TY - GEN A1 - Fügenschuh, Armin A1 - Humpola, Jesco T1 - A Unified View on Relaxations for a Nonlinear Network Flow Problem N2 - We consider a nonlinear nonconvex network flow problem that arises, for example, in natural gas or water transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides flow conservation constraints in the nodes the flow must fulfill nonlinear nonconvex pressure loss constraints on the arcs subject to potential values (i.e., pressure levels) in both end nodes of each arc. The problem is how to numerically compute this flow and pressures. We review an existing approach of Maugis (1977) and extend it to the case of networks with active elements (for example, compressors). We further examine different ways of relaxations for the nonlinear network flow model. We compare different approaches based on nonlinear optimization numerically on a set of test instances. T3 - ZIB-Report - 13-31 KW - Nonlinear Network Flow KW - Mixed-Integer Nonlinear Programming KW - Relaxations Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18857 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Fügenschuh, Armin A1 - Klug, Torsten A1 - Schang, Thilo A1 - Schlechte, Thomas A1 - Schülldorf, Hanno T1 - The Freight Train Routing Problem N2 - We consider the following freight train routing problem (FTRP). Given is a transportation network with fixed routes for passenger trains and a set of freight trains (requests), each defined by an origin and destination station pair. The objective is to calculate a feasible route for each freight train such that a sum of all expected delays and all running times is minimal. Previous research concentrated on microscopic train routings for junctions or inside major stations. Only recently approaches were developed to tackle larger corridors or even networks. We investigate the routing problem from a strategic perspective, calculating the routes in a macroscopic transportation network of Deutsche Bahn AG. Here macroscopic refers to an aggregation of complex real-world structures are into fewer network elements. Moreover, the departure and arrival times of freight trains are approximated. The problem has a strategic character since it asks only for a coarse routing through the network without the precise timings. We give a mixed-integer nonlinear programming~(MINLP) formulation for FTRP, which is a multi-commodity flow model on a time-expanded graph with additional routing constraints. The model's nonlinearities are due to an algebraic approximation of the delays of the trains on the arcs of the network by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model~(MILP) by piecewise linear approximation. The latter is solved by a state of the art MILP solver for various real-world test instances. T3 - ZIB-Report - 13-36 KW - Mixed-Integer Nonlinear Programming KW - multi-commodity flows KW - freight train routing Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18991 ER - TY - GEN A1 - Humpola, Jesco A1 - Fügenschuh, Armin T1 - A New Class of Valid Inequalities for Nonlinear Network Design Problems N2 - We consider a nonlinear nonconvex network design problem that arises in the extension of natural gas transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides flow conservation constraints in the nodes the flow must fulfill nonlinear nonconvex pressure loss constraints on the arcs subject to potential values (i.e., pressure levels) in both end nodes of each arc. Assume that there does not exist a feasible flow that fulfills all physical constraints and meets the desired entry and exit amounts. Then a natural question is where to extend the network by adding pipes in the most economic way such that this flow becomes feasible. Answering this question is computationally demanding because of the difficult problem structure. We use mixed-integer nonlinear programming techniques that rely on an outer approximation of the overall problem, and a branching on decision variables. We formulate a new class of valid inequalities (or cutting planes) which reduce the overall solution time when added to the formulation. We demonstrate the computational merits of our approach on test instances. T3 - ZIB-Report - 13-06 KW - Network Design KW - Mixed-Integer Nonlinear Programming KW - Cutting Planes Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-17771 SN - 1438-0064 ER - TY - GEN A1 - Humpola, Jesco A1 - Fügenschuh, Armin A1 - Lehmann, Thomas T1 - A Primal Heuristic for MINLP based on Dual Information N2 - We present a novel heuristic algorithm to identify feasible solutions of a mixed-integer nonlinear programming problem arising in natural gas transportation: the selection of new pipelines to enhance the network's capacity to a desired level in a cost-efficient way. We solve this problem in a linear programming based branch-and-cut approach, where we deal with the nonlinearities by linear outer approximation and spatial branching. At certain nodes of the branching tree, we compute a KKT point for a nonlinear relaxation. Based on the information from the KKT point we alter some of the integer variables in a locally promising way. We describe this heuristic for general MINLPs and then show how to tailor the heuristic to exploit our problem-specific structure. On a test set of real-world instances, we are able to increase the chance of identifying feasible solutions by some order of magnitude compared to standard MINLP heuristics that are already built in the general-purpose MINLP solver SCIP. T3 - ZIB-Report - 13-49 KW - Mixed-Integer Nonlinear Programming KW - Relaxations KW - Heuristics KW - Duality KW - Nonlinear Network Design Applications Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-43110 SN - 1438-0064 ER -