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Mon, 16 Oct 2017 11:40:55 +0200Mon, 16 Oct 2017 11:40:55 +0200Applications of Differential-Algebraic Equations: Examples and Benchmarks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/184
Christoph Huck; Caren Tischendorf; Tom Streubel; Sarah Grundel; Christian Himpebookparthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/184Mon, 16 Oct 2017 11:40:55 +0200Transient Modeling and Simulation of Gas Pipe Networks with Characteristic Diagram Models for Compressors
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/185
One challenge for the simulation and optimization of real gas pipe networks is the treatment of compressors. Their behavior is usually described by characteristic diagrams reflecting the connection of the volumetric flow and the enthalpy change or shaft torque. Such models are commonly used for an optimal control of compressors and compressor stations [4, 7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1–3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system.Christoph Huck; Caren Tischendorfpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/185Mon, 16 Oct 2017 11:40:55 +0200Perturbation analysis of hyperbolic PDAEs describing flow networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/183
Christoph Huck; Caren Tischendorfpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/183Mon, 16 Oct 2017 11:40:54 +0200Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/179
For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H2-Lyapunov function and show that the boundary feedback constant can be chosen such that the H2-Lyapunov function and hence also the H2-norm of the difference between the non-stationary and the stationary state decays exponentially with time. Martin Gugat; Günter Leugering; Ke Wangarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/179Thu, 12 Oct 2017 17:20:07 +0200Domain Decomposition of an Optimal Control Problem for Semi-Linear Elliptic Equations on Metric Graphs with Application to Gas Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/175
We consider optimal control problems for the flow of gas in a pipe network.
The equations of motions are taken to be represented by a semi-linear model
derived from the fully nonlinear isothermal Euler gas equations. We formulate
an optimal control problem on a given network and introduce a time discretization
thereof. We then study the well-posedness of the corresponding
time-discrete optimal control problem. In order to further reduce the complexity,
we consider an instantaneous control strategy. The main part of the
paper is concerned with a non-overlapping domain decomposition of the
semi-linear elliptic optimal control problem on the graph into local problems
on a small part of the network, ultimately on a single edge.Günter Leugeringperiodicalhttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/175Thu, 12 Oct 2017 17:20:06 +0200The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/177
Martin Gugat; Stefan Ulbricharticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/177Thu, 12 Oct 2017 17:20:06 +0200On the relaxation approximation for 2 × 2 hyperbolic balance laws
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/172
The relaxation approximation for systems of conservation laws has been
studied intensively for example by [17, 5, 19, 24]. In this paper the corresponding
relaxation approximation for 2x2 systems of balance laws is studied. Our driving
example is gas flow in pipelines described by the isothermal Euler equations. We
are interested in the limiting behavior as the relaxation parameter tends to zero. We
give conditions where the relaxation converges to the states of the original system
and counterexamples for cases where the steady states depend on the space variable.Martin Gugatconferenceobjecthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/172Mon, 02 Oct 2017 18:32:42 +0200Uniqueness and Multiplicity of Market Equilibria on DC Power Flow Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/176
We consider uniqueness and multiplicity of market equilibria in a short-run setup where traded quantities of electricity are transported through a capacitated network in which power flows have to satisfy the classical lossless DC approximation. The firms face fluctuating demand and decide on their production, which is constrained by given capacities. Today, uniqueness of such market outcomes are especially important in more complicated multilevel models for measuring market (in)efficiency. Thus, our findings are important prerequisites for such studies. We show that market equilibria are unique on tree networks under mild assumptions and we also present a priori conditions under which equilibria are unique on cycle networks. On general networks, uniqueness fails to hold and we present simple examples for which multiple equilibria exist. However, we prove a posteriori criteria for the uniqueness of a given solution and characterize situations in which multiple solutions exist.Vanessa Krebs; Lars Schewe; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/176Mon, 02 Oct 2017 18:32:42 +0200Optimality Conditions for Switching Operator Differential Equations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/171
We consider optimal switching of hybrid abstract evolution equations. The framework includes switching semilinear partial differential equations of parabolic or hyperbolic type, discontinuous state resets, switching costs and allows switching of the principle parts of the equations. We present adjoint-based formulae for the gradient of the cost functional with respect to position and number of switching time points that lead to first order necessary conditions. Moreover, we discuss an alternate-direction approach for implementing descent methods. As an application we consider optimal open/close-switching of valves and on/off-switching control of compressors in a gas network modelled by a graph with simplified euler equations on edges and suitable coupling conditions at nodes.
Fabian Rüffler; Falk Hanteconferenceobjecthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/171Thu, 14 Sep 2017 14:34:04 +0200Boundary feedback stabilization of the isothermal Euler-equations with uncertain boundary data
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/170
In a gas transport system, the
customer behavior is uncertain.
