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 may result in simpler classes of optimization problems because 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. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant TRR 154].

We propose a decomposition based method for solving mixed‐integer nonlinear optimization problems with “black‐box” nonlinearities, where the latter, for example, 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 a 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.

A problem of image registration is considered in the context of optimal mass transportation. The properties and limitations of an optimal image transportation are analyzed. A modified formulation of this approach is proposed in order to overcome the morphing effect. Finally, a fast and simple scale-space approach for the new formulation is introduced, and numerical examples are presented.