TY - JOUR A1 - Kleinert, Thomas A1 - Manns, Julian A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - Presolving Linear Bilevel Optimization Problems JF - EURO Journal on Computational Optimization N2 - 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. KW - Linear Bilevel Optimization KW - Presolve KW - Computational Analysis Y1 - 2021 U6 - https://doi.org/10.1016/j.ejco.2021.100020 IS - 9 ER - TY - THES A1 - Kleinert, Thomas T1 - Algorithms for Mixed-Integer Bilevel Problems with Convex Followers N2 - Bilevel problems are optimization problems for which a subset of variables is constrained to be an optimal solution of another optimization problem. As such, bilevel problems are capable of modeling hierarchical decision processes. This is required by many real-world problems from a broad spectrum of applications such as energy markets, traffic planning, or critical infrastructure defense, to name only a few. However, the hierarchy of decisions makes bilevel optimization problems also very challenging to solve—both in theory and practice. This cumulative PhD thesis is concerned with computational bilevel optimization. In the first part, we summarize several solution approaches that we developed over the last years and highlight the significant computational progress that these methods provide. For linear bilevel problems, we review branch-and-bound methods, critically discuss their practical use, and propose valid inequalities to extend the methods to branch-and-cut approaches. Further, we demonstrate on a large test set that it is no longer necessary to use the well-known but error-prone big-M reformulation to solve linear bilevel problems. We also present a bilevel-specific heuristic that is based on a penalty alternating direction method. This heuristic is applicable to a broad class of bilevel problems, e.g., linear or mixed-integer quadratic bilevel problems. In a computational study, we show that the method computes optimal or close-to-optimal feasible points in a very short time and that it outperforms a state-of-the-art local method from the literature. Finally, we review global approaches for mixed-integer quadratic bilevel problems. In addition to a Benders-like decomposition, we present a multi-tree and a single-tree outer-approximation approach. A computational evaluation demonstrates that both variants outperform known benchmark algorithms. The second part of this thesis consists of reprints of our original articles and preprints. These articles contain all details and are referenced throughout the first part of the thesis. Y1 - 2021 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Ljubić, Ivana A1 - Schmidt, Martin T1 - A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization JF - EURO Journal on Computational Optimization N2 - 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. KW - Bilevel optimization KW - Mixed-integer programming KW - Applications KW - Branch-and-bound KW - Branch-and-cut Y1 - 2021 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Schmidt, Martin A1 - Plein, Fränk T1 - Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality JF - Optimization Letters N2 - 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. Y1 - 2020 IS - 15 SP - 1027 EP - 1040 ER - TY - JOUR A1 - Böttger, Tom A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - The Cost of Decoupling Trade and Transport in the European Entry-Exit Gas Market with Linear Physics Modeling JF - European Journal of Operational Research N2 - 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. KW - Entry-Exit Gas Market KW - Gas Market Design KW - Multilevel Optimization KW - Robust Optimization Y1 - 2020 ER - TY - INPR A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Why there is no need to use a big-M in linear bilevel optimization: A computational study of two ready-to-use approaches N2 - Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a "best-practice" has developed in the applied OR community, in which not all people want to develop tailored algorithms but "just use" bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush-Kuhn-Tucker (KKT) conditions of the lower-level problem - an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost. In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance - without having all the possible theoretical disadvantages of the big-M approach. KW - Bilevel optimization KW - Big-M KW - SOS-1 KW - Valid inequalities KW - Computational analysis Y1 - 2020 ER - TY - INPR A1 - Heitsch, Holger A1 - Henrion, René A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - On Convex Lower-Level Black-Box Constraints in Bilevel Optimization with an Application to Gas Market Models with Chance Constraints N2 - Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage. KW - Bilevel optimization KW - Black-box constraints KW - Chance constraints KW - Cutting planes KW - European gas market Y1 - 2021 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Grimm, Veronika A1 - Schmidt, Martin T1 - Outer Approximation for Global Optimization of Mixed-Integer