Strong local convergence properties of adaptive regularized methods for nonlinear least-squares

This paper studies adaptive regularized methods for nonlinear least-squares problems where the model of the objective function used at each iteration is either the Euclidean residual regularized by a quadratic term or the Gauss-Newton model regularized by a cubic term. For suitable choices of the regularization parameter the role of the regularization term is to … Read more

New updates of incomplete LU factorizations and applications to large nonlinear systems

In this paper, we address the problem of preconditioning sequences of large sparse nonsymmetric systems of linear equations and present two new strategies to construct approximate updates of factorized preconditioners. Both updates are based on the availability of an incomplete LU (ILU) factorization for one matrix of the sequence and differ in the approximation of … Read more

A preconditioning framework for sequences of diagonally modified linear systems arising in optimization

We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region … Read more

Efficient preconditioner updates for shifted linear systems

We present a new technique for building effective and low cost preconditioners for sequences of shifted linear systems (A+aI)x=b, where A is symmetric positive definite and a>0. This technique updates a preconditioner for A, available in the form of an LDL’ factorization, by modifying only the nonzero entries of the L factor in such a … Read more

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

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 … Read more

Regularization and Preconditioning of KKT Systems Arising in Nonnegative Least-Squares Problems

A regularized Newton-like method for solving nonnegative least-squares problems is proposed and analysed in this paper. A preconditioner for KKT systems arising in the method is introduced and spectral properties of the preconditioned matrix are analysed. A bound on the condition number of the preconditioned matrix is provided. The bound does not depend on the … Read more

An interior Newton-like method for nonnegative least-squares problems with degenerate solution

An interior point approach for medium and large nonnegative 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. Citation Technical Report 1/2005, Dipartimento di Energetica ‘S. Stecco’, Universita di Firenze, Italia Article Download View … Read more

Subspace trust-region methods for large bound-constrained nonlinear equations

Trust-region methods for solving large bound-constrained nonlinear systems are considered. They allow for spherical or elliptical trust-regions where the search of 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 … Read more