Martin Schmidt

Mixed-Integer Quadratic Optimization and Iterative Clustering Techniques for Semi-Supervised Support Vector Machines

  • Martin Schmidt
  • Jan Pablo Burgard
  • Maria Eduarda Pinheiro
    • Categories (Mixed) Integer Nonlinear Programming, Data Science Algorithms, Data Science Applications Tags , , ,

    Among the most famous algorithms for solving classification problems are support vector machines (SVMs), which find a separating hyperplane for a set of labeled data points. In some applications, however, labels are only available for a subset of points. Furthermore, this subset can be non-representative, e.g., due to self-selection in a survey. Semi-supervised SVMs tackle … Read more

    A PDE-Constrained Generalized Nash Equilibrium Approach for Modeling Gas Markets with Transport

  • Veronika Grimm
  • Michael Hintermueller
  • Olivier Huber
  • Lars Schewe
  • Martin Schmidt
  • Gregor Zöttl
    • Categories Game Theory, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We investigate a class of generalized Nash equilibrium problems (GNEPs) in which the objectives of the individuals are interdependent and the shared constraint consists of a system of partial differential equations. This setup is motivated by the modeling of strategic interactions of competing firms, which explicitly take into account the dynamics of transporting a commodity, … Read more

    Gas Transport Network Optimization: PDE-Constrained Models

  • Falk M. Hante
  • Martin Schmidt
    • Categories Applications - Science and Engineering, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically … Read more

    Gas Transport Network Optimization: Mixed-Integer Nonlinear Models

  • Falk M. Hante
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering Tags , , ,

    Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia’s 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important … Read more

    A Brief Introduction to Robust Bilevel Optimization

  • Yasmine Beck
  • Ivana Ljubić
  • Martin Schmidt
    • Categories Robust Optimization Tags , ,

    Bilevel optimization is a powerful tool for modeling hierarchical decision making processes. However, the resulting problems are challenging to solve – both in theory and practice. Fortunately, there have been significant algorithmic advances in the field so that we can solve much larger and also more complicated problems today compared to what was possible to … Read more

    A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems

  • Richard Krug
  • Günter Leugering
  • Alexander Martin
  • Martin Schmidt
  • Dieter Weninger
    • Categories (Mixed) Integer Nonlinear Programming, Network Optimization, Optimization of Systems modeled by PDEs Tags , , , ,

    We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel … Read more

    An Exact Method for Nonlinear Network Flow Interdiction Problems

  • Martin Schmidt
  • Johannes Thürauf
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Network Optimization Tags , , ,

    We study network flow interdiction problems with nonlinear and nonconvex flow models. The resulting model is a max-min bilevel optimization problem in which the follower’s problem is nonlinear and nonconvex. In this game, the leader attacks a limited number of arcs with the goal to maximize the load shed and the follower aims at minimizing … Read more

    On the Relation Between Affinely Adjustable Robust Linear Complementarity and Mixed-Integer Linear Feasibility Problems

  • Christian Biefel
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, Complementarity and Variational Inequalities Tags , , ,

    We consider adjustable robust linear complementarity problems and extend the results of Biefel et al.~(2022) towards convex and compact uncertainty sets. Moreover, for the case of polyhedral uncertainty sets, we prove that computing an adjustable robust solution of a given linear complementarity problem is equivalent to solving a properly chosen mixed-integer linear feasibility problem. ArticleDownload … Read more

    A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

  • Julia Grübel
  • Richard Krug
  • Martin Schmidt
  • Winnifried Wollner
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Global Optimization Tags , , , ,

    We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate … Read more

    A Penalty Branch-and-Bound Method for Mixed-Integer Quadratic Bilevel Problems

  • Horländer Andreas
  • Martin Schmidt
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Complementarity and Variational Inequalities Tags , , ,

    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 … Read more