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 <bibref>Binder2001</bibref>: 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., <bibref>Bock1984</bibref>,<bibref>Biegler1984</bibref>, to (re)formulate the control problem by outer convexification <bibref>Sager2009</bibref>, to use global optimization methods by under- and overestimators, e.g., <bibref>Esposito2000</bibref>,<bibref>Papamichail2004</bibref>,<bibref>Chachuat2006</bibref>, to optimize the time-points for a given switching structure, e.g., <bibref>Kaya2003</bibref>,<bibref>Gerdts2006</bibref>,<bibref>Sager2009</bibref>, to consider a static optimization problem instead of the transient behavior, e.g., <bibref>Grossmann2005</bibref>, to approximate nonlinearities by piecewise-linear functions, e.g., <bibref>Martin2006</bibref>, or by approximating the combinatorial decisions by continuous formulations, as in <bibref>Burgschweiger2009</bibref> for drinking water networks.

We do not want to discuss these methods here, but rather refer to <bibref>Sager2009</bibref>,<bibref>Sager2009b</bibref> 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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