Nonlinear Systems and Least-Squares

Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations

  • Ernesto G. Birgin
  • José Mario Martínez
    • Categories Nonlinear Systems and Least-Squares Tags , , ,

    Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. … Read more

    Strong Evaluation Complexity of An Inexact Trust-Region Algorithm for Arbitrary-Order Unconstrained Nonconvex Optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , ,

    A trust-region algorithm using inexact function and derivatives values is introduced for solving unconstrained smooth optimization problems. This algorithm uses high-order Taylor models and allows the search of strong approximate minimizers of arbitrary order. The evaluation complexity of finding a $q$-th approximate minimizer using this algorithm is then shown, under standard conditions, to be $\mathcal{O}\big(\min_{j\in\{1,\ldots,q\}}\epsilon_j^{-(q+1)}\big)$ … Read more

    Spectral Residual Method for Nonlinear Equations on Riemannian Manifolds

  • Hugo Lara
  • Harry Oviedo
    • Categories Nonlinear Systems and Least-Squares Tags , , , ,

    In this paper, the spectral algorithm for nonlinear equations (SANE) is adapted to the problem of finding a zero of a given tangent vector field on a Riemannian manifold. The generalized version of SANE uses, in a systematic way, the tangent vector field as a search direction and a continuous real–valued function that adapts this … Read more

    On the abs-polynomial expansion of piecewise smooth functions

  • Andreas Griewank
  • Caren Tischendorf
  • Tom Streubel
    • Categories Nonlinear Systems and Least-Squares, Nonsmooth Optimization

    Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\bd \!- \!1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|\cdot|$ apart from arithmetic operations and $\bd$ … Read more

    A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Least-Squares Problems

  • Elhoucine Bergou
  • Youssef Diouane
  • Vyacheslav Kungurtsev
  • Clément W. Royer
    • Categories Constrained Nonlinear Optimization, Nonlinear Systems and Least-Squares

    Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty … Read more

    Solving nonlinear systems of equations via spectral residual methods: stepsize selection and applications

  • Enrico Meli
  • Benedetta Morini
  • Margherita Porcelli
  • Cristina Sgattoni
    • Categories Mechanical Engineering, Nonlinear Systems and Least-Squares

    Spectral residual methods are derivative-free and low-cost per iteration procedures for solving nonlinear systems of equations. They are generally coupled with a nonmonotone linesearch strategy and compare well with Newton-based methods for large nonlinear systems and sequences of nonlinear systems. The residual vector is used as the search direction and choosing the steplength has a … Read more

    A robust method based on LOVO functions for solving least squares problems

  • E. V. Castelani
  • R. Lopes
  • W. V. I. Shirabayashi
  • Francisco N. C. Sobral
    • Categories Nonlinear Systems and Least-Squares, Nonsmooth Optimization, Parallel Algorithms Tags , , ,

    The robust adjustment of nonlinear models to data is considered in this paper. When data comes from real experiments, it is possible that measurement errors cause the appearance of discrepant values, which should be ignored when adjusting models to them. This work presents a Lower Order-value Optimization (LOVO) version of the Levenberg-Marquardt algorithm, which is … Read more

    Continuous selections of solutions for locally Lipschitzian equations

  • A.V. Arutyunov
  • Alexey F. Izmailov
  • S.E. Zhukovskiy
    • Categories Complementarity and Variational Inequalities, Nonlinear Systems and Least-Squares, Nonsmooth Optimization

    This paper answers in affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given … Read more

    A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion

  • Stefania Bellavia
  • Jacek Gondzio
  • Margherita Porcelli
    • Categories Nonlinear Systems and Least-Squares, Semi-definite Programming Tags , , ,

    A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) … Read more

    Hybrid methods for nonlinear least squares problems

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , , , , , ,

    This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function $F(x) = (1/2) f^T(x) f(x)$, where $x \in R^n$ and $f \in R^m$, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved … Read more