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.
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.