Scholar
[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 ArtLink to Google Scholar
[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 CongressLink to Google Scholar
[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--16Link to Google Scholar
[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--8392Link to Google Scholar
[Esposito2000]W.R. Esposito; C.A. Floudas (2000): Deterministic Global Optimization in Nonlinear Optimal Control Problems. Journal of Global Optimization, 17, 97--126Link to Google Scholar
[Gerdts2006]M. Gerdts (2006): A variable time transformation method for mixed-integer optimal control problems. Optimal Control Applications and Methods, 27, 169--182Link to Google Scholar
[Grossmann2005]I.E. Grossmann; P.A. Aguirre; M. Barttfeld (2005): Optimal synthesis of complex distillation columns using rigorous models. Computers \& Chemical Engineering, 29, 1203--1215Link to Google Scholar
[Kaya2003]C.Y. Kaya; J.L. Noakes (2003): A Computational Method for Time-Optimal Control. Journal of Optimization Theory and Applications, 117, 69--92Link to Google Scholar
[Martin2006]A. Martin; M. M\"oller; S. Moritz (2006): Mixed integer models for the stationary case of gas network optimization. Mathematical Programming, 105, 563--582Link to Google Scholar
[Papamichail2004]Papamichail, I.; Adjiman, C.S. (2004): Global optimization of dynamic systems. Computers \& Chemical Engineering, 28, 403--415Link to Google Scholar
[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--149Link to Google Scholar
[Sager2009b]S. Sager (2009): Reformulations and Algorithms for the Optimization of Switching Decisions in Nonlinear Optimal Control. Journal of Process Control, 19, 1238--1247Link to Google Scholar

Subcategories

This category has the following 11 subcategories, out of 11 total.