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Approximate norm descent methods for constrained nonlinear systems


Authors: Benedetta Morini, Margherita Porcelli and Philippe L. Toint
Journal: Math. Comp.
MSC (2010): Primary 65H10, 90C06, 90C56
DOI: https://doi.org/10.1090/mcom/3251
Published electronically: May 11, 2017
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Abstract: We address the solution of convex-constrained nonlinear systems of equations where the Jacobian matrix is unavailable or its computation/
storage is burdensome. In order to efficiently solve such problems, we propose a new class of algorithms which are ``derivative-free'' both in the computation of the search direction and in the selection of the steplength. Search directions comprise the residuals and quasi-Newton directions while the steplength is determined by using a new linesearch strategy based on a nonmonotone approximate norm descent property of the merit function. We provide a theoretical analysis of the proposed algorithm and we discuss several conditions ensuring convergence to a solution of the constrained nonlinear system. Finally, we illustrate its numerical behaviour also in comparison with existing approaches.


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Additional Information

Benedetta Morini
Affiliation: Dipartimento di Ingegneria Industriale, Università degli Studi di Firenze, viale G.B. Morgagni 40, 50134 Firenze, Italy
Email: benedetta.morini@unifi.it

Margherita Porcelli
Affiliation: Dipartimento di Ingegneria Industriale, Università degli Studi di Firenze, viale G.B. Morgagni 40, 50134 Firenze, Italy
Email: margherita.porcelli@unifi.it

Philippe L. Toint
Affiliation: Namur Center for Complex Systems (naXys), University of Namur, 61, rue de Bruxelles, B-5000 Namur, Belgium
Email: philippe.toint@unamur.be

DOI: https://doi.org/10.1090/mcom/3251
Keywords: Nonlinear systems of equations, convex constraints, numerical algorithms, convergence theory
Received by editor(s): July 27, 2016
Received by editor(s) in revised form: December 16, 2016
Published electronically: May 11, 2017
Additional Notes: The work of the first two authors was supported by Gruppo Nazionale per il Calcolo Scientifico (GNCS-INdAM) of Italy
Part of this research was conducted during a visit supported by GNCS-INdAM of the third author to the Università degli Studi di Firenze
Article copyright: © Copyright 2017 American Mathematical Society