TY - INPR A1 - Grübel, Julia A1 - Huber, Olivier A1 - Hümbs, Lukas A1 - Klimm, Max A1 - Schmidt, Martin A1 - Schwartz, Alexandra T1 - Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium N2 - 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. KW - Energy markets KW - Nonconvex games KW - Existence KW - Equilibrium computation KW - Perfect competition Y1 - 2021 ER - TY - INPR A1 - Halbig, Katrin A1 - Hümbs, Lukas A1 - Rösel, Florian A1 - Schewe, Lars A1 - Weninger, Dieter T1 - Computing optimality certificates for convex mixed-integer nonlinear problems N2 - Every optimization problem has a corresponding verification problem which verifies whether a given optimal solution is in fact optimal. In the literature there are a lot of such ways to verify optimality for a given solution, e.g., the branch-and-bound tree. To simplify this task, Baes et al. introduced optimality certificates for convex mixed-integer nonlinear programs and proved that these are bounded in the number of integer variables. We introduce an algorithm to compute the certificates and conduct computational experiments. Through the experiments we show that the optimality certificates can be surprisingly small. Y1 - 2021 ER - TY - INPR A1 - Hümbs, Lukas A1 - Martin, Alexander A1 - Schewe, Lars T1 - Exploiting complete linear descriptions for decentralized power market problems with integralities N2 - It is well known that linear prices supporting a competitive equilibrium exist in the case of convex markets, however, in the presence of integralities this is open and hard to decide in general. We present necessary and sufficient conditions for the existence of such prices for decentralized market problems where market participants have integral decision variables and their feasible sets are given in complete linear description. We utilize total unimodularity and the aforementioned conditions to show that such linear prices exist and present some applications. Furthermore, we compute competitive equilibria for two classes of decentralized market problems arising in energy markets and show that competitive equilibria may exist regardless of integralities. KW - Mixed-integer programming KW - Competitive equilibrium KW - Linear prices KW - Total unimodularity Y1 - 2019 N1 - Former title: "On the existence of solutions to market equilibrium problems with integralities" ER -