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A Local-Search Algorithm for Steiner Forest (2017)
Groß, Martin ; Gupta, Anupam ; Kumar, Amit ; Matuschke, Jannik ; Schmidt, Daniel R. ; Schmidt, Melanie ; Verschae, José
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.
Algorithmic Results for Potential-Based Flows: Easy and Hard Cases (2017)
Groß, Martin ; Marc E., Pfetsch ; Schewe, Lars ; Schmidt, Martin ; Skutella, Martin
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.
Transmission and generation investment in electricity markets: The effects of market splitting and network fee regimes (2016)
Grimm, Veronika ; Martin, Alexander ; Schmidt, Martin ; Weibelzahl, Martin ; Zöttl, Gregor
We propose an equilibrium model that allows to analyze the long-run impact of the electricity market design on transmission line expansion by the regulator and investment in generation capacity by private firms in liberalized electricity markets. The model incorporates investment decisions of the transmission system operator and private firms in expectation of an energy-only market and cost-based redispatch. In different specifications we consider the cases of one vs. multiple price zones (market splitting) and analyze different approaches to recover network cost—in particular lump sum, generation capacity based, and energy based fees. In order to compare the outcomes of our multilevel market model with a first best benchmark, we also solve the corresponding integrated planner problem. Using two test networks we illustrate that energy-only markets can lead to suboptimal locational decisions for generation capacity and thus imply excessive network expansion. Market splitting heals these problems only partially. These results are valid for all considered types of network tariffs, although investment slightly differs across those regimes.
Towards Simulation Based Mixed-Integer Optimization with Differential Equations (2018)
Gugat, Martin ; Leugering, Günter ; Martin, Alexander ; Schmidt, Martin ; Sirvent, Mathias ; Wintergerst, David
We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with an approximate feasible global optimal solution of the mixed integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks.
MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems (2017)
Gugat, Martin ; Leugering, Günter ; Martin, Alexander ; Schmidt, Martin ; Sirvent, Mathias ; Wintergerst, David
We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.
Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems with Mixed Two-Point Boundary Conditions (2021)
Krug, Richard ; Leugering, Günter ; Martin, Alexander ; Schmidt, Martin ; Weninger, Dieter
In this article, we continue our work (Krug et al., 2021) on time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations in that we now consider mixed two-point boundary value problems and provide an in-depth well-posedness analysis. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.
Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems (2020)
Krug, Richard ; Leugering, Günter ; Martin, Alexander ; Schmidt, Martin ; Weninger, Dieter
In this article, we extend the time-domain decomposition method described by Lagnese and Leugering (2003) to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process.
Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications (2016)
Hante, Falk ; Leugering, Günter ; Martin, Alexander ; Schewe, Lars ; Schmidt, Martin
We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St.~Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique.
Mathematische Modellierung, Simulation und Optimierung von Gastransportnetzwerken (2015)
Domschke, Pia ; Groß, Martin ; Hiller, Benjamin ; Hante, Falk ; Schewe, Lars ; Schmidt, Martin
Nonoverlapping Domain Decomposition for Optimal Control Problems governed by Semilinear Models for Gas Flow in Networks (2017)
Leugering, Günter ; Martin, Alexander ; Schmidt, Martin ; Sirvent, Mathias
We consider optimal control problems for gas flow in pipeline networks. The equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs - ultimately on single edges. We prove convergence of the domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system.
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