A Framework for Data-driven Explainability in Mathematical Optimization - Aigner, Kevin-Martin; Goerigk, Marc; Hartisch, Michael; Liers, Frauke; Miehlich, Arthur (The 38th Annual AAAI Conference on Artificial Intelligence, 2024)
Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the perception of optimization software as a black box. Although well understood by scientists, this lacks easy accessibility for practitioners. Hence, we advocate for introducing the explainability of a solution as another evaluation criterion, next to its objective value, which enables us to find trade-off solutions between these two criteria. Explainability is attained by comparing against (not necessarily optimal) solutions that were implemented in similar situations in the past. Thus, solutions are preferred that exhibit similar features. Although we prove that already in simple cases the explainable model is NP-hard, we characterize relevant polynomially solvable cases such as the explainable shortest-path problem. Our numerical experiments on both artificial as well as real-world road networks show the resulting Pareto front. It turns out that the cost of enforcing explainability can be very small.<br
Keywords: explainability; mathematical optimization; data-driven optimization; bicriteria optimization

Data-driven Distributionally Robust Optimization over Time - Aigner, Kevin-Martin; Bärmann, Andreas; Braun, Kristin; Liers, Frauke; Pokutta, Sebastian; Schneider, Oskar; Sharma, Kartikey; Tschuppik, Sebastian (INFORMS Journal on Optimization, 2023)
Stochastic Optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. As the latter is often unknown, Distributionally Robust Optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy which converges at a rate of O(T / T), where T is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulation
Keywords: distributionally robust optimization; gradient descent; learning over time

Data-driven Optimization under Uncertainty for Power Networks - Aigner, Kevin-Martin (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2024)
This cumulative doctoral thesis presents 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. Finally, we present a novel algorithmic framework for optimization under uncertainty over time. The approach uses online learning and scenario observations arriving as a data stream to learn more about the uncertain parameters. We provide a dynamic regret bound for our solutions and illustrate the broad applicability of our approac
Keywords: Mixed-integer nonlinear programming; Optimal Power Flow; Data-driven Optimization; Robust Optimization; Optimization under Uncertainty; Mathematical Optimization; Power System Analysis

Datenbasierte Optimierung unter Unsicherheit für Stromnetzwerke - Aigner, Kevin-Martin (2024)

Feature Selection for Data-driven Explainable Optimization - Aigner, Kevin-Martin; Goerigk, Marc; Hartisch, Michael; Liers, Frauke; Miehlich, Arthur; Rösel, Florian (2025)
Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: a solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to explain a solution. In mathematical optimization, feature selection has received little attention yet. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size.
https://cris.fau.de/converis/publicweb/Publication/339395864
Keywords: Feature selection; Explainable optimization; Data-driven optimization

Modellierung des Wärmesektors und dessen Flexibilität für Simulationen gekoppelter Energiesysteme - Luna-Jaspe Roa, Natalia; Aigner, Kevin-Martin; Kolb, Sebastian; Karl, Jürgen (EnInnov2024 - 18. Symposium Energieinnovation 2024, 2024)
Die Defossilisierung des Wärmesektors ist eine der größten Herausforderungen der europäischen Energiewende. Ohne drastische Transformation werden stark öl- und gasabhängige Länder wie Deutschland ihre Klimaziele nicht erreichen. Mögliche Entwicklungsszenarien der Literatur sehen die Elektrifizierung des Wärmesektors durch Wärmepumpen als eine der am besten geeigneten Lösungen. Doch nicht nur der Wärmesektor wird in Zukunft hohe Stromnachfragen generieren, auch andere Sektoren wie der Verkehrssektor werden stark elektrifiziert werden. In diesem Kontext spielt die Möglichkeit zur Flexibilisierung dieser Stromnachfragen eine wichtige Rolle, insbesondere mit Blick auf den Ausbau der fluktuierenden erneuerbaren Energien. Diese Arbeit analysiert das Flexibilitätspotential des Wohngebäudesektors durch die Nutzung der thermischen Gebäudemasse. Außerdem wird untersucht, wie die zur Verfügung stehende Flexibilität optimal eingesetzt werden kann. Wohngebäude können durch die Nutzung der thermischen Gebäudemasse einen Beitrag zur Flexibilisierung des Energiesystems leisten. Das Flexibilitätspotential in der Zukunft ist von der Sanierungstiefe abhängig. Die Effekte von Flexibilitätsmaßnahmen auf die Kosten und die Netzentlastung sind zusätzlich von den gewählten Optimierungsbedingungen abhängig.<br
Keywords: Simulationsmodell; Wärmesektor; thermische Gebäudemasse; Flexibilisierung; Optimierung

