TY - INPR A1 - Hante, Falk A1 - Krug, Richard A1 - Schmidt, Martin T1 - Time-Domain Decomposition for Mixed-Integer Optimal Control Problems N2 - We consider mixed-integer optimal control problems, whose optimality conditions involve global combinatorial optimization aspects for the corresponding Hamiltonian pointwise in time. We propose a time-domain decomposition, which makes this problem class accessible for mixed-integer programming using parallel-in-time direct discretizations. The approach is based on a decomposition of the optimality system and the interpretation of the resulting subproblems as suitably chosen mixed-integer optimal control problems on subintervals in time. An iterative procedure then ensures continuity of the states at the boundaries of the subintervals via co-state information encoded in virtual controls. We prove convergence of this iterative scheme for discrete-continuous linear-quadratic problems and present numerical results both for linear-quadratic as well as nonlinear problems. KW - Mixed-integer optimal control problems KW - Time-domain decomposition KW - Mixed-integer nonlinear optimization KW - Convergence Y1 - 2021 ER - TY - INPR A1 - Gugat, Martin A1 - Krug, Richard A1 - Martin, Alexander T1 - Transient gas pipeline flow: Analytical examples, numerical simulation and a comparison to the quasi-static approach N2 - The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a pde model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based upon multi-period optimization of steady states does not lead to approximations that converge to the optimal state. Y1 - 2021 U6 - https://doi.org/10.1007/s11081-021-09690-4 ER - TY - INPR A1 - Grübel, Julia A1 - Krug, Richard A1 - Schmidt, Martin A1 - Wollner, Winnifried T1 - A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities N2 - We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems. KW - Mixed-Integer Nonlinear Optimization KW - Global Optimization KW - Lipschitz Optimization KW - Bilevel Optimization KW - Gas Networks Y1 - 2022 ER - TY - INPR A1 - Krug, Richard A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems N2 - We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and which may result in simpler classes of optimization problems since not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research. KW - Gas transport networks KW - Mixed-integer nonlinear optimization KW - Alternating direction methods KW - Graph decomposition KW - Penalty methods Y1 - 2022 ER -