A class of trust-region methods for large scale bound-constrained systems of nonlinear equations is presented. The methods in this class follow the so called affine-scaling approach and can efficiently handle large scale problems. At each iteration, a suitably scaled region around the current approximate solution is defined and, within such a region, the norm of … Read more
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 … Read more
We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on … Read more
We consider penalized-likelihood reconstruction for X-ray computed tomography of objects that contain small metal structures. To reduce the beam hardening artefacts induced by these structures, we derive the reconstruction algorithm from a projection model that takes into account the photon emission spectrum and nonlinear variation of attenuation to photon energy. This algorithm requires excessively long … Read more
Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. In this paper, we propose a general form of three-term conjugate gradient methods which always generate a sufficient descent direction. We give a sufficient condition for the global convergence of the proposed general method. … Read more
We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for an unconstrained minimization problem. We show the behavioral importance of this proximal point model for habit’s formation in Decision and Making Sciences. ArticleDownload View PDF
The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage … Read more
The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take … Read more
This view of the development of algorithms for nonlinear optimization is based on the research that has been of particular interest to the author since 1959, including several of his own contributions. After a brief survey of classical methods, which may require good starting points in order to converge successfully, the huge impact of variable … Read more
Newton’s method is a classical method for solving a nonlinear equation $F(z)=0$. We derive inexact Newton steps that lead to an inexact Newton method, applicable near a solution. The method is based on solving for a particular $F'(z_{k’})$ during $p$ consecutive iterations $k=k’,k’+1,\dots,k’+p-1$. One such $p$-cycle requires $2^p-1$ solves with the matrix $F'(z_{k’})$. If matrix … Read more