Robust Approximation of Chance Constrained DC Optimal Power Flow under Decision-Dependent Uncertainty - Aigner, Kevin-Martin; Clarner, Jan-Patrick; Liers, Frauke; Martin, Alexander (European Journal of Operational Research, 2022)
We propose a mathematical optimization model and its solution for joint chance constrained DC Optimal Power Flow. In this application, it is particularly important that there is a high probability of transmission limits being satisfied, even in the case of uncertain or fluctuating feed-in from renewable energy sources. In critical network situations where the network risks overload, renewable energy feed-in has to be curtailed by the transmission system operator (TSO). The TSO can reduce the feed-in in discrete steps at each network node. The proposed optimization model minimizes curtailment while ensuring that there is a high probability of transmission limits being maintained. The latter is modeled via (joint) chance constraints that are computationally challenging. Thus, we propose a solution approach based on the robust safe approximation of these constraints. Hereby, probabilistic constraints are replaced by robust constraints with suitably defined uncertainty sets constructed from historical data. The uncertainty sets are calculated by encompassing randomly drawn scenarios using the scenario approach proposed by Margellos et al. (IEEE Transactions on Automatic Control, 59 (2014)). The ability to discretely control the power feed-in then leads to a robust optimization problem with decision-dependent uncertainties, i.e. the uncertainty sets depend on decision variables. We propose an equivalent mixed-integer linear reformulation for box uncertainties with the exact linearization of bilinear terms. Finally, we present numerical results for different test cases from the Nesta archive, as well as for a real network. We consider the discrete curtailment of solar feed-in, for which we use real-world weather and network data. The experimental tests demonstrate the effectiveness of this method and run times are very fast. Moreover, on average the calculated robust solutions lead only to a small increase in curtailment, when compared to nominal solutions.
Keywords: OR in energy; optimal power flow; chance constraints; robust optimization; decision-dependent uncertainty; robust approximation; safe approximation; scenario approach; mixed-integer linear optimization

Robust DC Optimal Power Flow with Modeling of Solar Power Supply Uncertainty via R-Vine Copulas - Aigner, Kevin-Martin; Schaumann, Peter; von Loeper, Freimut; Martin, Alexander; Schmidt, Volker; Liers, Frauke (Optimization and Engineering, 2022)
We present a robust approximation of joint chance constrained DC Optimal Power Flow in combination with a model-based prediction of uncertain power supply via R-vine copulas. It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in. This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (European Journal of Operational Research, (2021)). The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets. The resulting robust optimization problem has a known equivalent tractable reformulation.%, which can be solved with standard software. To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes. The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model. Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts. The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas. We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.<br
Keywords: chance constrained programming; optimal power flow; robust optimization; multiple prediction interval; R-vine copula; conditional uncertainty set

Robust Optimization in Nanoparticle Technology: A Proof of Principle by Quantum Dot Growth in a Residence Time Reactor - Dienstbier, Jana; Aigner, Kevin-Martin; Rolfes, Jan; Peukert, Wolfgang; Segets, Doris; Pflug, Lukas; Liers, Frauke (Computers & Chemical Engineering, 2021)
Knowledge-based determination of the best-possible experimental setups for nanoparticle design is highly challenging. Additionally, such processes are accompanied by noticeable uncertainties. Therefore, protection against these uncertainties is needed. Robust optimization helps determining such best possible processes. The latter guarantees quality requirements regardless of how the uncertainties, e.g. with regard to variations in raw materials, heat and mass transport characteristics, material properties and (growth) rates, manifest within predefined ranges. To approach this huge task, in this paper we exemplarily model a particle synthesis process with seeded growth by population balance equations and study different growth kinetics. We determine the mean residence time maximizing the product mass subject to a guaranteed yield. Additionally, we hedge against uncertain growth rates and derive an algorithmically tractable reformulation for the robustified problem. We evaluate our approach for seeded growth synthesis of zinc oxide quantum dots and demonstrate computationally that a guaranteed yield is met for all growth rates within previously defined regions. The protection against uncertainties only reduces the maximum amount of product that can be obtained by a negligible margin.<br
Keywords: particle design; robust optimization; process optimization; reformulation

Scenario Reduction for Distributionally Robust Optimization - Aigner, Kevin-Martin; Denzler, Sebastian; Liers, Frauke; Pokutta, Sebastian; Sharma, Kartikey (2025)
Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the proposed approximation significantly reduces solution time while maintaining high solution quality with only minor error
https://cris.fau.de/converis/publicweb/Publication/336975332
Keywords: distributionally robust optimization; scenario reduction; scenario clustering; approximation bounds; mixed-integer programming; portfolio optimization

Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality - Aigner, Kevin-Martin; Burlacu, Robert; Liers, Frauke; Martin, Alexander (Informs Journal on Computing, 2023)
We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs. The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second-order cone programs as well as primal and dual cutting planes. For example objective cuts and no-good-cuts help to compute good feasible solutions as where outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems where discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP solvers. Our method outperforms all other algorithms.<br
Keywords: AC Optimal Power Flow; Mixed-Integer Nonlinear Programming; Second-Order Cone Programming; Piecewise Linear Relaxation; Discrete Decisions; Generator Switching; Transmission Line Switching; Curtailment of Renewables