Category:AMPL

This category lists all problems for which AMPL code is provided. AMPL does not support differential equations, hence all models are a (finite-dimensional) discretization in one sense or another of a 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 this category are only one out of many possible ways to formulate the underlying MIOCP problems.

There are compilers available to process optimica models and automatically generate AMPL output, e.g., by applying a Radau collocation.

Recently, the TACO Toolkit for AMPL Control Optimization is a new effort aimed at modeling optimal control problems in AMPL. A small set of extensions allows to decouple the choice of a discretization scheme from the actual AMPL model. MUSCOD-II (Heidelberg) will be the first solver to support AMPL models using the TACO extensions.

References

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