Quadratic Programming

Piecewise Parametric Structure in the Pooling Problem – from Sparse Strongly-Polynomial Solutions to NP-Hardness

  • Radu Baltean-Lugojan
  • Ruth Misener
    • Categories Combinatorial Optimization, Quadratic Programming Tags , , , , , ,

    The standard pooling problem is a NP-hard sub-class of non-convex quadratically-constrained optimization problems that commonly arises in process systems engineering applications. We take a parametric approach to uncovering topological structure and sparsity of the standard pooling problem in its p-formulation. The structure uncovered in this approach validates Professor Christodoulos A. Floudas’ intuition that pooling problems … Read more

    A fresh CP look at mixed-binary QPs: New formulations and relaxations

  • Immanuel M. Bomze
  • Peter J.C. Dickinson
  • Abdel Lisser
  • Jianqiang Cheng
    • Categories Constrained Nonlinear Optimization, Quadratic Programming Tags , , , , , ,

    Triggered by Burer’s seminal characterization from 2009, many copositive (CP) reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely )positive cones are replaced by tractable approximations. While the widely used approximation hierarchies have the disadvantage to use positive-semidefinite (psd) matrices of orders … Read more

    The complexity of simple models – a study of worst and typical hard cases for the Standard Quadratic Optimization Problem

  • Immanuel M. Bomze
  • Werner Schachinger
  • Reinhard Ullrich
    • Categories Game Theory, Global Optimization Theory, Quadratic Programming

    In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which are closely related to practical difficulties in solving StQPs globally. … Read more

    A Second-Order Cone Based Approach for Solving the Trust Region Subproblem and Its Variants

  • Nam Ho-Nguyen
  • Fatma Kılınç-Karzan
    • Categories Constrained Nonlinear Optimization, Quadratic Programming, Second-Order Cone Programming Tags , , , ,

    We study the trust region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone based approach to derive an exact … Read more

    A robust Lagrangian-DNN method for a class of quadratic optimization problems

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
    • Categories Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , ,

    The Lagrangian-doubly nonnegative (DNN) relaxation has recently been shown to provide effective lower bounds for a large class of nonconvex quadratic optimization problems (QOPs) using the bisection method combined with first-order methods by Kim, Kojima and Toh in 2016. While the bisection method has demonstrated the computational efficiency, determining the validity of a computed lower … Read more

    A Dual Gradient-Projection Method for Large-Scale Strictly Convex Quadratic Problems

  • Nicholas I. M. Gould
  • Daniel P. Robinson
    • Categories Quadratic Programming Tags , ,

    The details of a solver for minimizing a strictly convex quadratic objective function subject to general linear constraints is presented. The method uses a gradient projection algorithm enhanced with subspace acceleration to solve the bound-constrained dual optimization problem. Such gradient projection methods are well-known, but are typically employed to solve the primal problem when only … Read more

    Generalized Conjugate Gradient Methods for $\ell_1$ Regularized Convex Quadratic Programming with Finite Convergence

  • Xiaojun Chen
  • Zhaosong Lu
    • Categories Convex Optimization, Nonsmooth Optimization, Quadratic Programming Tags , , , ,

    The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper we propose some generalized CG (GCG) methods for solving the $\ell_1$-regularized (possibly not strongly) convex QP that terminate at an optimal solution in a finite number of iterations. At each iteration, our methods first … Read more

    Robust Sensitivity Analysis of the Optimal Value of Linear Programming

  • Samuel Burer
  • Guanglin Xu
    • Categories Quadratic Programming, Robust Optimization, Semi-definite Programming Tags , , , , ,

    We propose a framework for sensitivity analysis of linear programs (LPs) in minimiza- tion form, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modeled in a compact, convex uncertainty set. This framework unifies and extends multiple approaches for LP sensitivity analysis in the literature and has close ties … Read more

    The Uncapacitated Single Allocation p-Hub Median Problem with Stepwise Cost Function

  • Christoph Buchheim
  • Borzou Rostami
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences, Quadratic Programming Tags , ,

    In this paper, we address a new version of the Uncapacitated Single Allocation p-Hub Median Problem (USApHMP) in which transportation costs on each edge are given by piecewise constant cost functions. In the classical USApHMP, transportation costs are modelled as linear functions of the transport volume, where a fixed discount factor on hub-hub connections is … Read more

    Solving the Probabilistic Traveling Salesman Problem by Linearising a Quadratic Approximation

  • Uwe Clausen
  • J. Fabian Meier
    • Categories Integer Programming, Quadratic Programming, Transportation

    The Probabilistic Traveling Salesman Problem, introduced in 1985 by Jaillet, is one of the fundamental stochastic versions of the Traveling Salesman Problem: After the tour is chosen, each vertex is deleted with given probability 1-p. The eliminated vertices are bypassed which leads to shorter tours. The aim is to minimize the expected tour length. The … Read more