convex quadratic programming

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

    Active-Set Methods for Convex Quadratic Programming

  • Anders Forsgren
  • Philip E. Gill
  • Elizabeth Wong
    • Categories Convex Optimization, Quadratic Programming Tags , , , , , ,

    Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate … Read more

    Globally Convergent Primal-Dual Active-Set Methods with Inexact Subproblem Solves

  • Frank E. Curtis
  • Zheng Han
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Quadratic Programming Tags , , , , ,

    We propose primal-dual active-set (PDAS) methods for solving large-scale instances of an important class of convex quadratic optimization problems (QPs). The iterates of the algorithms are partitions of the index set of variables, where corresponding to each partition there exist unique primal-dual variables that can be obtained by solving a (reduced) linear system. Algorithms of … Read more

    How the augmented Lagrangian algorithm can deal with an infeasible convex quadratic optimization problem

  • Alice Chiche
  • Jean Charles Gilbert
    • Categories Convex and Nonsmooth Optimization, Quadratic Programming Tags , , , , , , , , ,

    This paper analyses the behavior of the augmented Lagrangian algorithm when it deals with an infeasible convex quadratic optimization problem. It is shown that the algorithm finds a point that, on the one hand, satisfies the constraints shifted by the smallest possible shift that makes them feasible and, on the other hand, minimizes the objective … Read more

    A comparison of reduced and unreduced KKT systems arising from Interior Point methods

  • Benedetta Morini
  • Valeria Simoncini
  • Mattia Tani
    • Categories Quadratic Programming Tags , , ,

    We address the iterative solution of symmetric KKT systems arising in the solution of convex quadratic programming problems. Two strictly related and well established formulations for such systems are studied with particular emphasis on the effect of preconditioning strategies on their relation. Constraint and augmented preconditioners are considered, and the choice of the augmentation Matrix … Read more

    Spectral estimates for unreduced symmetric KKT systems arising from Interior Point methods

  • Benedetta Morini
  • Valeria Simoncini
  • Mattia Tani
    • Categories Quadratic Programming Tags , , , ,

    We consider symmetrized KKT systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods. Their coefficient matrices usually have 3×3 block structure and, under suitable conditions on both the quadratic programming problem and the solution, they are nonsingular in the limit. We present new spectral estimates for these … Read more

    Active Set Methods with Reoptimization for Convex Quadratic Integer Programming

  • Christoph Buchheim
  • Long Trieu
    • Categories (Mixed) Integer Nonlinear Programming, Convex Optimization, Quadratic Programming Tags , ,

    We present a fast branch-and-bound algorithm for solving convex quadratic integer programs with few linear constraints. In each node, we solve the dual problem of the continuous relaxation using an infeasible active set method proposed by Kunisch and Rendl to get a lower bound; this active set algorithm is well suited for reoptimization. Our algorithm … Read more

    Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections

  • Stefania Bellavia
  • Valentina De Simone
  • Daniela di Serafino
  • Benedetta Morini
    • Categories Quadratic Programming Tags , , , ,

    This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective … Read more

    A Globally Convergent Primal-Dual Active-Set Framework for Large-Scale Convex Quadratic Optimization

  • Frank E. Curtis
  • Daniel P. Robinson
  • Zheng Han
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    We present a primal-dual active-set framework for solving large-scale convex quadratic optimization problems (QPs). In contrast to classical active-set methods, our framework allows for multiple simultaneous changes in the active- set estimate, which often leads to rapid identification of the optimal active-set regardless of the initial estimate. The iterates of our framework are the active-set … Read more

    Primal-dual relationship between Levenberg-Marquardt and central trajectories for linearly constrained convex optimization

  • Roger Behling
  • Clóvis Caesar Gonzaga
  • Gabriel Haeser
    • Categories Convex Optimization, Quadratic Programming Tags , , , , , ,

    We consider the minimization of a convex function on a compact polyhedron defined by linear equality constraints and nonnegative variables. We define the Levenberg-Marquardt (L-M) and central trajectories starting at the analytic center and using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of … Read more