No abstract

This is an author's version published in: http://oatao.univ-toulouse.fr/24842To cite this version: Bellavia, Stefania and Gratton, Serge and Riccietti, Elisa A Levenberg-Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients. (2018) Numerische Mathematik, 140 (3). 791-825. ISSN 0029-599X Official URL: https://doi. AbstractIn this paper we consider large scale nonlinear least-squares problems for which function and gradient are evaluated with dynamic accuracy and propose a Levenberg-Marquardt method for solving such problems. More precisely, we consider the case in which the exact function to optimize is not available or its evaluation is computationally demanding, but approximations of it are available at any prescribed accuracy level. The proposed method relies on a control of the accuracy level, and imposes an improvement of function approximations when the accuracy is detected to be too low to proceed with the optimization process. We prove global and local convergence and complexity of our procedure and show encouraging numerical results on test problems arising in data assimilation and machine learning. Mathematics Subject ClassificationAchieving the Cauchy decrease is a sufficient condition to get global convergence of the method, so one can rely on approximated solutions of problem (2.1), see [11]. A solution of (2.1) can alternatively be found solving the optimality conditions(2.3)Then, we approximately solve (2.3), i.e. we compute a step p such that J δ k (x k ) T J δ k (x k ) + λ k I p = −g δ k (x k ) + r k ,