Maria Macconi

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.

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.

Initial-value methods for second-order singularly perturbed boundary-value problems

In this paper, we present a numerical method for solving linear and nonlinear second-order singularly perturbed boundary-value-problems. For linear problems, the method comes from the well-known WKB method. The required approximate solution is obtained by solving the reduced problem and one or two suitable initial-value problems, directly deduced from the given problem. For nonlinear problems, the quasilinearization method is applied. Numerical results are given showing the accuracy and feasibility of the proposed method.

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

New initial-value method for singularly perturbed boundary-value problems

An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.

Parallel initial-value algorithms for singularly perturbed boundary-value problems

Initial-value methods for linear and semilinear singularly perturbed boundary-value problems are examined with a view to designing and implementing algorithms on parallel architectures. Practical experiments on a CRAY Y-MP 8/432 multiprocessor have been performed, showing the reliability and performance of several proposed parallel schemes.

A Gauss-Newton method for solving bound-constrained underdetermined nonlinear systems

An iterative method for solving bound-constrained underdetermined nonlinear systems is presented. The procedure consists of a Gauss--Newton method embedded into a trust–region strategy. Global and fast local convergence results are established. A specific implementation of the method is given along with its application to nonlinear systems of equalities and inequalities.

Constrained Dogleg methods for nonlinear systems with simple bounds

We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problems.

A switching-method for nonlinear systems

A new iterative method is proposed for the solution of nonlinear systems. The method does not use explicit derivative informations and at each iteration automatically selects one of two distinct iterative schemes: a direct search method and a damped approximate Newtons method. So, the method is referred as Switching-Method. It is shown that the method is a global method with quadratic convergence. Numerical results show the very good practical performance of the method.

Numerical solution of second-order nonlinear singularly perturbed boundary-value problems by initial-value methods

Nonlinear singularly perturbed boundary-value problems are considered, with one or two boundary layers but no turning points. The theory of differential inequalities is used to obtain a numerical procedure for quasilinear and semilinear problems. The required solution is approximated by combining the solutions of suitable auxiliary initial-value problems easily deduced from the given problem. From the numerical results, the method seems accurate and solutions to problems with extremely thin layers can be obtained at reasonable cost.