Martin Schmidt

Complementarity-Based Nonlinear Programming Techniques for Optimal Mixing in Gas Networks

  • Falk M. Hante
  • Martin Schmidt
    • Categories Applications - Science and Engineering, Complementarity and Variational Inequalities, Nonlinear Optimization Tags , , , ,

    We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we … Read more

    Algorithmic Results for Potential-Based Flows: Easy and Hard Cases

  • Martin Gross
  • Marc E. Pfetsch
  • Lars Schewe
  • Martin Schmidt
  • Martin Skutella
    • Categories Applications - Science and Engineering, Integer Programming, Network Optimization Tags , , , , ,

    Potential-based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize … Read more

    A Decomposition Method for MINLPs with Lipschitz Continuous Nonlinearities

  • Martin Schmidt
  • Mathias Sirvent
  • Winnifried Wollner
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Systems governed by Differential Equations Optimization Tags , , , ,

    Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test … Read more

    A Multilevel Model of the European Entry-Exit Gas Market

  • Veronika Grimm
  • Lars Schewe
  • Martin Schmidt
  • Gregor Zöttl
    • Categories (Mixed) Integer Nonlinear Programming, Applications - OR and Management Sciences Tags , ,

    In entry-exit gas markets as they are currently implemented in Europe, network constraints do not affect market interaction beyond the technical capacities determined by the TSO that restrict the quantities individual firms can trade at the market. It is an up to now unanswered question to what extent existing network capacity remains unused in an … Read more

    MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems

  • Martin Gugat
  • Günter Leugering
  • Martin Schmidt
  • Mathias Sirvent
  • David Wintergerst
  • Alexander Martin
    • Categories (Mixed) Integer Nonlinear Programming, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , ,

    We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations … Read more

    Optimization Techniques for Tree-Structured Nonlinear Problems

  • Jens Hübner
  • Martin Schmidt
  • Marc C. Steinbach
    • Categories Nonlinear Optimization, Stochastic Programming Tags , , , ,

    Robust model predictive control approaches and other applications lead to nonlinear optimization problems defined on (scenario) trees. We present structure-preserving Quasi-Newton update formulas as well as structured inertia correction techniques that allow to solve these problems by interior-point methods with specialized KKT solvers for tree-structured optimization problems. The same type of KKT solvers could be … Read more

    Optimal Price Zones of Electricity Markets: A Mixed-Integer Multilevel Model and Global Solution Approaches

  • Veronika Grimm
  • Frauke Liers
  • Martin Schmidt
  • Thomas Kleinert
  • Gregor Zöttl
    • Categories (Mixed) Integer Nonlinear Programming, Applications - OR and Management Sciences, Global Optimization Tags , , , ,

    Mathematical modeling of market design issues in liberalized electricity markets often leads to mixed-integer nonlinear multilevel optimization problems for which no general-purpose solvers exist and which are intractable in general. In this work, we consider the problem of splitting a market area into a given number of price zones such that the resulting market design … Read more

    Computing Feasible Points for Binary MINLPs with MPECs

  • Lars Schewe
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Complementarity and Variational Inequalities, Integer Programming Tags , , ,

    Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use … Read more

    Towards Simulation Based Mixed-Integer Optimization with Differential Equations

  • Martin Gugat
  • Günter Leugering
  • Martin Schmidt
  • Mathias Sirvent
  • David Wintergerst
  • Alexander Martin
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Systems governed by Differential Equations Optimization Tags , , , ,

    We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinearity. We prove that … Read more

    Solving Highly Detailed Gas Transport MINLPs: Block Separability and Penalty Alternating Direction Methods

  • Björn Geißler
  • Antonio Morsi
  • Lars Schewe
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , , ,

    Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network as well as accurate physical and technical modeling is required in order to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for … Read more