Motivated by this situation, we consider
a boundary stabilization problem
for the flow through a gas pipeline,
where the outflow at one
end of the pipe
%that is governed by the customer's behavior
is uncertain.
The control action is located
at the
other end of the pipe.
The feedback law is
a classical
Neumann velocity feedback with a feedback parameter $k>0$.
We show that
as long as the
$H^1$-norm of the function that describes the noise in the customer's
behavior decays exponentially with
a rate that is sufficiently large,
the velocity of the gas can be stabilized exponentially
fast in the sense that a suitably chosen Lyapunov function decays exponentially.
For the exponential stability it is sufficient
that the feedback parameter $k$ is sufficiently large
and the stationary state to which the system is stabilized is
sufficiently small.
The stability result is local, that is it holds for initial
states that are sufficiently close to the stationary state.
This result is an example for the exponential boundary feedback stabilization of
a quasilinear hyperbolic system with uncertain boundary data.
The analysis is based upon the choice of a suitably Lyapunov function.
The decay of this Lyapunov function implies that
also the $L^2$-norm of the
difference of the system state and the stationary state decays exponentially.Martin Gugat; Rüdiger Schultzpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/170Thu, 14 Sep 2017 11:14:12 +0200Complementarity-Based Nonlinear Programming Techniques for Optimal Mixing in Gas Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/169
We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems and show that mixed-integer reformulations of pooling-type mixing models already render small-size instances intractable. Therefore we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting complementarity-based reformulation of the mixing equations. We test this approach numerically by comparing such a reformulation with a more compact complementarity-based one that does not feature such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport. Our numerical results show that both complementarity-based models outperform the mixed-integer reformulation significantly and that the complementarity-based model with beneficial regularity properties can be solved more reliable.Falk Hante; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/169Tue, 12 Sep 2017 10:56:10 +0200GasLib – A Library of Gas Network Instances
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/168
The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML format. In this paper, we explain this format and present the instances that are available in the library.Jesco Humpola; Imke Joormann; Nikolaos Kanelakis; Djamal Oucherif; Marc E. Pfetsch; Lars Schewe; Martin Schmidt; Robert Schwarz; Mathias Sirventpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/168Tue, 12 Sep 2017 10:56:09 +0200On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/167
The directional differentiability of the solution mapping for a class of variational inequali-
ties of the second kind inspired by applications in fluid mechanics and moving free boundary
problems is investigated. The result is particularly relevant for the model predictive control
or optimal control of such variational inequalities in that it can be used to derive stationarity
conditions and efficient numerical methods.Michael Hintermüller; Thomas Surowiecpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/167Mon, 11 Sep 2017 13:53:00 +0200A Bundle-Free Implicit Programming Approach for a Class of MPECs in Function Space
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/166
Using a standard first-order optimality condition for nonsmooth optimization prob-
lems, a general framework for a descent method is developed. This setting is applied to
a class of mathematical programs with equilibrium constraints in function space from
which a new algorithm is derived. Global convergence of the algorithm is demonstrated
in function space and the results are then illustrated by numerical experiments.Michael Hintermüller; Thomas Surowiecarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/166Mon, 11 Sep 2017 13:52:59 +0200Optimal sensor placement: A robust approach
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/164
We address the problem of optimally placing sensor networks for convection-diffusion
processes where the convective part is perturbed. The problem is formulated as an optimal control
problem where the integral Riccati equation is a constraint and the design variables are sensor
locations. The objective functional involves a term associated to the trace of the solution to the
Riccati equation and a term given by a constrained optimization problem for the directional derivative
of the previous quantity over a set of admissible perturbations. The paper addresses the existence
of the derivative with respect to the convective part of the solution to the Riccati equation, the
well-posedness of the optimization problem and finalizes with a range of numerical tests.Michael Hintermüller; Carlos Rautenberg; Masoumeh Mohammadi; Martin Kanitsarpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/164Mon, 11 Sep 2017 13:52:58 +0200On the Complexity of Instationary Gas Flows
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/162
We study a simplistic model of instationary gas flows consisting of a sequence of k stationary gas flows. We present efficiently solvable cases and NP-hardness results, establishing complexity gaps between stationary and instationary gas flows (already for k=2) as well as between instationary gas s-t-flows and instationary gas b-flows.Martin Groß; Marc E. Pfetsch; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/162Tue, 29 Aug 2017 18:24:19 +0200Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/161
This paper is concerned with the distributed optimal control of a time-discrete Cahn–
Hilliard/Navier–Stokes system with variable densities. It focuses on the double-obstacle potential
which yields an optimal control problem for a family of coupled systems in each time instant of a
variational inequality of fourth order and the Navier–Stokes equation. By proposing a suitable time-
discretization, energy estimates are proved and the existence of solutions to the primal system and of
optimal controls is established for the original problem as well as for a family of regularized problems.
