fORum Seminar

The Bipartite Boolean Quadric Polytope with Multiple-Choice Constraints

The well-known bipartite boolean quadric polytope (BQP) is defined as the convex hull of all quadric incidence vectors over a bipartite graph. Here, we consider the case where there is a partition on one of the two bipartite node sets such that at most one node per subset of the partition can be chosen. In this talk, results of the polyhedral study and the relation to the pooling problem are presented. Furthermore, a similar polytope is introduced that will be analyzed in future work.

Haoyue (Claire) Zhang⁽¹⁾

Facility Location Problem under Uncertainty with Service Level Constraints

In facility location, most models assume customer demands to be deterministic. However, in practice, there is often a large degree of uncertainty about future demands, especially given the strategic nature of location problems where decisions have to be made for the next twenty or thirty years. Stochastic programming models are widely applied to solve FLPs under uncertainty. However, since all the demand has to be satisfied in every scenario, the models sometimes give conservative results. This presentation is about including service levels as chance constraints in the stochastic programming model, allowing for a certain probability and a certain percent of the demand to be unsatisfied. In the model, the α-service level is applied both locally and globally while the β-service level constraints take the expected value, as well as the maximum value of the excess into account. To decide which combination of service levels “works the best”, we carry out experiments and tests with combinations of different settings on randomly generated data sets.

Gas Network Control by Simulation-based Reinforcement Learning

Optimal control problems for gas flow in pipeline networks are usually tackled in mathematics within three steps: first, modelling the problem as a MINLP, second, approximating it to receive a MIP, third, optimizing the approximated problem. Imagining a discretized time horizon, we leave the determination of control variables in every time step as a decision to a machine learning approach. In contrast to standard methods, we leverage the optimization as a simulation framework for the resulting easier problem. This technique combines the fields of artificial intelligence as well as mathematical optimization and is more accurate in the modelling of nonlinearities, as well as regarding the functionality of gas network controls. In cooperation with an industry partner, we apply a reinforcement learning technique called (categorical) deep Q-network (CDQN) to control gas subnetworks. The learning of the agent and the improvement of its control is achieved via Q-learning, a special case of approximate dynamic programming by Bellman that incorporates future, as well as present states to accomplish the overall best result. This thesis contains a description of the original model, as well as an explanation of the used CDQN approach, and closes with computational results on a real-world gas subnetwork.

A nonlocal PDE-constrained optimisation model for containment of infectious diseases

Nonpharmaceutical interventions have proven crucial in the containment and prevention of Covid-19 outbreaks. In particular public health policy makers have to assess the effects of strategies such as social distancing and isolation to avoid exceeding social and economical costs. In this work, we study an optimal control approach for parameter selection applied to a dynamical density functional theory model. This is applied in particular to a spatially-dependent SIRD model where social distancing and isolation of infected persons are explicitly taken into account. Special attention is paid when the strength of these measures is considered as a function of time and their effect on the overall infected compartment. A first order optimality system is presented, and numerical simulations are presented using a proximal method. This work could potentially provide some mathematical insights into the management of disease outbreaks.