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We present a new method for large-scale nonnegative regularization, based on a quadratically and nonnegatively constrained quadratic problem. Such problems arise for example in the regularization of ill-posed problems in image restoration where, in addition, some of the matrices involved are very ill-conditioned. The method is an interior-point iteration… (More)

We present a matrix{free algorithm for the large{scale trust{region sub-problem. Our algorithm relies on matrix{vector products only and does not require matrix factorizations. We recast the trust{region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter. We then nd the optimal solution of the trust{region… (More)

We consider large-scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill-posed problem, and the right-hand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization… (More)

A MATLAB 6.0 implementation of the LSTRS method is presented. LSTRS was described in Rojas et al. [2000]. LSTRS is designed for large-scale quadratic problems with one norm constraint. The method is based on a reformulation of the trust-region subproblem as a parameterized eigenvalue problem, and consists of an iterative procedure that finds the… (More)

- L. ELDÉN, P. C. HANSEN, M. ROJAS
- 2005

The minimization of linear functionals defined on the solutions of discrete ill-posed problems arises, e.g., in the computation of confidence intervals for these solutions. In 1990, Eldén proposed an algorithm for this minimization problem based on a para-metric programming reformulation involving the solution of a sequence of trust-region problems, and… (More)

We describe a MATLAB implementation [6] of the method LSTRS [5] for the large-scale trust-region subproblem: min 1 2 x T Hx + g T x subject to (s.t.) x 2 ≤ ∆, (1) where H is an n × n, real, large, symmetric matrix, g is an n-dimensional real vector, and ∆ is a positive scalar. Problem (??) arises in connection with the trust-region globalization strategy in… (More)

In a recent paper [Rojas, Santos, Sorensen: ACM ToMS 34 (2008), Article 11] an efficient method for solving the Large-Scale Trust-Region Subproblem was suggested which is based on recasting it in terms of a parameter dependent eigenvalue problem and adjusting the parameter iteratively. The essential work at each iteration is the solution of an eigenvalue… (More)

We describe an optimization method for large-scale nonnegative regularization. The method is an interior-point iteration that requires the solution of a large-scale and possibly ill-conditioned parameterized trust-region subproblem at each step. The method relies on recently developed techniques for the large-scale trust-region subproblem. We present… (More)

- D. C. SORENSEN, M. ROJAS
- 2010

The problem min x x, s.t. b − Ax ≤ arises in the regularization of discrete forms of ill-posed problems when an estimate of the noise level in the data is available. After deriving necessary and sufficient optimality conditions for this problem, we propose two different classes of algorithms: a factorization-based algorithm for small to medium problems, and… (More)