cutting planes

Gaining traction – On the convergence of an inner approximation scheme for probability maximization

  • Csaba I. Fábián
    • Categories Convex Optimization, Stochastic Programming Tags ,

    We analyze an inner approximation scheme for probability maximization. The approach was proposed in Fabian, Csizmas, Drenyovszki, Van Ackooij, Vajnai, Kovacs, Szantai (2018) Probability maximization by inner approximation, Acta Polytechnica Hungarica 15:105-125, as an analogue of a classic dual approach in the handling of probabilistic constraints. Even a basic implementation of the maximization scheme proved … Read more

    Multi-Row Intersection Cuts based on the Infinity Norm

  • Ricardo Fukasawa
  • Laurent Poirrier
  • Alinson S. Xavier
    • Categories (Mixed) Integer Linear Programming Tags , ,

    When generating multi-row intersection cuts for Mixed-Integer Linear Optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet-defining for the corner relaxation, the number of potential candidates is still very large, specially for instances of large size. In this paper, we introduce a subset … Read more

    Generalized Chvatal-Gomory closures for integer programs with bounds on variables

  • Sanjeeb Dash
  • Oktay Gunluk
  • Dabeen Lee
    • Categories (Mixed) Integer Linear Programming, Cutting Plane Approaches Tags , , ,

    Integer programming problems that arise in practice often involve decision variables with one or two sided bounds. In this paper, we consider a generalization of Chvatal-Gomory inequalities obtained by strengthening Chvatal-Gomory inequalities using the bounds on the variables. We prove that the closure of a rational polyhedron obtained after applying the generalized Chvatal-Gomory inequalities is … Read more

    On the Relation between the Extended Supporting Hyperplane Algorithm and Kelley’s Cutting Plane Algorithm

  • Ambros Gleixner
  • Felipe Serrano
  • Robert Schwarz
    • Categories (Mixed) Integer Nonlinear Programming, Convex Optimization Tags , ,

    Recently, Kronqvist et al.rediscovered the supporting hyperplane algorithm of Veinott and demonstrated its computational benefits for solving convex mixed-integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley’s cutting plane algorithm applied to a particular … Read more

    On the depth of cutting planes

  • Laurent Poirrier
  • James Yu
    • Categories (Mixed) Integer Linear Programming, Branch and Cut Algorithms, Cutting Plane Approaches Tags , ,

    We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical strength of cutting planes. In particular, we show that its value lies … Read more

    Generating feasible points for mixed-integer convex optimization problems by inner parallel cuts

  • Christoph Neumann
  • Oliver Stein
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches Tags , , ,

    In this article we introduce an inner parallel cutting plane method (IPCP) to compute good feasible points along with valid cutting planes for mixed-integer convex optimization problems. The method iteratively generates polyhedral outer approximations of an enlarged inner parallel set (EIPS) of the continuously relaxed feasible set. This EIPS possesses the crucial property that any … Read more

    Scoring positive semidefinite cutting planes for quadratic optimization via trained neural networks

  • Radu Baltean-Lugojan
  • Pierre Bonami
  • Ruth Misener
  • Andrea Tramontani
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Quadratic Programming Tags , , , , ,

    Semidefinite programming relaxations complement polyhedral relaxations for quadratic optimization, but global optimization solvers built on polyhedral relaxations cannot fully exploit this advantage. This paper develops linear outer-approximations of semidefinite constraints that can be effectively integrated into global solvers. The difference from previous work is that our proposed cuts are (i) sparser with respect to the … Read more

    Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

  • Robert Hildebrand
  • Matthias Köppe
  • Yuan Zhou
    • Categories Cutting Plane Approaches Tags , ,

    In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The … Read more

    Inexact cutting planes for two-stage mixed-integer stochastic programs

  • Ward Romeijnders
  • Niels van der Laan
    • Categories Cutting Plane Approaches, Stochastic Programming Tags , , ,

    We propose a novel way of applying cutting plane techniques to two-stage mixed-integer stochastic programs. Instead of using cutting planes that are always valid, our idea is to apply inexact cutting planes to the second-stage feasible regions that may cut away feasible integer second-stage solutions for some scenarios and may be overly conservative for others. … Read more

    A Branch-and-Cut Algorithm for Solving Mixed-integer Semidefinite Optimization Problems

  • Ken Kobayashi
  • Yuichi Takano
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Semi-definite Programming Tags , , ,

    This paper is concerned with a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite constraint is relaxed, and the resultant mixed-integer linear optimization problem is repeatedly solved with valid inequalities for the relaxed constraint. We prove convergence properties of the algorithm. Moreover, to speed up the computation, we … Read more