Global Optimization of Mixed-Integer ODE Constrained Network Problems Using the Example of Stationary Gas Transport

Habeck, Oliver; Pfetsch, Marc E.; Ulbrich, Stefan

doi:10.1137/17m1152668

Cited by 3 publications

(5 citation statements)

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“…The possible modes are: compressor on/bypass closed, compressor off/bypass open and compressor off/bypass closed. Control valves, resistors, and short pipes are modeled as described in [24] and [33]. As objective function we chose the maximization of the sum of pressures. Several other objective choices exist; see, for example, [18, 19]. Moreover, the computational results for other objective functions in [18] show a similar impact of our combinatorial models as seen here.…”

Section: Numerical Resultssupporting

confidence: 73%

“…• As objective function we chose the maximization of the sum of pressures. Several other objective choices exist; see, for example, [18,19]. Moreover, the computational results for other objective functions in [18] show a similar impact of our combinatorial models as seen here.…”

Section: Model Implementationsupporting

confidence: 73%

“…Remark The fact that potential‐based flows are acyclic does not only hold for this algebraic model for stationary gas flows. It also applies to the stationary model based on ordinary differential equations used in [19]. Moreover, we also used variant CB and a preliminary version of FLC to derive the computational results there.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The ideas of this article were used for the computations of [19], which was prepared independently of [3][4][5]21], and we provide the background in this article. Apart from the mentioned literature, we are not aware of any other works concerning combinatorial models for acyclic flows.…”

Section: Literature Reviewmentioning

confidence: 99%

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Combinatorial acyclicity models for potential‐based flows

Habeck

Pfetsch

2021

Networks

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Potential-based flows constitute a basic model to represent physical behavior in networks. Under natural assumptions, the flow in such networks must be acyclic. The goal of this article is to exploit this property for the solution of corresponding optimization problems. To this end, we introduce several combinatorial models for acyclic flows, based on binary variables for flow directions. We compare these models and introduce a particular model that tries to capture acyclicity together with the supply/demand behavior. We analyze properties of this model, including variable fixing rules. Our computational results show that the usage of the corresponding constraints speeds up solution times by about a factor of 3 on average and a speed-up of a factor of almost 5 for the time to prove optimality.

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Section: Numerical Resultssupporting

confidence: 73%

Section: Model Implementationsupporting

confidence: 73%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

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Combinatorial acyclicity models for potential‐based flows

Habeck

Pfetsch

2021

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“…A selection of notable publications include articles from Ismayilova [63], Kunisch and Meinlschmidt [69], Kunisch, Trautmann, and Vexler [70], Pfaff and Ulbrich [98], Schmitt and Ulbrich [106], and Zuazua [130]. Related research in the narrower context of hyperbolic equations describing the gas flow through pipes, in particular Euler equations, is due to Gugat, Dick, and Leugering [52] and Gugat and Ulbrich [53,54]; see also [56,57]. Questions of controllability, observability, and stabilization of hyperbolic equations are discussed for instance by Coron [32], Coron and Bastin [33], Li and Zhou [78], and Triggiani [120].…”

Section: A First Small Step Towards Optimal Control Introduction Over...mentioning

confidence: 99%

A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

Groh

2024

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Diese Dissertation befasst sich mit der Analyse eines gekoppelten Systems der Form (Eu) ′ (t) + φ 1 (t, u(t), v(t)) = q 1 (t), v ′′ (t) − ∆v(t) + φ 2 (t, u(t), v(t)) = q 2 (t).(1.1)Es besteht aus einer semilinearen abstrakten differential-algebraischen Gleichung (DAE) und einer semilinearen hyperbolischen partiellen Differentialgleichung zweiter Ordnung. Beide Gleichungen sind durch die nichtlinearen Kopplungsoperatoren φ 1 und φ 2 miteinander gekoppelt.Gekoppelte Systeme dieser Art können als spezielle abstrakte DAEs und als Verallgemeinerungen von partiell differential-algebraischen Gleichungen aufgefasst werden. Sie spielen in vielen Anwendungen wie der Modellierung von multiphysikalischen Systemen, bei der Simulation von Schaltkreisen, oder der Optimalsteuerung von Gasnetzwerken eine Rolle.In der vorliegenden Arbeit diskutieren wir zunächst nur die abstrakte DAE und führen sogenannte Matrix-induzierte lineare Operatoren ein. Wir übertragen unter Nutzung dieser Operatoren einen Entkopplungsansatz für DAEs auf den unendlichdimensionalen Fall der vorliegenden abstrakten DAE. In Kombination mit einer neuartigen Index-1-Charakterisierung für semilineare abstrakte DAEs gelingt es uns, die inhärente gewöhnliche Differentialgleichung und die algebraischen Nebenbedingungen aus der abstrakten DAE zu extrahieren und Existenz und Eindeutigkeit von Lösungen zu zeigen. Anschließend verbinden wir die entwickelten Ansätze zur Behandlung von derartigen abstrakten DAEs mit bereits bekannten Ansätzen für die Analyse von hyperbolischen Gleichungen zweiter Ordnung, und schaffen einen einheitlichen Rahmen, in dem wir das gekoppelte System (1.1) diskutieren können. Mithilfe eines Fixpunktansatzes zeigen wir Existenz und Eindeutigkeit von lokalen und globalen Lösungen zu diesem gekoppelten System. Zu guter Letzt formulieren wir ein Optimalsteuerungsproblem, in dem das System (1.1) als Nebenbedingung auftritt. Wir zeigen die Existenz einer optimalen globalen Steuerung und einer globalen Minimalstelle. vii I wish to say thanks to Prof. Dr. Barbara Zwicknagl, my mentor within the Berlin Mathematical School. Her professional advice and her kindness helped more than once to push through difficult moments. I also wish to thank Prof. Dr. Axel Kröner for his support and motivating words. Gratitude is equally owed to Prof. Dr. Etienne Emmrich who acted as reviewer for this thesis. I would also like to thank some of the most important staff members for technology, service, and administration at the Faculty of Mathematics and Natural Sciences as well as the Department of Mathematics of the Humboldt-Universität zu Berlin. I would like to thank

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Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

Severitt

Manns

2023

INFORMS Journal on Computing

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We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an [Formula: see text] algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the [Formula: see text] algorithm and a preprocessing procedure, which can be employed to accelerate the [Formula: see text] algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated [Formula: see text] algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated [Formula: see text] algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1294 .

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Global Optimization of Mixed-Integer ODE Constrained Network Problems Using the Example of Stationary Gas Transport

Cited by 3 publications

References 30 publications

Combinatorial acyclicity models for potential‐based flows

Combinatorial acyclicity models for potential‐based flows

A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

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