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Standard model predictive control for real-time operation of industrial production processes may be inefficient in the presence of substantial uncertainties. To avoid overly conservative disturbance corrections while ensuring safe operation, random influences should be taken into account explicitly. We propose a multistage stochastic programming approach within the model predictive control framework and apply it to a distillation process with a feed tank buffering external sources. A preliminary comparison to a probabilistic constraints approach is given and first computational results for the distillation process are presented.
Das aus den Medien bekannte umstrittene Ziegenproblem (auch Drei-Türen-Problem genannt) wird vollständig analysiert und gelöst. In der Streitfrage spielen sprachliche Mehrdeutigkeiten der Problemformulierung eine wesentliche Rolle; zudem werden Zufallsereignisse mit willkürlicher Information über deren Ergebnisse verwechselt. Tatsächlich erweisen sich beide strittigen Lösungen als teilweise richtige Bestandteile der Gesamtlösung. Die Argumentation wird in allgemeinverständlicher Sprache geführt und anschliessend durch eine formale mathematische Betrachtung ergänzt.
Scenario tree models of stochastic programs arise naturally under standard nonanticipativity assumptions. We demonstrate how tree-sparse programs cover the general case, with \emph{arbitrary} information constraints. Detailed examples and intuitive interpretations illuminate the basic thoughts behind the abstract but elementary construction.
Interior point methods for multistage stochastic programs involve KKT systems with a characteristic global block structure induced by dynamic equations on the scenario tree. We generalize the recursive solution algorithm proposed in an earlier paper so that its linear complexity extends to a refined tree-sparse KKT structure. Then we analyze how the block operations can be specialized to take advantage of problem-specific sparse substructures. Savings of memory and operations for a financial engineering application are discussed in detail.
Operative planning in gas networks with prescribed binary decisions yields large scale nonlinear programs defined on graphs. We study the structure of the KKT systems arising in interior methods and present a customized direct solution algorithm. Computational results indicate that the algorithm is suitable for optimization in small and medium-sized gas networks.
Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We investigate in detail the interplay between objective and constraints in a number of single-period variants, including semi-variance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multi-period models based on scenario trees. A key property is the possibility to remove surplus money in future decisions, yielding approximate downside risk minimization.
The recently imposed new gas market liberalization rules in Germany lead to a change of business of gas network operators.
While previously network operator and gas vendor where united, they were forced to split up into independent companies.
The network has to be open to any other gas trader at the same conditions, and free network capacities have to be identified and publicly offered in a non-discriminatory way.
We show that these new paradigms lead to new and challenging mathematical optimization problems.
In order to solve them and to provide meaningful results for practice, all aspects of the underlying problems, such as combinatorics, stochasticity, uncertainty, and nonlinearity, have to be addressed.
With such special-tailored solvers, free network capacities and topological network extensions can, for instance, be determined.
The recently imposed new gas market liberalization rules in Germany lead to a change of business of gas network operators. While previously network operator and gas vendor were united, they were forced to split up into independent companies. The network has to be open to any other gas trader at the same conditions, and free network capacities have to be identified and publicly offered in a non-discriminatory way. We discuss how these changing paradigms lead to new and challenging mathematical optimization problems. This includes the validation of nominations, that asks for the decision if the network’s capacity is sufficient to transport a specific amount of flow, the verification of booked capacities and the detection of available freely allocable capacities, and the topological extension of the network with new pipelines or compressors in order to increase its capacity. In order to solve each of these problems and to provide meaningful results for the practice, a mixture of different mathematical aspects have to be addressed, such as combinatorics, stochasticity, uncertainty, and nonlinearity. Currently, no numerical solver is available that can deal with such blended problems out-of-the-box. The main goal of our research is to develop such a solver, that moreover is able to solve instances of realistic size. In this article, we describe the main ingredients of our prototypical software implementations.
\def\KukaRob {{\sf KUKA IR\,761}} {\small Industrial robots have greatly enhanced the performance of automated manufacturing processes during the last decades. International competition, however, creates an increasing demand to further improve both the accuracy of off-line programming and the resulting cycle times on production lines. To meet these objectives, validated dynamic robot models are required. We describe in detail the development of a generic dynamic model, specialize it to an actual industrial robot \KukaRob, and discuss the problem of dynamic calibration. Efficient and robust trajectory optimization algorithms are then presented which, when integrated into a CAD system, are suitable for routine application in an industrial environment. Our computational results for the \KukaRob\ robot performing a real life transport maneuver show that considerable gains in productivity can be achieved by minimizing the cycle time.}
The operative planning problem in natural gas distribution networks is addressed. An optimization model focusing on the governing PDE and other nonlinear aspects is presented together with a suitable discretization for transient optimization in large networks by SQP methods. Computational results for a range of related dynamic test problems demonstrate the viability of the approach.