Integer Programming

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

    A random search method for finding ‘K ≥ 2’ number of ranked optimal solution to an assignment problem

  • Ali Al-Hasani
  • Andrew C. Eberhard
  • Masar Al-Rabeeah
  • Santosh Kumar
    • Categories 0-1 Programming, Integer Programming

    A need for an optimal solution for a given mathematical model is well known and many solution approaches have been developed to identify efficiently an optimal solution in a given situation. For example, one such class of mathematical models with industrial applications have been classified as mathematical programming models (MPM). The main idea behind these … Read more

    Chvatal rank in binary polynomial optimization

  • Alberto Del Pia
  • Silvia Di Gregorio
    • Categories 0-1 Programming, Cutting Plane Approaches, Global Optimization Theory Tags , , , , ,

    Recently, several classes of cutting planes have been introduced for binary polynomial optimization. In this paper, we present the first results connecting the combinatorial structure of these inequalities with their Chvatal rank. We show that almost all known cutting planes have Chvatal rank 1. All these inequalities have an associated hypergraph that is beta-acyclic, thus, … Read more

    Strong IP Formulations Need Large Coefficients

  • Christopher Hojny
    • Categories 0-1 Programming, Integer Programming Tags , , ,

    The development of practically well-behaving integer programming formulations is an important aspect of solving linear optimization problems over a set $X \subseteq \{0,1\}^n$. In practice, one is often interested in strong integer formulations with additional properties, e.g., bounded coefficients to avoid numerical instabilities. This article presents a lower bound on the size of coefficients in … Read more

    Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

  • Hongbo Dong
  • Yunqi Luo
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Second-Order Cone Programming

    Decades of advances in mixed-integer linear programming (MILP) and recent development in mixed-integer second-order-cone programming (MISOCP) have translated very mildly to progresses in global solving nonconvex mixed-integer quadratically constrained programs (MIQCP). In this paper we propose a new approach, namely Compact Disjunctive Approximation (CDA), to approximate nonconvex MIQCP to arbitrary precision by convex MIQCPs, which … Read more

    A branch and price algorithm for the resource constrained home health care vehicle routing problem

  • Neda Tanoumand
  • Tonguç Ünlüyurt
    • Categories Integer Programming, Linear Programming Tags , , ,

    We consider the vehicle routing problem with resource constraints motivated by a home health care application. We propose a branch and price algorithm to solve the problem. In our problem, we consider different types of patients that require a nurse or a health aid or both. The patients can be serviced by the appropriate vehicles … 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

    Submodularity and valid inequalities in nonlinear optimization with indicator variables

  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches Tags , , ,

    We propose a new class of valid inequalities for mixed-integer nonlinear optimization problems with indicator variables. The inequalities are obtained by lifting polymatroid inequalities in the space of the 0–1 variables into conic inequalities in the original space of variables. The proposed inequalities are shown to describe the convex hull of the set under study … Read more

    A Tutorial on Formulating QUBO Models

  • Fred Glover
  • Gary Kochenberger
    • Categories 0-1 Programming, Combinatorial Optimization, Meta Heuristics Tags ,

    The field of Combinatorial Optimization (CO) is one of the most important areas in the general field of optimization, with important applications found in every industry, including both the private and public sectors. It is also one of the most active research areas pursued by the research communities of Operations Research, Computer Science, and Analytics … Read more

    Strong Mixed-Integer Formulations for Power System Islanding and Restoration

  • Ignacio Aravena
  • Shmuel Oren
  • Deepak Rajan
  • Georgios Patsakis
    • Categories Branch and Cut Algorithms, Integer Programming, Polyhedra Tags , , , , ,

    The Intentional Controlled Islanding (ICI) and the Black Start Allocation (BSA) are two examples of problems in the power systems literature that have been formulated as Mixed Integer Programs (MIPs). A key consideration in both of these problems is that each island must have at least one energized generator. In this paper, we provide three … Read more