Quadratic Bilevel Problems JF - Mathematical Programming (Series B) N2 - Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches. KW - Bilevel optimization KW - Outer approximation KW - Quadratic programming KW - Convex mixed-integer nonlinear optimization Y1 - 2019 ER - TY - JOUR A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Liers, Frauke A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - Optimal Price Zones of Electricity Markets: A Mixed-Integer Multilevel Model and Global Solution Approaches JF - Optimization Methods and Software N2 - Mathematical modeling of market design issues in liberalized electricity markets often leads to mixed-integer nonlinear multilevel optimization problems for which no general-purpose solvers exist and which are intractable in general. In this work, we consider the problem of splitting a market area into a given number of price zones such that the resulting market design yields welfare-optimal outcomes. This problem leads to a challenging multilevel model that contains a graph-partitioning problem with multi-commodity flow connectivity constraints and nonlinearities due to proper economic modeling. Furthermore, it has highly symmetric solutions. We develop different problem-tailored solution approaches. In particular, we present an extended KKT transformation approach as well as a generalized Benders approach that both yield globally optimal solutions. These methods, enhanced with techniques such as symmetry breaking and primal heuristics, are evaluated in detail on academic as well as on realistic instances. It turns out that our approaches lead to effective solution methods for the difficult optimization tasks presented here, where the problem-specific generalized Benders approach performs considerably better than the methods based on KKT transformation. KW - Multilevel Optimization KW - Mixed-Integer Nonlinear Optimization KW - Graph Partitioning KW - Generalized Benders Decomposition KW - Electricity Market Design} Y1 - 2017 IS - 34(2) SP - 406 EP - 436 ER - TY - JOUR A1 - Ambrosius, Mirjam A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Liers, Frauke A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - Endogenous Price Zones and Investment Incentives in Electricity Markets: An Application of Multilevel Optimization with Graph Partitioning JF - Energy Economics N2 - In the course of the energy transition, load and supply centers are growing apart in electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. This paper focuses on electricity markets that operate under a zonal pricing market design. For a fixed number of zones, we endogenously derive the optimal configuration of price zones and available transfer capacities on a network in order to optimally govern investment and production decisions in the long run. In a multilevel mixed-integer nonlinear model that contains a graph partitioning problem on the first level, we determine welfare-maximizing price zones and available transfer capacities for a given electricity market and analyze their impact on market outcomes. Using a generalized Benders decomposition approach developed in Grimm et al. (2019) and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that even for a small number of price zones, an optimal configuration of zones induces a welfare level that almost approaches the first best. KW - Electricity Markets KW - Price Zones KW - Investment Incentives KW - Multilevel Optimization KW - Graph Partitioning Y1 - 2018 IS - 92 ER - TY - JOUR A1 - Egerer, Jonas A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - The Impact of Neighboring Markets on Renewable Locations, Transmission Expansion, and Generation Investment JF - European Journal of Operational Research N2 - Many long-term investment planning models for liberalized electricity markets either optimize for the entire electricity system or focus on confined jurisdictions, abstracting from adjacent markets. In this paper, we provide models for analyzing the impact of the interdependencies between a core electricity market and its neighboring markets on key long-run decisions. This we do both for zonal and nodal pricing schemes. The identification of welfare optimal investments in transmission lines and renewable capacity within a core electricity market requires a spatially restricted objective function, which also accounts for benefits from cross-border electricity trading. This leads to mixed-integer nonlinear multilevel optimization problems with bilinear nonconvexities for which we adapt a Benders-like decomposition approach from the literature. In a case study, we use a stylized six-node network to disentangle different effects of optimal regional (as compared to supra-regional) investment planning. Regional planning alters investment in transmission and renewable capacity in the core region, which affects private investment in generation capacity also in adjacent regions and increases welfare in the core region at the cost of system welfare. Depending on the congestion-pricing scheme, the regulator of the core