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On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

Stefania Bellavia (stefania.bellavia***at***unifi.it)
Maria Macconi (maria.macconi***at***unifi.it)
Sandra Pieraccini (pieraccini***at***calvino.polito.it)

Abstract: 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 the linear model of F is trusted to be an adequate representation of the merit function ||F||. Both spherical and elliptical trust-regions are allowed. An inexact dogleg method is used to obtain an approximate minimizer of the linear model within the trust-region and the feasible set. Thus, strictly feasible iterates are formed and a strictly feasible approximation of the solution is ensured. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix and locally fast convergence is showed under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds. In fact, a rather general class of scaling matrices is allowed in actual algorithms. An algorithm based on the standard spherical trust-region and the pioneer scaling matrix given by Coleman and Li is implemented in a Matlab code and its numerical performance is shown.

Keywords: bound-constrained equations, affine scaling, trust-region methods, dogleg methods, inexact Newton methods, global convergence.

Category 1: Nonlinear Optimization (Nonlinear Systems and Least-Squares )

Citation: Pubblicazione n. 5/2009. Dipartimento di Energetica ``Sergio Stecco'', Universit\`a di Firenze, ITALIA

Download: [PDF]

Entry Submitted: 12/28/2009
Entry Accepted: 12/28/2009
Entry Last Modified: 12/28/2009

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