The latter correspond to Moreau–Yosida type approximations of the double-obstacle potential. The
consistency of these approximations is shown and first order optimality conditions for the regularized
problems are derived. Through a limit process with respect to the regularization parameter, a
stationarity system for the original problem is established. The resulting system corresponds to a
function space version of C-stationarity which is a special notion of stationarity for MPECs.Tobias Keil; Michael Hintermüller; Donat Wegnerarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/161Tue, 29 Aug 2017 17:57:43 +0200Analytical aspects of spatially adapted total variation regularization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/160
In this paper we study the structure of solutions of the one dimensional weighted total
variation regularisation problem, motivated by its application in signal recovery tasks. We study
in depth the relationship between the weight function and the creation of new discontinuities in
the solution. A partial semigroup property relating the weight function and the solution is shown
and analytic solutions for simply data functions are computed. We prove that the weighted total
variation minimisation problem is well-posed even in the case of vanishing weight function, despite
the lack of coercivity. This is based on the fact that the total variation of the solution is bounded
by the total variation of the data, a result that it also shown here. Finally the relationship to the
corresponding weighted fidelity problem is explored, showing that the two problems can produce
completely different solutions even for very simple data functions.Konstantinos Papafitsoros; Michael Hintermüller; Carlos Rautenbergpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/160Tue, 29 Aug 2017 17:57:42 +0200Application of Chance Constrained Optimization to Gas Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/158
We consider optimization problems with a joint probabilistic constraint under
normally distributed uncertain parameters. The parametric constraints
are replaced by one constraint stating that the probability of being feasible
shall exceed or be equal to a prescribed threshold. In order to apply the
concept to gas network optimization under uncertain boundary flows, which
corresponds to the demand of customers, we derive an analytic gradient formula.
The integral corresponding to the probability can be parameterized by
spherical radial decomposition. For this parameterization gradient formulas
are known under convexity assumptions of the parametric constraints in the
parameter. For the application in gas networks that we have in mind, the
convexity assumption of the parametric constraints is not satisfied. Therefore,
we weaken it to convexity of the region of feasible parameters for a fixed
optimization variable. We proceed to show that the assumptions needed for
the gradient formula are met in the gas network optimization problem on
a tree. For the numerical implementation we propose a multilevel sampling
algorithm that uses a coarse approximation of the chance constraint to generate
a warm start for the expensive approximation with fine sampling. The
numerical results illustrate that this approach significantly reduces the computation
time.David Wintergerstpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/158Tue, 29 Aug 2017 17:57:41 +0200General Bounds for Incremental Maximization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/156
We propose a theoretical framework to capture incremental s
olutions to cardinality con-
strained maximization problems. The defining characterist
ic of our framework is that the
cardinality/support of the solution is bounded by a value
k
∈
N
that grows over time, and
we allow the solution to be extended one element at a time. We i
nvestigate the best-possible
competitive ratio of such an incremental solution, i.e., th
e worst ratio over all
k
between the
incremental solution after
k
steps and an optimum solution of cardinality
k
. We define a
large class of problems that contains many important cardin
ality constrained maximization
problems like maximum matching, knapsack, and packing/cov
ering problems. We provide a
general 2
.
618-competitive incremental algorithm for this class of pr
oblems, and show that no
algorithm can have competitive ratio below 2
.
18 in general.
In the second part of the paper, we focus on the inherently inc
remental greedy algorithm
that increases the objective value as much as possible in eac
h step. This algorithm is known
to be 1
.
58-competitive for submodular objective functions, but it
has unbounded competitive
ratio for the class of incremental problems mentioned above
. We define a relaxed submod-
ularity condition for the objective function, capturing pr
oblems like maximum (weighted)
(
b
-)matching and a variant of the maximum flow problem. We show t
hat the greedy algo-
rithm has competitive ratio (exactly) 2
.
313 for the class of problems that satisfy this relaxed
submodularity condition.
Note that our upper bounds on the competitive ratios transla
te to approximation ratios
for the underlying cardinality constrained problems.Aaron Bernstein; Yann Disser; Martin Großarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/156Mon, 07 Aug 2017 11:55:45 +0200A Local-Search Algorithm for Steiner Forest
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/155
In the
Steiner Forest
problem, we are given a graph and a collection of source-sink
pairs, and the
goal is to find a subgraph of minimum total length such that all
pairs are connected. The problem is
APX-Hard and can be
2
-approximated by, e.g., the elegant primal-dual algorithm
of Agrawal, Klein, and
Ravi from 1995.
We give a local-search-based constant-factor approximati
on for the problem. Local search brings in
new techniques to an area that has for long not seen any improv
ements and might be a step towards
a combinatorial algorithm for the more general survivable n
etwork design problem. Moreover, local
search was an essential tool to tackle the dynamic MST/Stein
er Tree problem, whereas dynamic Steiner
Forest is still wide open.
