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Algorithms based on trust regions have been shown to be robust methods for unconstrained optimization problems. All existing methods, either based on the dogleg strategy or Hebden-Mor6 iterations, require solution of system of linear equations. In large scale optimization this may be prohibitively expensive. It is shown in this paper that an approximate solution of the trust region problem may be found by the preconditioned conjugate gradient method. This may be regarded as a generalized dogleg technique where we asymptotically take the inexact quasi-Newton step. We also show that we have the same convergence properties as existing methods based on the dogleg strategy using an approximate Hessian.

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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 that requires the solution of a large-scale and possibly ill-conditioned parameterized trust-region subproblems at each step. The method uses recently developed techniques for the large-scale trust-region subproblem. We describe the method and present preliminary numerical results on test problems and image restoration problems.