region follows different strategies to increase welfare causing distributional effects among stakeholders. KW - Neighboring Markets KW - Renewables KW - Network Expansion KW - Multilevel Optimization KW - Benders Decomposition Y1 - 2019 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method JF - INFORMS Journal on Computing N2 - Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, etc. Often, these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization one develops primal heuristics in order to obtain feasible points of good quality quickly or to enhance the search process of exact global methods. However, there are comparably few heuristics for bilevel problems. In this paper, we develop such a primal heuristic for bilevel problems with mixed-integer linear or quadratic upper level and linear or quadratic lower level. The heuristic is based on a penalty alternating direction method, which allows for a theoretical analysis. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem and extensively test the method on a test set of more than 2800 instances - which is one of the largest computational test sets ever used in bilevel programming. The study illustrates the very good performance of the proposed method, both in terms of running times and solution quality. This renders the method a suitable sub-routine in global bilevel solvers as well as a reasonable standalone approach. KW - Bilevel optimization KW - Mixed-integer bilevel optimization KW - Stationary points KW - Penalty methods KW - Alternating direction methods Y1 - 2019 U6 - https://doi.org/10.1287/ijoc.2019.0945 IS - 33 (1) SP - 198 EP - 215 ER - TY - JOUR A1 - Grübel, Julia A1 - Kleinert, Thomas A1 - Krebs, Vanessa A1 - Orlinskaya, Galina A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - On Electricity Market Equilibria with Storage: Modeling, Uniqueness, and a Distributed ADMM JF - Computers & Operations Research N2 - We consider spot-market trading of electricity including storage operators as additional agents besides producers and consumers. Storages allow for shifting produced electricity from one time period to a later one. Due to this, multiple market equilibria may occur even if classical uniqueness assumptions for the case without storages are satisfied. For models containing storage operators, we derive sufficient conditions that ensure uniqueness of generation and demand. We also prove uniqueness of the market equilibrium for the special case of a single storage operator. Nevertheless, in case of multiple storage operators, uniqueness fails to hold in general, which we show by illustrative examples. We conclude the theoretical discussion with a general ex-post condition for proving the uniqueness of a given solution. In contrast to classical settings without storages, the computation of market equilibria is much more challenging since storage operations couple all trading events over time. For this reason, we propose a tailored parallel and distributed alternating direction method of multipliers (ADMM) for efficiently computing spot-market equilibria over long time horizons. We first analyze the parallel performance of the method itself. Finally, we show that the parallel ADMM clearly outperforms solving the respective problems directly and that it is capable of solving instances with more than 42 million variables in less than 13 minutes. Y1 - 2019 U6 - https://doi.org/10.1016/j.cor.2019.104783 IS - 114 ER - TY - INPR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Plein, Fränk A1 - Schmidt, Martin T1 - There's No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization T2 - Operations Research N2 - One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level's dual feasible set such that no bilevel-optimal solution is cut off. In practice, heuristics are often used to find a big-M although it is known that these approaches may fail. In this paper, we consider the hardness of two proxies for the above mentioned concept of a bilevel-correct big-M. First, we prove that verifying that a given big-M does not cut off any feasible vertex of the lower level's dual polyhedron cannot be done in polynomial time unless P=NP. Second, we show that verifying that a given big-M does not cut off any optimal point of the lower level's dual problem (for any point in the projection of the high-point relaxation onto the leader's decision space) is as hard as solving the original bilevel problem. KW - Bilevel optimization KW - Mathematical programs with complementarity constraints (MPCC) KW - Bounding polyhedra KW - Big-M KW - Hardness Y1 - 2019 IS - 68(6) SP - 1716 EP - 1721 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning JF - Discrete Optimization N2 - We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in [25] that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of [25] by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model. KW - Network design KW - Graph partitioning KW - Multilevel optimization KW - Mixed-integer optimization KW - Electricity market design Y1 - 2018 IS - 33 SP - 43 EP - 69 ER -