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Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems subject to convex constraints

Nicholas Gould(nick.gould***at***stfc.ac.uk)
Tyrone Rees(tyrone.rees***at***stfc.ac.uk)
Jennifer Scott(jennifer.scott***at***stfc.ac.uk)

Abstract: Given a twice-continuously differentiable vector-valued function $r(x)$, a local minimizer of $\|r(x)\|_2$ within a convex set is sought. We propose and analyse tensor-Newton methods, in which $r(x)$ is replaced locally by its second-order Taylor approximation. Convergence is controlled by regularization of various orders. We establish global convergence to a constrained first-order critical point of $\|r(x)\|_2$, and provide function evaluation bounds that agree with the best-known bounds for methods using second derivatives. Numerical experiments comparing tensor-Newton methods with regularized Gauss-Newton and Newton methods demonstrate the practical performance of the newly proposed method in the unconstrained case.

Keywords: Nonlinear least squares, Tensor-Newton methods, Convergence analysis, Evaluation complexity analysis

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

Citation: Technical report RAL-TR-2019-001

Download: [PDF]

Entry Submitted: 01/17/2019
Entry Accepted: 01/17/2019
Entry Last Modified: 01/17/2019

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