In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is termed the extended trust region subproblem in this paper, to distinguish it from the ordinary trust region subproblem where the constraint is a single ellipsoi...

Joint estimation of spin density, R∗ 2 decay and offresonance frequency maps is very useful in many magnetic resonance imaging (MRI) applications. The standard multi-echo approach can achieve high accuracy but requires a long acquisition time for sampling multiple k-space frames. There are many approaches to accelerate the acquisition. Among them, singleor multi-shot trajectory based sampling h...

We introduce an inexact Gauss-Newton trust-region method for solving bound-constrained nonlinear least-squares problems where, at each iteration, a trust-region subproblem is approximately solved by the Conjugate Gradient method. Provided a suitable control on the accuracy to which we attempt to solve the subproblems, we prove that the method has global and asymptotic fast convergence properties.

We describe an optimization method for large-scale nonnegative regularization. The method is an interiorpoint 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 preliminary numerical results on image restor...

We present a trust region method for minimizing a general diierentiable function restricted to an arbitrary closed set. We prove a global convergence theorem. The trust region method deenes diicult subproblems that are solvable in some particular cases. We analyze in detail the case where the domain is an Euclidean ball. For this case we present numerical experiments where we consider diierent ...

In this paper, a non-monotone adaptive trust region method for the system of non-linear equations is proposed, in part, which is based on the technique in [9]. The local and global convergence properties of non-monotone adaptive trust region method are proved under favorable conditions. Some numerical experiments show that the method is effective.

In this note, we recall two solutions to alleviate the catastrophic cancellations that occur when comparing function values in descent algorithms. The automatic finite differencing approach [4] was shown useful to trust region and line search variants. The main original contribution is to successfully adapt the line search strategy [6] for use within trust region like algorithms.

In this paper we address the stable numerical solution of nonlinear ill-posed systems by a trust-region method. We show that an appropriate choice of the trust-region radius gives rise to a procedure that has the potential to approach a solution of the unperturbed system. This regularizing property is shown theoretically and validated numerically.

An interior-point trust-region algorithm is proposed for minimization of general (perhaps, non-convex) quadratic objective function over the domain obtained as the intersection of a symmetric cone with an affine subspace. The algorithm uses a trust-region model to ensure descent on a suitable merit function. Convergence to first-order and second-order optimality conditions is proved. Numerical ...

In this paper, we present a nonmonotone adaptive trust region method for unconstrained optimization based on conic model. The new method combines nonmonotone technique and a new way to determine trust region radius at each iteration. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments show that our algorithm is effective.