Martin Schmidt

The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities

  • Martin Schmidt
  • Mathias Sirvent
  • Winnifried Wollner
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Global Optimization Tags , , ,

    It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit … Read more

    Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning

  • Thomas Kleinert
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Applications - OR and Management Sciences, Global Optimization Tags , , , ,

    We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in [25] that does not include a network design problem. The two classical discrete optimization … Read more

    Robust Optimal Discrete Arc Sizing for Tree-Shaped Potential Networks

  • Martin Robinius
  • Lars Schewe
  • Martin Schmidt
  • Johannes Thürauf
  • Detlef Stolten
  • Lara Welder
    • Categories (Mixed) Integer Linear Programming, Applications - Science and Engineering, Robust Optimization Tags , , , , ,

    We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to … Read more

    Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization

  • Volker Mehrmann
  • Martin Schmidt
  • Jeroen J. Stolwijk
    • Categories Applications - Science and Engineering, Network Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and … Read more

    Uniqueness of Market Equilibria on Networks with Transport Costs

  • Vanessa Krebs
  • Martin Schmidt
    • Categories Applications - OR and Management Sciences, Convex and Nonsmooth Optimization, Network Optimization Tags , , , ,

    We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a … Read more

    Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches

  • Veronika Grimm
  • Julia Grübel
  • Lars Schewe
  • Martin Schmidt
  • Gregor Zöttl
    • Categories Applications - OR and Management Sciences, Finance and Economics, Nonlinear Optimization Tags , , , , ,

    This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a … Read more

    Nonoverlapping Domain Decomposition for Optimal Control Problems governed by Semilinear Models for Gas Flow in Networks

  • Günter Leugering
  • Martin Schmidt
  • Mathias Sirvent
  • Alexander Martin
    • Categories Applications - Science and Engineering, Systems governed by Differential Equations Optimization Tags , , , ,

    We consider optimal control problems for gas flow in pipeline networks. The equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order … Read more

    Uniqueness and Multiplicity of Market Equilibria on DC Power Flow Networks

  • Vanessa Krebs
  • Lars Schewe
  • Martin Schmidt
    • Categories Applications - OR and Management Sciences, Convex Optimization, Network Optimization Tags , , , , ,

    We consider uniqueness and multiplicity of market equilibria in a short-run setup where traded quantities of electricity are transported through a capacitated network in which power flows have to satisfy the classical lossless DC approximation. The firms face fluctuating demand and decide on their production, which is constrained by given capacities. Today, uniqueness of such … Read more

    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