Conference Agenda

We present a comprehensive exploration of data-driven optimization under uncertainty in the context of power system analysis. Critical mathematical challenges related to the operation of power grids are addressed, accompanied by innovative solution approaches. The primary focus is on several extensions of the optimal power flow problem which is the predominantly used model in the literature to optimize the power distribution in an electricity network. We study nonconvex mixed-integer nonlinear programs arising in power system analysis, which we solve by the construction and successive refinement of piecewise linear relaxations. Our work introduces various problem-specific and generally applicable algorithmic enhancements to obtain an efficient implementation that outperforms state-of-the-art solvers. Another focus are stochastic mixed-integer linear optimal power flow problems with probabilistic constraints. The solution approach is based on the robust safe approximation of the computationally intractable chance constraints. To construct the approximative problems, suitably defined confidence sets from historical data are computed. We derive a tractable reformulation of the resulting problems and prove quality guarantees about the robustness of the calculated solutions. Numerical experiments on benchmark instances with real weather and network data demonstrate the quality of our solutions. Further improvements are achieved by combining stochastic programming with a model-based prediction of uncertainties.

The Capacitated Facility Location Problem (CFLP) is a core problem in location science. We show that the combinatorial element of the MIP formulation manifests itself to different degrees in the level of interdependence between facilities serving similar subsets of customers. This induces an implied separation of the sets of candidates and customers into regions within which location decisions interdepend more strongly. Although these regions are easily identifiable in visual representations of allocation decisions or the spatial distribution of candidates and customers, detecting them solely from decision vectors is challenging. We show that spectral biclustering, a pattern recognition technique, can be used to retrieve implied regions from integer-infeasible solutions. This opens novel directions for the development of exact and heuristic solution procedures.

Instantaneous dynamic equilibria (IDE) are an equilibrium concept for flows over time with deterministic queueing wherein individual flow particles make selfish decisions based on current information. This can be used as a model for car traffic where each driver initially chooses her route in such a way as to minimize her travel time under the current congestion state of the road network and then continuously adapts her route while driving.

We first prove existence of such equilibria under quite mild assumptions. We then show that a natural extension approach can be used to compute IDE in single-commodity networks within finite time. On the other hand we provide a multi-commodity instance where there exists a finite time horizon that cannot be reached by such an extension based algorithm within finite time. Moreover, we show that several natural decision problems involving IDE are NP-hard. Finally, we study the quality of IDE with regards to makespan as well as total travel time.

For single-commodity networks we give both upper and a lower bounds while for multi-commodity networks we provide an instance wherein IDE never terminate.

Solving real-world combinatorial optimization problems with traditional operations research methods can be a costly and time-consuming endeavor, often requiring the development of completely new methods or significant modifications to existing techniques. In this talk, we explore several methods that use deep reinforcement learning to automate the development of problem-specific solution approaches. Rather than focusing on end-to-end solution generation, we investigate the use of machine learning to learn heuristic components for high-level search procedures. By automating the design of these components, the overall solution approach can be easily customized to the characteristics of specific problem instances, potentially lowering the barriers to entry for the use of optimization technologies across a wide range of use cases.