Stefania Bellavia

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.

Inexact interior-point method

In this paper, we introduce an inexact interior-point algorithm for a constrained system of equations. The formulation of the problem is quite general and includes nonlinear complementarity problems of various kinds. In our convergence theory, we interpret the inexact interior-point method as an inexact Newton method. This enables us to establish a global convergence theory for the proposed algorithm. Under the additional assumption of the invertibility of the Jacobian at the solution, the superlinear convergence of the iteration sequence is proved.

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.

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

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 convergence properties of the new regularized Euclidean residual method for solving general nonlinear least-squares and nonlinear equation problems are investigated. This method, derived from a proposal by Nesterov [Optim. Methods Softw., 22 (2007), pp. 469-483], uses a model of the objective function consisting of the unsquared Euclidean linearized residual regularized by a quadratic term. At variance with previous analysis, its convergence properties are here considered without assuming uniformly nonsingular globally Lipschitz continuous Jacobians nor an exact subproblem solution. It is proved that the method is globally convergent to first-order critical points and, under stronger assumptions, to roots of the underlying system of nonlinear equations. The rate of convergence is also shown to be quadratic under stronger assumptions.

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

Trust-region methods for solving large bound-constrained nonlinear systems are considered. These allow for spherical or elliptical trust regions where the search for an approximate solution is restricted to a low-dimensional space. A general formulation for these methods is introduced and global and superlinear/quadratic convergence is shown under standard assumptions. Viable approaches for implementation in conjunction with Krylov methods are discussed and the practical performance of the resulting algorithms is shown.

Nonsymmetric Preconditioner Updates in Newton-Krylov Methods for Nonlinear Systems

Newton-Krylov methods, a combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners, and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners may destroy the “Jacobian-free” (or matrix-free) setting where the single Jacobian-vector product can be provided without forming and storing the element of the true Jacobian. In this paper, we propose and analyze a preconditioning technique for sequences of nonsymmetric Jacobian matrices based on the update of an earlier preconditioner. The proposed strategy can be implemented in a matrix-free manner. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with the standard ILU-preconditioned Newton-Krylov approaches.

An interior global method for nonlinear systems with simple bounds

The problem of solving a nonlinear system of equations subject to simple bounds is addressed and a new globally convergent method is developed and analyzed. The globalization process employs a trust-region strategy and possibly bends the trust-region solution by a new interior point modification of the projection onto the feasible set. The steps used are obtained by a combination of the chopped Cauchy step and the possibly modified trust-region solution. Thus, strictly feasible iterates are formed. The proposed method is shown to be globally and fast locally convergent. The practical viability of our approach is shown by a concrete implementation and numerical experience on well-known problems. The obtained results indicate that the method works well in practice.

Efficient Preconditioner Updates for Shifted Linear Systems

We present a technique for building effective and low cost preconditioners for sequences of shifted linear systems