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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.
Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve - both in theory and practice. Fortunately, there have been significant algorithmic advances in the field of bilevel optimization so that we can solve much larger and also more complicated problems today compared to what was possible to solve two decades ago. This results in more and more challenging bilevel problems that researchers try to solve today. This survey gives a detailed overview of one of these more challenging classes of bilevel problems: bilevel optimization under uncertainty. We review the classic ways of addressing uncertainties in bilevel optimization using stochastic or robust techniques. Moreover, we highlight that the sources of uncertainty in bilevel optimization are much richer than for usual, i.e., single-level, problems since not only the problem's data can be uncertain but also the (observation of the) decisions of the two players can be subject to uncertainty. We thus also review the field of bilevel optimization under limited observability, the area of problems considering only near-optimal decisions, and discuss intermediate solution concepts between the optimistic and pessimistic cases. Finally, we also review the rich literature on applications studied using uncertain bilevel problems such as in energy, for interdiction games and security applications, in management sciences, and networks.
Developing solution methods for discrete bilevel problems is known to be a challenging task - even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a Γ-robust approach, we present an extended formulation and a multi-follower formulation to model this type of problem. For both settings, we provide a generic branch-and-cut framework. Specifically, we investigate interdiction problems with a monotone Γ-robust follower and we derive problem-tailored cuts, which extend existing techniques that have been proposed for the deterministic case. For the Γ-robust knapsack interdiction problem, we computationally evaluate and compare the performance of the proposed algorithms for both modeling approaches.
Many applications of bilevel optimization contain a leader facing a follower whose reaction deviates from the one expected by the leader due to some kind of bounded rationality. We consider bilinear bilevel problems with follower's response uncertainty due to limited observability regarding the leader's decision and exploit robust optimization to model the decision making of the follower. We show that the robust counterpart of the lower level allows to tackle the problem via the lower level's KKT conditions.
These are lecture notes on bilevel optimization. The class of bilevel optimization problems is formally introduced and motivated using examples from different fields. Afterward, the main focus is on how to solve linear and mixed-integer linear bilevel optimization problems. To this end, we first consider various single-level reformulations of bilevel optimization problems with linear or convex follower problems, discuss geometric properties of linear bilevel problems, and study different algorithms for solving linear bilevel problems. Finally, we consider mixed-integer linear bilevel problems, discuss the main obstacles for deriving exact as well as effective solution methods, and derive a branch-and-bound method for solving these problems.
It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to ɛ-feasibility regarding its nonlinear constraints for an arbitrarily small but positive ɛ, the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater's constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of ɛ-feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.
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.
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.
We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important real-world problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the ROADEF challenge 2020 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem.
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.