bfgs

Preconditioned Barzilai-Borwein Methods for Multiobjective Optimization Problems

  • Jian Chen
    • Categories Multi-Criteria Optimization Tags , , ,

    Preconditioning is a powerful approach for solving ill-conditioned problems in optimization, where a preconditioning matrix is used to reduce the condition number and speed up the convergence of first-order method. Unfortunately, it is impossible to capture the curvature of all objective functions with a single preconditioning matrix in multiobjective optimization. Instead, second-order methods for multiobjective … Read more

    Global convergence of a BFGS-type algorithm for nonconvex multiobjective optimization problems

  • Leandro Fonseca Prudente
  • D. R. Souza
    • Categories Multi-Criteria Optimization Tags , , , , , ,

    We propose a modified BFGS algorithm for multiobjective optimization problems with global convergence, even in the absence of convexity assumptions on the objective functions. Furthermore, we establish the superlinear convergence of the method under usual conditions. Our approach employs Wolfe step sizes and ensures that the Hessian approximations are updated and corrected at each iteration … Read more

    A structured quasi-Newton algorithm for optimizing with incomplete Hessian information

  • Mihai Anitescu
  • Naiyuan Chiang
  • Cosmin G. Petra
    • Categories Unconstrained Optimization Tags , , ,

    We present a structured quasi-Newton algorithm for unconstrained optimization problems that have unavailable second-order derivatives or Hessian terms. We provide a formal derivation of the well-known BFGS secant update formula that approximates only the missing Hessian terms, and we propose a line-search quasi-Newton algorithm based on a modification of Wolfe conditions that converges to first-order … Read more

    Block BFGS Methods

  • Wenbo Gao
  • Donald Goldfarb
    • Categories Unconstrained Optimization Tags , , ,

    We introduce a quasi-Newton method with block updates called Block BFGS. We show that this method, performed with inexact Armijo-Wolfe line searches, converges globally and superlinearly under the same convexity assumptions as BFGS. We also show that Block BFGS is globally convergent to a stationary point when applied to non-convex functions with bounded Hessian, and … Read more

    Nonsmooth Optimization via BFGS

  • Adrian Lewis
  • Michael L. Overton
    • Categories Nonsmooth Optimization Tags , , , , ,

    We investigate the BFGS algorithm with an inexact line search when applied to nonsmooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, assuming only absolute continuity. We also prove that the line search terminates … Read more

    Behavior of BFGS with an Exact Line Search on Nonsmooth Examples

  • Adrian Lewis
  • Michael L. Overton
    • Categories Nonsmooth Optimization Tags , , , , ,

    We investigate the behavior of the BFGS algorithm with an exact line search on nonsmooth functions. We show that it may fail on a simple polyhedral example, but that it apparently always succeeds on the Euclidean norm function, spiraling into the origin with a Q-linear rate of convergence; we prove this in the case of … Read more

    Duality in quasi-newton methods and new variational characterizations of the DFP and BFGS updates

  • Osman Guler
  • Filiz Gurtuna
  • Olena Shevchenko
    • Categories Unconstrained Optimization Tags , , , ,

    It is known that quasi-Newton updates can be characterized by variational means, sometimes in more than one way. This paper has two main goals. We first formulate variational problems appearing in quasi-Newton methods within the space of symmetric matrices. This simplies both their formulations and their subsequent solutions. We then construct, for the first time, … Read more

    Variational Problems in Quasi-Newton Methods

  • Osman Guler
  • Filiz Gurtuna
  • Olena Shevchenko
    • Categories Unconstrained Optimization Tags , , , , ,

    It has been known since the early 1970s that the Hessian matrices in quasi-Newton methods can be updated by variational means, in several different ways. The usual formulation of these variational problems uses a coordinate system, and the symmetry of the Hessian matrices are enforced as explicit constraints. As a result, the variational problems seem … Read more

    iNEOS : An Interactive Environment for Nonlinear Optimization

  • Jean-Pierre Goux
  • Marcel Good
  • Jorge Nocedal
  • Victor Pereyra
    • Categories Nonlinear Optimization, Optimization of Systems modeled by PDEs, Problem Solving Environments Tags , , , , ,

    In this paper we describe iNEOS, an Internet-based environment which facilitates the solution of complex nonlinear optimization problems. It enables a user to easily invoke a remote optimization code without having to supply the model to be optimized. An interactive communication between client and server is established and maintainted using CORBA. We test the system … Read more