Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Abstract

List of Figures List of Tables 1.3 Scientific Contribution and Results challenge is to find formulations and corresponding algorithms that are able to tackle transient gas transport optimization problems. Note that the first challenge seeks for general algorithms that are then used for stationary gas transport optimization, whereas the second challenge is tailored to the application of transient gas transport optimization. Our roadmap to cope with the challenges is inspired by two guidelines. We want to decompose our problems and we want to use mixed-integer linear programs. The simple reason for the first point is that decomposing problems into smaller ones has a successful history and is common sense for solving big problems. Second, mixed-integer linear programs underwent an enormous amount of research in the last decades. Consequently, fast, stable, robust, and reliable solvers as Gurobi, Cplex, or Scip are available and we use these solvers, in our case Gurobi, as a workhorse; see Gurobi [80], Cplex [36], and Maher et al. [110], respectively. 1.3 Scientific Contribution and Results For the first challenge, we consider the case where the constraint functions in (1.1c) are not given analytically. We build up a new hierarchy of assumptions that restricts the knowledge about the nonlinear functions. Depending on the assumptions, we develop three new global decomposition algorithms that decompose the difficult constraints (1.1c), i.e., we rely on our first guideline. Specifically, we build upon relaxation strategies that are motivated by developments in the field of (mixed-integer) (non)linear programming and Lipschitz optimization. As a result, mixed-integer linear programs according to our second guideline are obtained. We prove correct and finite terminations and clarify the limitations of our approaches. Note that the presented framework is a general approach for problems in the form of (1.1). In particular, we consider stationary gas transport optimization, i.e., Problem (1.1) with the Euler equations in the form of ODEs in (1.1c). Moreover, we show promising numerical results for the real-world gas network of Greece. Additionally, we consider the second challenge of transient gas transport optimization, i.e., Problem (1.1) with PDEs (1.1c). We use an instantaneous control algorithm and adapt it to the Euler equations for the first time. Specifically, we decompose the problem into single time steps and present new discretization techniques that ensure the mixed-integer linear program property. In other words, we build upon our guidelines again. Finally, we

The cost of not knowing enough: mixed-integer optimization with implicit Lipschitz nonlinearities

“…In this paper, we focus on a specific setting that can be observed in many applications (see below for some problem-specific references), namely that the Lipschitzian MINLP under consideration can be decomposed into a mixed-integer linear (MILP) part and a nonlinear part. Our working hypothesis in this and also the preceding works [20][21][22][23] is that the MILP part can be solved comparably fast and reliable whereas the nonlinearity really hampers the solution process-at least in combination with the MILP part of the problem. See also [24][25][26][27], where this working hypothesis is followed as well.…”