Integer Programming

On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

  • Matthias Köppe
  • Yuan Zhou
    • Categories (Mixed) Integer Linear Programming, Cutting Plane Approaches Tags , ,

    We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory–Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing … Read more

    A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

  • Natashia Boland
  • Brian Dandurand
  • Andrew C. Eberhard
  • Fabricio Oliveira
  • Jeffrey Christiansen
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Parallel Algorithms Tags , , , ,

    We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. The dual solution approach needs … Read more

    Special cases of the quadratic shortest path problem

  • Hao Hu
  • Renata Sotirov
    • Categories (Mixed) Integer Nonlinear Programming, Combinatorial Optimization, Network Optimization Tags , ,

    The quadratic shortest path problem (QSPP) is the problem of finding a path in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was … Read more

    Path Cover and Path Pack Inequalities for the Capacitated Fixed-Charge Network Flow Problem

  • Alper Atamturk
  • Simge Küçükyavuz
  • Birce Tezel
    • Categories (Mixed) Integer Linear Programming, Cutting Plane Approaches Tags , , ,

    Capacitated fixed-charge network flows are used to model a variety of problems in telecommunication, facility location, production planning and supply chain management. In this paper, we investigate capacitated path substructures and derive strong and easy-to-compute path cover and path pack inequalities. These inequalities are based on an explicit characterization of the submodular inequalities through a … Read more

    Global Solution Strategies for the Network-Constrained Unit Commitment Problem With AC Transmission Constraints

  • Anya Castillo
  • Carl D. Laird
  • Jean-Paul Watson
  • Jianfeng Liu
    • Categories Global Optimization, Integer Programming, Nonlinear Optimization Tags , ,

    We propose a novel global solution algorithm for the network-constrained unit commitment problem that incorporates a nonlinear alternating current model of the transmission network, which is a nonconvex mixed-integer nonlinear programming (MINLP) problem. Our algorithm is based on the multi-tree global optimization methodology, which iterates between a mixed-integer lower-bounding problem and a nonlinear upper-bounding problem. … Read more

    Generalized average shadow prices and bottlenecks

  • Alejandro Crema
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences Tags , , ,

    We present a generalization of the average shadow price in 0-1-Mixed Integer Linear Programming problems and its relation with bottlenecks including the analysis relative to the coefficients matrix of resource constraints. A mathematical programming approach to find the strategy for investment in resources is presented. Citation Escuela de Computación, Facultad de Ciencias, Universidad Central de … Read more

    The (not so) Trivial Lifting in Two Dimensions

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

    When generating cutting-planes for mixed-integer programs from multiple rows of the simplex tableau, the usual approach has been to relax the integrality of the non-basic variables, compute an intersection cut, then strengthen the cut coefficients corresponding to integral non-basic variables using the so-called trivial lifting procedure. Although of polynomial-time complexity in theory, this lifting procedure … Read more

    Extension Complexity Lower Bounds for Mixed-Integer Extended Formulations

  • Robert Hildebrand
  • Robert Weismantel
  • Rico Zenklusen
    • Categories (Mixed) Integer Linear Programming, Combinatorial Optimization Tags ,

    We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $\Omega(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical … Read more

    A Complete Characterization of Disjunctive Conic Cuts for Mixed Integer Second Order Cone Optimization

  • Pietro Belotti
  • Julio C. Góez
  • Imre Pólik
  • Ted K. Ralphs
  • Tamás Terlaky
    • Categories (Mixed) Integer Nonlinear Programming, Second-Order Cone Programming Tags , ,

    We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization problem. We extend our prior work on disjunctive conic cuts, which has thus far been restricted to the case in which the intersection of the hyperplanes and … Read more

    Lattice closures of polyhedra

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

    Given $P\subset\R^n$, a mixed-integer set $P^I=P\cap (\Z^{t}\times\R^{n-t}$), and a $k$-tuple of $n$-dimensional integral vectors $(\pi_1, \ldots, \pi_k)$ where the last $n-t$ entries of each vector is zero, we consider the relaxation of $P^I$ obtained by taking the convex hull of points $x$ in $P$ for which $ \pi_1^Tx,\ldots,\pi^T_kx$ are integral. We then define the $k$-dimensional … Read more