(Mixed) Integer Nonlinear Programming

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

    A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function

  • Kevin C Furman
  • Ignacio E. Grossmann
  • Nicolas Sawaya
    • Categories (Mixed) Integer Nonlinear Programming

    Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete–continuous optimization problems. Often, a tight algebraic representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly … Read more

    Complete Description of Matching Polytopes with One Linearized Quadratic Term for Bipartite Graphs

  • Matthias Walter
    • Categories (Mixed) Integer Nonlinear Programming, Polyhedra Tags

    We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of matchings, extended by a binary number indicating whether the matching contains two specific edges. This polytope is associated to the quadratic matching problem with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein … 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

    An Adaptive Discretization MINLP Algorithm for Optimal Synthesis of Decentralized Energy Supply Systems

  • Björn Bahl
  • André Bardow
  • Arie M.C.A. Koster
  • Marco E. Lübbecke
  • Sebastian Goderbauer
  • Philip Voll
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering Tags , , , ,

    Decentralized energy supply systems (DESS) are highly integrated and complex systems designed to meet time-varying energy demands, e.g., heating, cooling, and electricity. The synthesis problem of DESS addresses combining various types of energy conversion units, choosing their sizing and operations to maximize an objective function, e.g., the net present value. In practice, investment costs and … Read more

    Three ideas for a Feasibility Pump for nonconvex MINLP

  • Pietro Belotti
  • Timo Berthold
    • Categories (Mixed) Integer Nonlinear Programming Tags , ,

    We describe an implementation of the Feasibility Pump heuristic for nonconvex MINLPs. Our implementation takes advantage of three novel techniques, which we discuss here: a hierarchy of procedures for obtaining an integer solution, a generalized definition of the distance function that takes into account the nonlinear character of the problem, and the insertion of linearization … Read more

    Minimization of Akaike’s Information Criterion in Linear Regression Analysis via Mixed Integer Nonlinear Program

  • Keiji Kimura
  • Hayato Waki
    • Categories (Mixed) Integer Nonlinear Programming Tags ,

    Akaike’s information criterion (AIC) is a measure of the quality of a statistical model for a given set of data. We can determine the best statistical model for a particular data set by the minimization of the AIC. Since we need to evaluate exponentially many candidates of the model by the minimization of the AIC, … Read more

    Ellipsoidal Mixed-Integer Representability

  • Alberto Del Pia
  • Jeffrey Poskin
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory

    Representability results for mixed-integer linear systems play a fundamental role in optimization since they give geometric characterizations of the feasible sets that can be formulated by mixed-integer linear programming. We consider a natural extension of mixed-integer linear systems obtained by adding just one ellipsoidal inequality. The set of points that can be described, possibly using … Read more

    On Decomposability of Multilinear Sets

  • Alberto Del Pia
  • Aida Khajavirad
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Polyhedra

    In this paper, we consider the Multilinear set defined as the set of binary points satisfying a collection of multilinear equations. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is … Read more

    On Approximation Algorithms for Concave Mixed-Integer Quadratic Programming

  • Alberto Del Pia
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory

    Concave Mixed-Integer Quadratic Programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an ε-approximate solution to a Concave Mixed-Integer Quadratic Programming problem. The running time of the proposed algorithm is polynomial in the size of the problem … Read more