Stefania Bellavia

A preconditioning framework for sequences of diagonally modified linear systems arising in optimization

  • Stefania Bellavia
  • Valentina De Simone
  • Daniela di Serafino
  • Benedetta Morini
    • Categories Nonlinear Optimization Tags , , ,

    We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region … Read more

    Efficient preconditioner updates for shifted linear systems

  • Stefania Bellavia
  • Valentina De Simone
  • Daniela di Serafino
  • Benedetta Morini
    • Categories Nonlinear Systems and Least-Squares Tags , ,

    We present a new technique for building effective and low cost preconditioners for sequences of shifted linear systems (A+aI)x=b, where A is symmetric positive definite and a>0. This technique updates a preconditioner for A, available in the form of an LDL’ factorization, by modifying only the nonzero entries of the L factor in such a … Read more

    On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

  • Stefania Bellavia
  • Sandra Pieraccini
  • Maria Macconi
    • Categories Nonlinear Systems and Least-Squares Tags , , , , ,

    A class of trust-region methods for large scale bound-constrained systems of nonlinear equations is presented. The methods in this class follow the so called affine-scaling approach and can efficiently handle large scale problems. At each iteration, a suitably scaled region around the current approximate solution is defined and, within such a region, the norm of … Read more

    Constrained Dogleg Methods for nonlinear systems with simple bounds

  • Stefania Bellavia
  • Sandra Pieraccini
  • Maria Macconi
    • Categories Nonlinear Systems and Least-Squares Tags , , , ,

    We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is … Read more

    Regularization and Preconditioning of KKT Systems Arising in Nonnegative Least-Squares Problems

  • Stefania Bellavia
  • Jacek Gondzio
  • Benedetta Morini
    • Categories Bound-constrained Optimization Tags , , , ,

    A regularized Newton-like method for solving nonnegative least-squares problems is proposed and analysed in this paper. A preconditioner for KKT systems arising in the method is introduced and spectral properties of the preconditioned matrix are analysed. A bound on the condition number of the preconditioned matrix is provided. The bound does not depend on the … Read more

    An interior Newton-like method for nonnegative least-squares problems with degenerate solution

  • Stefania Bellavia
  • Benedetta Morini
  • Maria Macconi
    • Categories Nonlinear Optimization Tags , , , ,

    An interior point approach for medium and large nonnegative linear least-squares problems is proposed. Global and locally quadratic convergence is shown even if a degenerate solution is approached. Viable approaches for implementation are discussed and numerical results are provided. CitationTechnical Report 1/2005, Dipartimento di Energetica ‘S. Stecco’, Universita di Firenze, ItaliaArticleDownload View PDF

    Subspace trust-region methods for large bound-constrained nonlinear equations

  • Stefania Bellavia
  • Benedetta Morini
    • Categories Nonlinear Optimization Tags , , ,

    Trust-region methods for solving large bound-constrained nonlinear systems are considered. They allow for spherical or elliptical trust-regions where the search of an approximate solution is restricted to a low dimensional space. A general formulation for these methods is introduced and global and superlinear/quadratic convergence is shown under standard assumptions. Viable approaches for implementation in conjunction … Read more