Nonlinear Systems and Least-Squares

Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints

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

    We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e.\ problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend or improve all known upper and lower complexity bounds … Read more

    Subset selection in sparse matrices

  • Alberto Del Pia
  • Santanu S. Dey
  • Robert Weismantel
    • Categories Global Optimization Theory, Nonlinear Systems and Least-Squares Tags , , ,

    In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using mainly tools from discrete geometry, we show that some sparsity conditions on the original data matrix allow … Read more

    Local convergence analysis of the Levenberg-Marquardt framework for nonzero-residue nonlinear least-squares problems under an error bound condition

  • Roger Behling
  • Douglas S. Gonçalves
  • Sandra Augusta Santos
    • Categories Nonlinear Systems and Least-Squares

    The Levenberg-Marquardt method (LM) is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares prob- lems. In this paper, we consider local convergence issues of the LM method when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full-rank of the Jacobian in a … Read more

    A stochastic Levenberg-Marquardt method using random models with complexity results and application to data assimilation

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

    Globally convergent variants of the Gauss-Newton algorithm are often the methods of choice to tackle nonlinear least-squares problems. Among such frameworks, Levenberg-Marquardt and trust-region methods are two well-established, similar paradigms. Both schemes have been studied when the Gauss-Newton model is replaced by a random model that is only accurate with a given probability. Trust-region schemes … Read more

    Quasi-Newton approaches to Interior Point Methods for quadratic problems

  • Jacek Gondzio
  • Francisco N. C. Sobral
    • Categories Nonlinear Systems and Least-Squares, Quadratic Programming Tags , , , ,

    Interior Point Methods (IPM) rely on the Newton method for solving systems of nonlinear equations. Solving the linear systems which arise from this approach is the most computationally expensive task of an interior point iteration. If, due to problem’s inner structure, there are special techniques for efficiently solving linear systems, IPMs enjoy fast convergence and … Read more

    On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems

  • Stefania Bellavia
  • Elisa Riccietti
    • Categories Nonlinear Systems and Least-Squares Tags , , , ,

    In this paper we address the stable numerical solution of ill-posed nonlinear least-squares problems with small residual. We propose an elliptical trust-region reformulation of a Levenberg-Marquardt procedure. Thanks to an appropriate choice of the trust-region radius, the proposed procedure guarantees an automatic choice of the free regularization parameters that, together with a suitable stopping criterion, … Read more

    Local attractors of newton-type methods for constrained equations and complementarity problems with nonisolated solutions

  • Andreas Fischer
  • Alexey F. Izmailov
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Nonlinear Systems and Least-Squares

    For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well-de ned … Read more

    Combining Multi-Level Real-time Iterations of Nonlinear Model Predictive Control to Realize Squatting Motions on Leo

  • Christian Kirches
  • Ivan Koryakovskiy
  • Manuel Kudruss
  • Katja Mombaur
  • Heike Vallery
    • Categories Constrained Nonlinear Optimization, Control Applications, Nonlinear Systems and Least-Squares Tags , , , , ,

    Today’s humanoid robots are complex mechanical systems with many degrees of freedom that are built to achieve locomotion skills comparable to humans. In order to synthesize whole-body motions, real-tme capable direct methods of optimal control are a subject of contemporary research. To this end, Nonlinear Model Predictive Control is the method of choice to realize … Read more

    A note on preconditioning weighted linear least squares, with consequences for weakly-constrained variational data assimilation

  • Serge Gratton
  • Philippe L. Toint
  • Selime Gürol
  • Ehouarn Simon
    • Categories Basic Sciences Applications, Nonlinear Systems and Least-Squares Tags , ,

    The effect of preconditioning linear weighted least-squares using an approximation of the model matrix is analyzed, showing the interplay of the eigenstructures of both the model and weighting matrices. A small example is given illustrating the resulting potential inefficiency of such preconditioners. Consequences of these results in the context of the weakly-constrained 4D-Var data assimilation … Read more

    On the use of the saddle formulation in weakly-constrained 4D-VAR data assimilation

  • Serge Gratton
  • Philippe L. Toint
  • Selime Gürol
  • Ehouarn Simon
    • Categories Basic Sciences Applications, Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , , , ,

    This paper discusses the practical use of the saddle variational formulation for the weakly-constrained 4D-VAR method in data assimilation. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationaly coherent variants of the algorithm … Read more