Universität Trier
We propose an algorithm for solving bilevel problems with mixed-integer convex-quadratic upper level as well as convex-quadratic and continuous lower level. The method is based on a classic branch-and-bound procedure, where branching is performed on the integer constraints and on the complementarity constraints resulting from the KKT reformulation of the lower-level problem. However, instead of branching on constraints as usual, suitably chosen penalty terms are added to the objective function in order to create new subproblems in the tree. We prove the correctness of the method and present its applicability by some first numerical results.
Mixed complementarity problems are of great importance in practice since they appear in various fields of applications like energy markets, optimal stopping, or traffic equilibrium problems. However, they are also very challenging due to their inherent, nonconvex structure. In addition, recent applications require the incorporation of integrality constraints. Since complementarity problems often model some kind of equilibrium, these recent applications ask for equilibrium points that additionally satisfy certain integer conditions. Obviously, this makes the problem even harder to solve. The solution approach used most frequently in the literature is to recast the complementarity conditions as disjunctive constraints using additional binary variables and big-M constraints. However, both latter aspects create issues regarding the tractability and correctness of the reformulation. In this paper, we follow the opposite route and restate the integrality conditions as complementarity constraints, leading to purely continuous reformulations that can be tackled by local solvers. We study these reformulations theoretically and provide a numerical study that shows that continuous reformulations are useful in practice both in terms of solution times and solution quality.
We compare various flexible tariffs that have been proposed to cost-effectively govern a prosumer's electricity management - in particular time-of-use (TOU), critical-peak-pricing (CPP), and a real-time-pricing tariff (RTP). As the outside option, we consider a fixed-price tariff (FP) that restricts the specific characteristics of TOU, CPP, and RTP, so that the flexible tariffs are at least as profitable for the prosumer as the fixed-price tariff. We propose bilevel models to determine the optimal interplay between the retailer's tariff design and the prosumer's decisions on using the storage, on consumption, and on electricity purchases from as well as electricity sales to the grid. The single-level reformulations of the considered bilevel models are computationally highly challenging optimization problems since they, e.g., combine bilinearities and mixed-integer aspects for modeling certain tariff structures. Based on a computational study using real-world data, we find that RTP increases retailer profits, however, leads to the largest price volatility for the prosumer. TOU and CPP only yield mild additional retailer profits and, due to the multiplicity of optimal plans on the part of the prosumer, imply uncertain revenues for the retailer.
Portfolio optimization is an ongoing hot topic of mathematical optimization and management science. Due to the current financial market environment with low interest rates and volatile stock markets, it is getting more and more important to extend portfolio optimization models by other types of investments than classical assets. In this paper, we present a mixed-integer multistage stochastic model that includes investment opportunities in irreversible and long-term infrastructure projects in the context of renewable energies, which are also subject to policy risk. On realistic time scales for investment problems of this type, the resulting instances are by far too large to be solved with today's most evolved optimization software. Thus, we present a tailored moving-horizon approach together with suitable approximations and simplifications of the model. We evaluate these approximations and simplifications in a computational sensitivity analysis and derive a final model that can be tackled on a realistic instance by our moving-horizon approach.
Mixed-integer supply chain models typically are very large but are also very sparse and can be decomposed into loosely coupled blocks. In this paper, we use general-purpose techniques to obtain a block decomposition of supply chain instances and apply a tailored penalty alternating direction method, which exploits the structural properties of the decomposed instances. We further describe problem-specific enhancements of the algorithm and present numerical results on real-world instances that illustrate the applicability of the approach.