Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems

The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst-case iteration complexity rate, and a guaran...

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Published in:	Journal of optimization theory and applications Vol. 185; no. 3; pp. 927 - 944
Main Authors:	Bergou, El Houcine, Diouane, Youssef, Kungurtsev, Vyacheslav
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.06.2020 Springer Nature B.V Springer Verlag
Subjects:	Algorithms Applications of Mathematics Basic converters Calculus of Variations and Optimal Control; Optimization Complexity Convergence Engineering General Mathematics Inverse problems Iterative methods Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation Levenberg–Marquardt method Inverse problems Global and local convergence Worst-case complexity bound 90C06 49M05 49M15 90C60
ISSN:	0022-3239, 1573-2878
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Abstract	The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst-case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under suitable assumptions. We introduce a novel Levenberg-Marquardt method that matches, simultaneously, the state of the art in all of these convergence properties with a single seamless algorithm. Numerical experiments confirm the theoretical behavior of our proposed algorithm.
AbstractList	The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst-case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under suitable assumptions. We introduce a novel Levenberg-Marquardt method that matches, simultaneously, the state of the art in all of these convergence properties with a single seamless algorithm. Numerical experiments confirm the theoretical behavior of our proposed algorithm.
Author	Diouane, Youssef Kungurtsev, Vyacheslav Bergou, El Houcine
Author_xml	– sequence: 1 givenname: El Houcine surname: Bergou fullname: Bergou, El Houcine organization: MaIAGE, INRAE, Université Paris-Saclay, KAUST – sequence: 2 givenname: Youssef orcidid: 0000-0002-6609-7330 surname: Diouane fullname: Diouane, Youssef email: youssef.diouane@isae-supaero.fr organization: ISAE-SUPAERO, Université de Toulouse – sequence: 3 givenname: Vyacheslav surname: Kungurtsev fullname: Kungurtsev, Vyacheslav organization: Department of Computer Science, Faculty of Electrical Engineering, Czech Technical University in Prague
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Cites_doi	10.1090/qam/10666 10.1080/10556788.2016.1179737 10.1137/140974687 10.1137/090780882 10.1137/1.9780898719857 10.1145/355934.355936 10.1002/qj.94 10.1137/0111030 10.1080/02331930801951256 10.1007/s00607-004-0083-1 10.1080/1055678021000049345 10.1137/1.9780898717921 10.1007/978-3-7091-6217-0_18 10.1007/s00186-012-0419-0 10.1007/s10957-011-9907-2 10.1007/s101070100263 10.1007/s10957-010-9731-0 10.1017/S033427000000120X 10.1007/s10589-005-3078-8
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Keywords	Levenberg–Marquardt method Inverse problems Global and local convergence Worst-case complexity bound 90C06 49M05 49M15 90C60
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Snippet	The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified...
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SubjectTerms	Algorithms Applications of Mathematics Basic converters Calculus of Variations and Optimal Control; Optimization Complexity Convergence Engineering General Mathematics Inverse problems Iterative methods Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation
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Title	Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems
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