Abstract Bilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an $$ \varepsilon $$ ε -approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error $$ \varepsilon $$ ε , univariate $$ \varepsilon $$ ε -approximations require fewer simplices than bivariate $$ \varepsilon $$ ε -approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation.

Abstract We introduce EETTlib, an instance library for the Energy‐Efficient Train Timetabling problem. The task in this problem is to adjust a given timetable draft such that the energy consumption of the resulting railway traffic is minimized. To this end, the departure times of the trains can be slightly, and their velocity profiles on each trip can be modified. We provide real‐world data originating from two research projects in this field, one with Deutsche Bahn AG, the most important railway company in Germany, the other with VAG Verkehrs‐Aktiengesellschaft, the operator of public transport in the city of Nürnberg, Germany. In both cases, our library contains representative data on the relevant operational constraints and supports various possible choices for the objective function with respect to energy‐efficiency. The resulting benchmark instances can be used by the scheduling and timetabling community to improve their models and algorithms. They are available under https://www.eettlib.fau.de .

In this paper, we demonstrate how to learn the objective function of a decision maker while only observing the problem input data and the decision maker’s corresponding decisions over multiple rounds. Our approach is based on online learning techniques and works for linear objectives over arbitrary sets for which we have a linear optimization oracle and as such generalizes previous work based on KKT-system decomposition and dualization approaches. The applicability of our framework for learning linear constraints is also discussed briefly. Our algorithm converges at a rate of O(1/sqrt(T)), and we demonstrate its effectiveness and applications in preliminary computational results.