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This paper addresses the optimal design of resilient systems, in which components can fail. The system can react to failures and its behavior is described by general mixed integer nonlinear programs, which allows for applications to many (technical) systems. This then leads to a three-level optimization problem. The upper level designs the system minimizing a cost function, the middle level represents worst-case failures of components, i.e., interdicts the system, and the lowest level operates the remaining system. We describe new inequalities that characterize the set of resilient solutions and allow to reformulate the problem. The reformulation can then be solved using a nested branch-and-cut approach. We discuss several improvements, for instance, by taking symmetry into account and strengthening cuts. We demonstrate the effectiveness of our implementation on the optimal design of water networks, robust trusses, and gas networks, in comparison to an approach in which the failure scenarios are directly
included into the model.
For an infeasible network flow system with supplies and demands, we consider the problem of finding a minimum irreducible infeasible subsystem cover, i.e., a smallest set of constraints that must be dropped to obtain a feasible system. The special cases of covers which only contain flow balance constraints (node cover) or only flow bounds (arc cover) are investigated as well. We show strong NP-hardness of all three variants. Furthermore, we show that finding minimum arc covers for assignment problems is still hard and as hard to approximate as the set covering problem. However, the minimum arc cover problem is polynomially solvable for networks on cactus graphs. This leads to the development of two different fixed parameter algorithms with respect to the number of elementary cycles connected at arcs and the treewidth, respectively. The latter can be adapted for node covers and the general case.
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.
In this paper we propose a new approach for finding global solutions of mixed-integer nonlinear optimization problems with ordinary differential equation constraints on networks. Instead of using a first discretize then optimize approach, we combine spatial and variable branching with appropriate discretizations of the differential equations to derive relaxations of the original problem. To construct the relaxations we derive convex under- and concave over-estimators for the ODE solution operators using numerical discretization schemes. Thereby, we make use of the underlying network structure, where the solutions of the ODEs only need to be known at a finite number of points. This property enables us to adaptively refine the discretization and relaxation without introducing new variables. The incorporation into a spatial branch-and-bound process allows to compute global epsilon-optimal solutions or decide infeasibility. We prove that this algorithm terminates finitely under some natural assumptions. We then show how this approach works for the example of stationary gas transport and provide some illustrative computational examples.
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.
We consider a general class of binary packing problems with a convex quadratic knapsack constraint.
We prove that these problems are APX-hard to approximate and present constant-factor approximation
algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored
to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the
golden ratio; (2) a greedy strategy; (3) a randomized rounding method
leading to an approximation algorithm for the more general case with multiple convex quadratic
constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three
algorithms for problem instances arising in the context of real-world gas transport networks.
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
paper 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.
Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.
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.