## Nonlinear Stepsize Control, Trust Regions and Regularizations for Unconstrained Optimization

A general class of algorithms for unconstrained optimization is introduced, which subsumes the classical trust-region algorithm and two of its newer variants, as well as the cubic and quadratic regularization methods. A unified theory of global convergence to first-order critical points is then described for this class. An extension to projection-based trust-region algorithms for nonlinear … Read more

## Trust-region and other regularisations of linear least-squares problems

We consider methods for regularising the least-squares solution of the linear system Ax = b. In particular, we propose iterative methods for solving large problems in which a trust-region bound ||x||

## Adjoint Broyden a la GMRES

It is shown that a compact storage implementation of a quasi-Newton method based on the adjoint Broyden update reduces in the affine case exactly to the well established GMRES procedure. Generally, storage and linear algebra effort per step are small multiples of n k, where n is the number of variables and k the number … Read more

## Nonlinear programming without a penalty function or a filter

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a barrier or a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, … Read more

## An Accelerated Newton Method for Equations with Semismooth Jacobians and Nonlinear Complementarity Problems: Extended Version

We discuss local convergence of Newton’s method to a singular solution $x^*$ of the nonlinear equations $F(x) = 0$, for $F:\R^n \rightarrow \R^n$. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution $x^*$ from a starlike domain around $x^*$ for $F$ twice Lipschitz continuously differentiable and $x^*$ satisfying … Read more

## Kantorovich’s Majorants Principle for Newton’s Method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate … Read more

## Regularization Using a Parameterized Trust Region Subproblem

We present a new method for regularization of ill-conditioned problems, such as those that arise in image restoration or mathematical processing of medical data. The method extends the traditional {\em trust-region subproblem}, \TRS, approach that makes use of the {\em L-curve} maximum curvature criterion, a strategy recently proposed to find a good regularization parameter. We … Read more

## Social Cognitive Maps, Swarm Collective Perception and Distributed Search on Dynamic Landscapes

Swarm Intelligence (SI) is the property of a system whereby the collective behaviors of (unsophisticated) entities interacting locally with their environment cause coherent functional global patterns to emerge. SI provides a basis with wich it is possible to explore collective (or distributed) problem solving without centralized control or the provision of a global model. To … Read more

## Varying the Population Size of Artificial Foraging Swarms on Time Varying Landscapes

Swarm Intelligence (SI) is the property of a system whereby the collective behaviors of (unsophisticated) entities interacting locally with their environment cause coherent functional global patterns to emerge. SI provides a basis with wich it is possible to explore collective (or distributed) problem solving without centralized control or the provision of a global model. In … Read more

## On the Global Convergence of a Trust Region Method for Solving Nonlinear Constraints Infeasibility Problem

A framework for proving global convergence for a class of nonlinear constraints infeasibility problem is presented without assuming that the Jacobian has full rank everywhere. The underlying method is based on the simple sufficient reduction criteria where trial points are accepted provided there is a sufficient decrease in the constraints violation function. The proposed methods … Read more