Nonlinear Systems and Least-Squares

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

    New quasi-Newton method for solving systems of nonlinear equations

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

    In this report, we propose the new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but it is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires $O(n^2)$ … Read more

    A Levenberg-Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients

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

    In this paper we consider large scale nonlinear least-squares problems for which function and gradient are evaluated with dynamic accuracy and propose a Levenberg-Marquardt method for solving such problems. More precisely, we consider the case in which the exact function to optimize is not available or its evaluation is computationally demanding, but ap- proximations of … Read more

    Optimality of orders one to three and beyond: characterization and evaluation complexity in constrained nonconvex optimization

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

    Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second- and third-order criticality and its evaluation complexity is analyzed as a function of the choice (among existing methods) of an inner algorithm for solving subproblems in … Read more

    Quasi-Newton methods for constrained nonlinear systems: complexity analysis and application

  • Leopoldo Marini
  • Benedetta Morini
  • Margherita Porcelli
    • Categories Nonlinear Optimization, Nonlinear Systems and Least-Squares

    We address the solution of convex constrained nonlinear systems by new linesearch Quasi-Newton methods. These methods are based on a proper use of the projection map onto the constraint set and on a derivative-free and nonmonotone linesearch strategy. The convergence properties of the proposed methods are presented along with a worst-case iteration complexity bound. Several … Read more

    Approximate norm descent methods for constrained nonlinear systems

  • Benedetta Morini
  • Margherita Porcelli
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Nonlinear Systems and Least-Squares Tags , , ,

    We address the solution of convex-constrained nonlinear systems of equations where the Jacobian matrix is unavailable or its computation/storage is burdensome. In order to efficiently solve such problems, we propose a new class of algorithms which are “derivative-free” both in the computation of the search direction and in the selection of the steplength. Search directions … Read more