@unpublished{GruebelHuberHuembsetal.2021, author = {Gr{\"u}bel, Julia and Huber, Olivier and H{\"u}mbs, Lukas and Klimm, Max and Schmidt, Martin and Schwartz, Alexandra}, title = {Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium}, pages = {29}, year = {2021}, abstract = {Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems.}, language = {en} } @unpublished{DeSantisdeVriesSchmidtetal.2021, author = {De Santis, Marianna and de Vries, Sven and Schmidt, Martin and Winkel, Lukas}, title = {A Penalty Branch-and-Bound Method for Mixed-Binary Linear Complementarity Problems}, pages = {24}, year = {2021}, abstract = {Linear complementarity problems (LCPs) are an important modeling tool for many practically relevant situations but also have many important applications in mathematics itself. Although the continuous version of the problem is extremely well studied, much less is known about mixed-integer LCPs (MILCPs) in which some variables have to be integer-valued in a solution. In particular, almost no tailored algorithms are known besides reformulations of the problem that allow to apply general-purpose mixed-integer linear programming solvers. In this paper, we present, theoretically analyze, enhance, and test a novel branch-and-bound method for MILCPs. The main property of this method is that we do not ``branch'' on constraints as usual but by adding suitably chosen penalty terms to the objective function. By doing so, we can either provably compute an MILCP solution if one exists or compute an approximate solution that minimizes an infeasibility measure combining integrality and complementarity conditions. We enhance the method by MILCP-tailored valid inequalities, node selection strategies, branching rules, and warmstarting techniques. The resulting algorithm is shown to clearly outperform two benchmark approaches from the literature.}, language = {en} } @article{KrugLeugeringMartinetal.2021, author = {Krug, Richard and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Weninger, Dieter}, title = {Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems with Mixed Two-Point Boundary Conditions}, series = {Control and Cybernetics}, journal = {Control and Cybernetics}, pages = {20}, year = {2021}, abstract = {In this article, we continue our work (Krug et al., 2021) on time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations in that we now consider mixed two-point boundary value problems and provide an in-depth well-posedness analysis. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.}, language = {en} } @article{KleinertMannsSchmidtetal.2021, author = {Kleinert, Thomas and Manns, Julian and Schmidt, Martin and Weninger, Dieter}, title = {Presolving Linear Bilevel Optimization Problems}, series = {EURO Journal on Computational Optimization}, journal = {EURO Journal on Computational Optimization}, number = {9}, doi = {10.1016/j.ejco.2021.100020}, year = {2021}, abstract = {Linear bilevel optimization problems are known to be strongly NP-hard and the computational techniques to solve these problems are often motivated by techniques from single-level mixed-integer optimization. Thus, during the last years and decades many branch-and-bound methods, cutting planes, or heuristics have been proposed. On the other hand, there is almost no literature on presolving linear bilevel problems although presolve is a very important ingredient in state-of-the-art mixed-integer optimization solvers. In this paper, we carry over standard presolve techniques from single-level optimization to bilevel problems and show that this needs to be done with great caution since a naive application of well-known techniques does often not lead to correctly presolved bilevel models. Our numerical study shows that presolve can also be very beneficial for bilevel problems but also highlights that these methods have a more heterogeneous effect on the solution process compared to what is known from single-level optimization. As a side result, our numerical experiments reveal that there is an urgent need for better and more heterogeneous test instance libraries to further propel the field of computational bilevel optimization.}, language = {en} } @article{KrugLeugeringMartinetal.2020, author = {Krug, Richard and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Weninger, Dieter}, title = {Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems}, series = {SIAM Journal on Control and Optimization}, journal = {SIAM Journal on Control and Optimization}, pages = {28}, year = {2020}, abstract = {In this article, we extend the time-domain decomposition method described by Lagnese and Leugering (2003) to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process.}, language = {en} } @article{KleinertLabbeLjubićetal.2021, author = {Kleinert, Thomas and Labb{\´e}, Martine and Ljubić, Ivana and Schmidt, Martin}, title = {A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization}, series = {EURO Journal on Computational Optimization}, journal = {EURO Journal on Computational Optimization}, pages = {47}, year = {2021}, abstract = {Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980's but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.}, language = {en} } @article{KleinertLabbeSchmidtetal.2020, author = {Kleinert, Thomas and Labb{\´e}, Martine and Schmidt, Martin and Plein, Fr{\"a}nk}, title = {Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality}, series = {Optimization Letters}, journal = {Optimization Letters}, number = {15}, pages = {1027 -- 1040}, year = {2020}, abstract = {Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem's optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.}, language = {de} } @article{BoettgerGrimmKleinertetal.2020, author = {B{\"o}ttger, Tom and Grimm, Veronika and Kleinert, Thomas and Schmidt, Martin}, title = {The Cost of Decoupling Trade and Transport in the European Entry-Exit Gas Market with Linear Physics Modeling}, series = {European Journal of Operational Research}, journal = {European Journal of Operational Research}, pages = {27}, year = {2020}, abstract = {Liberalized gas markets in Europe are organized as entry-exit regimes so that gas trade and transport are decoupled. The decoupling is achieved via the announcement of technical capacities by the transmission system operator (TSO) at all entry and exit points of the network. These capacities can be booked by gas suppliers and customers in long-term contracts. Only traders who have booked capacities up-front can "nominate" quantities for injection or withdrawal of gas via a day-ahead market. To ensure feasibility of the nominations for the physical network, the TSO must only announce technical capacities for which all possibly nominated quantities are transportable. In this paper, we use a four-level model of the entry-exit gas market to analyze possible welfare losses associated with the decoupling of gas trade and transport. In addition to the multilevel structure, the model contains robust aspects to cover the conservative nature of the European entry-exit system. We provide several reformulations to obtain a single-level mixed-integer quadratic problem. The overall model of the considered market regime is extremely challenging and we thus have to make the main assumption that gas flows are modeled as potential-based linear flows. Using the derived single-level reformulation of the problem, we show that the feasibility requirements for technical capacities imply significant welfare losses due to unused network capacity. Furthermore, we find that the specific structure of the network has a considerable influence on the optimal choice of technical capacities. Our results thus show that trade and transport are not decoupled in the long term. As a further source of welfare losses and discrimination against individual actors, we identify the minimum prices for booking capacity at the individual nodes.}, language = {en} } @unpublished{GrimmNowakScheweetal.2020, author = {Grimm, Veronika and Nowak, Daniel and Schewe, Lars and Schmidt, Martin and Schwartz, Alexandra and Z{\"o}ttl, Gregor}, title = {A Tractable Multi-Leader Multi-Follower Peak-Load-Pricing Model with Strategic Interaction}, doi = {10.1007/s10107-021-01708-0}, pages = {35}, year = {2020}, abstract = {While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems.}, language = {en} } @unpublished{BiefelLiersRolfesetal.2020, author = {Biefel, Christian and Liers, Frauke and Rolfes, Jan and Schmidt, Martin}, title = {Affinely Adjustable Robust Linear Complementarity Problems}, pages = {20}, year = {2020}, abstract = {Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties - especially in the sense of robust optimization - is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and Γ-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. Our main results are valid for the case of an uncertain LCP vector. Here, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If, in addition, the certain LCP matrix is positive semidefinite, we prove polynomial-time solvability and uniqueness of robust solutions. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixed-integer programming formulations still remain open problems if the nominal LCP matrix is not positive definite.}, language = {en} }