Martin Schmidt

Mixed-Integer Bilevel Optimization with Nonconvex Quadratic Lower-Level Problems: Complexity and a Solution Method

  • Immanuel M. Bomze
  • Martin Schmidt
  • Horländer Andreas
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming Tags , , ,

    We study bilevel problems with a convex quadratic mixed-integer upper-level, integer linking variables, and a nonconvex quadratic, purely continuous lower-level problem. We prove $\Sigma_p^2$-hardness of this class of problems, derive an iterative lower- and upper-bounding scheme, and show its finiteness and correctness in the sense that it computes globally optimal points or proves infeasibility of … Read more

    Computational Methods for the Household Assignment Problem

  • Ulf Friedrich
  • Lucas Moschen
  • Ralf Münnich
  • Martin Schmidt
    • Categories Applications - OR and Management Sciences, Approximation Algorithms, Integer Programming Tags , , , ,

    We consider the problem of assigning the entries of a household data set to real-world address data. This household assignment problem occurs in the geo-referencing step of spatial microsimulation models. The resulting combinatorial optimization model is a maximum weight matching problem with additional side constraints. Even for real-world instances of medium size, such as the … Read more

    BOBILib: Bilevel Optimization (Benchmark) Instance Library

  • Johannes Thürauf
  • Thomas Kleinert
  • Ivana Ljubić
  • Ted K. Ralphs
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, Bilevel Optimization, Optimization Software Benchmark Tags , , , ,

    In this report, we present the BOBILib, a collection of more than 2600 instances of mixed integer bilevel linear optimization problems (MIBLPs). The goal of this library is to provide a large and well-curated set of test instances freely available for the research community so that new and existing algorithms in bilevel optimization can be … Read more

    Exact Augmented Lagrangian Duality for Nonconvex Mixed-Integer Nonlinear Optimization

  • Henri Lefebvre
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming Tags , ,

    In the context of mixed-integer nonlinear problems (MINLPs), it is well-known that strong duality does not hold in general if the standard Lagrangian dual is used. Hence, we consider the augmented Lagrangian dual (ALD), which adds a nonlinear penalty function to the classic Lagrangian function. For this setup, we study conditions under which the ALD … Read more

    Toll Setting with Robust Wardrop Equilibrium Conditions Under Budgeted Uncertainty

  • Yasmine Beck
  • Martine Labbé
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Robust Optimization, Transportation Tags , , , ,

    We consider two variants of the toll-setting problem in which a traffic authority uses tolls either to maximize revenue or to alleviate bottlenecks in the traffic network. The users of the network are assumed to act according to Wardrop’s user equilibrium so that the overall toll-setting problems are modeled as mathematical problems with equilibrium constraints. … Read more

    Stabilizing GNEP-Based Model Predictive Control: Quasi-GNEPs and End Constraints

  • Falk M. Hante
  • Martin Schmidt
  • Antonia Topalovic
    • Categories Other Topics Tags , , ,

    We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP) in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the … Read more

    Mixed-Integer Linear Optimization for Cardinality-Constrained Random Forests

  • Jan Pablo Burgard
  • Maria Eduarda Pinheiro
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, Optimization in Data Science Tags , , , ,

    Random forests are among the most famous algorithms for solving classification problems, in particular for large-scale data sets. Considering a set of labeled points and several decision trees, the method takes the majority vote to classify a new given point. In some scenarios, however, labels are only accessible for a proper subset of the given … Read more

    Heuristic Methods for Γ-Robust Mixed-Integer Linear Bilevel Problems

  • Yasmine Beck
  • Ivana Ljubić
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, Robust Optimization Tags , , ,

    Due to their nested structure, bilevel problems are intrinsically hard to solve–even if all variables are continuous and all parameters of the problem are exactly known. In this paper, we study mixed-integer linear bilevel problems with lower-level objective uncertainty, which we address using the notion of Γ-robustness. To tackle the Γ-robust counterpart of the bilevel … Read more

    Adjustable Robust Nonlinear Network Design Without Controllable Elements under Load Scenario Uncertainties

  • Johannes Thürauf
  • Julia Grübel
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Energy, Robust Optimization Tags , , , ,

    We study network design problems for nonlinear and nonconvex flow models without controllable elements under load scenario uncertainties, i.e., under uncertain injections and withdrawals. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, load scenarios within a given … Read more

    Using Disjunctive Cuts in a Branch-and-Cut Method to Solve Convex Integer Nonlinear Bilevel Problems

  • Horländer Andreas
  • Ivana Ljubić
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Bilevel Optimization, Cutting Plane Approaches Tags , , ,

    We present a branch-and-cut method for solving convex integer nonlinear bilevel problems, i.e., bilevel models with nonlinear but jointly convex objective functions and constraints in both the upper and the lower level. To this end, we generalize the idea of using disjunctive cuts to cut off integer-feasible but bilevel-infeasible points. These cuts can be obtained … Read more