It is easy to see that any constant factor local search algori
thm requires steps that add/drop many edges
together. We propose natural local moves which, at each step
, either (a) add a shortest path in the current
graph and then drop a bunch of inessential edges, or (b) add a s
et of edges to the current solution. This
second type of moves is motivated by the potential function w
e use to measure progress, combining the
cost of the solution with a penalty for each connected compon
ent. Our carefully-chosen local moves and
potential function work in tandem to eliminate bad local min
ima that arise when using more traditional
local moves.
Our analysis first considers the case where the local optimum
is a single tree, and shows optimality w.r.t.
moves that add a single edge (and drop a set of edges) is enough
to bound the locality gap. For the
general case, we show how to “project” the optimal solution o
nto the different trees of the local optimum
without incurring too much cost (and this argument uses opti
mality w.r.t. both kinds of moves), followed
by a tree-by-tree argument. We hope both the potential funct
ion, and our analysis techniques will be
useful to develop and analyze local-search algorithms in ot
her contexts.Martin Groß; Anupam Gupta; Amit Kumar; Jannik Matuschke; Daniel R. Schmidt; Melanie Schmidt; José Verschaepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/155Mon, 07 Aug 2017 11:55:44 +0200Scheduling Maintenance Jobs in Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/154
We investigate the problem of scheduling the maintenance
of edges in a network, motivated by the goal of minimizing outages in
transportation or telecommunication networks. We focus on maintaining
connectivity between two nodes over time; for the special case of path
networks, this is related to the problem of minimizing the busy time of
machines.
We show that the problem can be solved in polynomial time in arbitrary
networks if preemption is allowed. If preemption is restricted to integral
time points, the problem is NP-hard and in the non-preemptive case
we give strong non-approximability results. Furthermore, we give tight
bounds on the power of preemption, that is, the maximum ratio of the
values of non-preemptive and preemptive optimal solutions.
Interestingly, the preemptive and the non-preemptive problem can be
solved efficiently on paths, whereas we show that mixing both leads to a
weakly NP-hard problem that allows for a simple 2-approximation.Fidaa Abed; Lin Chen; Yann Disser; Martin Groß; Nicole Megow; Julie Meißner; Alexander T. Richter; Roman Rischkearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/154Mon, 07 Aug 2017 11:55:43 +0200Algorithmic Results for Potential-Based Flows: Easy and Hard Cases
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/153
Potential-based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize the flow between two designated nodes. We show that these problems can be solved for the single source and sink case by reducing the network to a single arc. However, if we additionally consider switches that allow to force the flow to 0 and decouple the potentials, these problems are NP-hard. Nevertheless, for particular series-parallel networks, one can use algorithms for the subset sum problem. Moreover, applying network presolving based on generalized series-parallel structures allows to significantly reduce the size of realistic energy networks.Martin Groß; Pfetsch Marc E.; Lars Schewe; Martin Schmidt; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/153Fri, 04 Aug 2017 10:59:54 +0200Total variation diminishing schemes in optimal control of scalar conservation laws
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/152
In this paper, optimal control problems subject to a nonlinear scalar conservation law are
studied. Such optimal control problems are challenging both at the continuous and at the discrete
level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore
discretization of the underlying optimal control problem should be designed with care. Here the
discretize-then-optimize approach is employed where first the full discretization of the objective
function as well as the underlying PDE is considered. Then, the derivative of the reduced objective
is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta
(TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods,
also called strong stability preserving (SSP), are originally designed to preserve total variation of
the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to
ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is
too strong. Also approximation properties that the discrete adjoint inherits from the discretization
of the state equation are studied. Moreover order conditions are derived. In addition, optimal
choices with respect to CFL constant are discussed and numerical experiments are presented.Soheil Hajian; Michael Hintermüller; Stefan Ulbrichpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/152Fri, 28 Jul 2017 16:35:44 +0200Solving Mixed-Integer Nonlinear Programs using Adaptively Refined Mixed-Integer Linear Programs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/151
We propose a method for solving mixed-integer nonlinear programs (MINLPs) to global optimality by discretization of occuring nonlinearities. The main idea is based on using piecewise linear functions to construct mixed-integer linear program (MIP) relaxations of the underlying MINLP. In order to find a global optimum of the given MINLP we develope an iterative algorithm which solves MIP relaxations that are adaptively refined. We are able to give convergence results for a wide range of MINLPs requiring only continuous nonlinearities with bounded domains and an oracle computing maxima of the nonlinearities on their domain. Moreover, the practicalness of our approach is shown numerically by an application from the field of gas network optimization.Robert Burlacu; Björn Geißler; Lars Schewepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/151Fri, 21 Jul 2017 17:24:37 +0200