Benedetta Morini

An affine scaling trust-region approach to bound-constrained nonlinear systems

This paper presents an iterative method for solving bound-constrained systems of nonlinear equations. It combines ideas from the classical trust-region Newton method for unconstrained nonlinear equations and the recent interior affine scaling approach for constrained optimization problems. The method generates feasible iterates and handles the bounds implicitly. It reduces to a standard trust-region method for unconstrained problems when there are no upper or lower bounds on the variables. Global and local fast convergence properties are obtained. The numerical performance of the method is shown on a large number of test problems.

A Globally Convergent Newton-GMRES Subspace Method for Systems of Nonlinear Equations

Newton--Krylov methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimension. Here we introduce a new hybrid Newton-GMRES method where a global strategy restricted to a low-dimensional subspace generated by GMRES is performed. The obtained process is consistent with preconditioning and with matrix-free implementation. Computational results indicate that our proposal enhances the classical backtracking inexact method.

Convergence behaviour of inexact Newton methods

In this paper we investigate local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations. Processes with modified relative residual control are considered, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided. For a special case the results are affine invariant.

STRSCNE: A Scaled Trust-Region Solver for Constrained Nonlinear Equations

In this paper a Matlab solver for constrained nonlinear equations is presented. The code, called STRSCNE, is based on the affine scaling trust-region method STRN, recently proposed by the authors. The approach taken in implementing the key steps of the method is discussed. The structure and the usage of STRSCNE are described and its features and capabilities are illustrated by numerical experiments. The results of a comparison with high quality codes for nonlinear optimization are shown.

Compututational Techniques for Real Logarithms of Matrices

In this work, we consider computing the real logarithm of a real matrix. We pay attention to general conditioning issues, provide careful implementation for several techniques including scaling issues, and finally test and compare the techniques on a number of problems. All things considered, our recommendation for a general purpose method goes to the Schur decomposition approach with eigenvalue grouping, followed by square roots and diagonal Padé approximants of the diagonal blocks. Nonetheless, in some cases, a well-implemented series expansion technique outperformed the other methods. We have also analyzed and implemented a novel method to estimate the Frech&eacutet derivative of the

An interior point Newton-like method for non-negative least-squares problems with degenerate solution

\log

TRESNEI, a Matlab trust-region solver for systems of nonlinear equalities and inequalities

, which proved very successful for condition estimation.

An Inexact Cayley Transform Method For Inverse Eigenvalue Problems

An interior point approach for medium and large non-negative 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. Copyright

Convergence of a Regularized Euclidean Residual Algorithm for Nonlinear Least-Squares

The Matlab implementation of a trust-region Gauss-Newton method for bound-constrained nonlinear least-squares problems is presented. The solver, called TRESNEI, is adequate for zero and small-residual problems and handles the solution of nonlinear systems of equalities and inequalities. The structure and the usage of the solver are described and an extensive numerical comparison with functions from the Matlab Optimization Toolbox is carried out.

Subspace Trust-Region Methods for Large Bound-Constrained Nonlinear Equations

The Cayley transform method is a Newton-like method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. In this paper, we develop an inexact version of the Cayley transform method. Our method can reduce the oversolving problem and it improves the efficiency with respect to the exact version. We show that the convergence rate of our method is superlinear and that a good tradeoff between the required inner and outer iterations can be obtained.