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Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods

Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods
Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods

We introduce LMLS and LMQR, two globally convergent Levenberg–Marquardt methods for finding zeros of Hölder metrically subregular mappings that may have non-isolated zeros. The first method unifies the Levenberg–Marquardt direction and an Armijo-type line search, while the second incorporates this direction with a non-monotone quadratic regularization technique. For both methods, we prove the global convergence to a first-order stationary point of the associated merit function. Furthermore, the worst-case global complexity of these methods are provided, indicating that an approximate stationary point can be computed in at most O(ε -2) function and gradient evaluations, for an accuracy parameter ε > 0. We also study the conditions for the proposed methods to converge to a zero of the associated mappings. Computing a moiety conserved steady state for biochemical reaction networks can be cast as the problem of finding a zero of a Hölder metrically subregular mapping. We report encouraging numerical results for finding a zero of such mappings derived from real-world biological data, which supports our theoretical foundations.

65K05, 68Q25, 90C26, Hölder metric subregularity, Levenberg–Marquardt methods, Nonlinear equation, biochemical reaction network kinetics, global convergence, non-isolated solutions, worst-case global complexity
1055-6788
Ahookhosh, Masoud
48e5bfa1-5dbf-4cf7-8c2e-549e39040524
Fleming, Ronan T.
7a001bd0-09f9-4592-91e7-bf61d5641505
Vuong, Phan Tu
52577e5d-ebe9-4a43-b5e7-68aa06cfdcaf
Ahookhosh, Masoud
48e5bfa1-5dbf-4cf7-8c2e-549e39040524
Fleming, Ronan T.
7a001bd0-09f9-4592-91e7-bf61d5641505
Vuong, Phan Tu
52577e5d-ebe9-4a43-b5e7-68aa06cfdcaf

Ahookhosh, Masoud, Fleming, Ronan T. and Vuong, Phan Tu (2020) Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods. Optimization Methods and Software. (doi:10.1080/10556788.2020.1712602).

Record type: Article

Abstract

We introduce LMLS and LMQR, two globally convergent Levenberg–Marquardt methods for finding zeros of Hölder metrically subregular mappings that may have non-isolated zeros. The first method unifies the Levenberg–Marquardt direction and an Armijo-type line search, while the second incorporates this direction with a non-monotone quadratic regularization technique. For both methods, we prove the global convergence to a first-order stationary point of the associated merit function. Furthermore, the worst-case global complexity of these methods are provided, indicating that an approximate stationary point can be computed in at most O(ε -2) function and gradient evaluations, for an accuracy parameter ε > 0. We also study the conditions for the proposed methods to converge to a zero of the associated mappings. Computing a moiety conserved steady state for biochemical reaction networks can be cast as the problem of finding a zero of a Hölder metrically subregular mapping. We report encouraging numerical results for finding a zero of such mappings derived from real-world biological data, which supports our theoretical foundations.

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Finding Zeros of Holder Metrically Subregular Mappings via Globally Convergent Levenberg-Marquardt Methods - Accepted Manuscript
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More information

Accepted/In Press date: 21 December 2019
e-pub ahead of print date: 21 January 2020
Published date: 21 January 2020
Keywords: 65K05, 68Q25, 90C26, Hölder metric subregularity, Levenberg–Marquardt methods, Nonlinear equation, biochemical reaction network kinetics, global convergence, non-isolated solutions, worst-case global complexity

Identifiers

Local EPrints ID: 437575
URI: http://eprints.soton.ac.uk/id/eprint/437575
ISSN: 1055-6788
PURE UUID: 850d3bca-c60c-4fbe-8742-af86aecd9cdd
ORCID for Phan Tu Vuong: ORCID iD orcid.org/0000-0002-1474-994X

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Date deposited: 06 Feb 2020 17:30
Last modified: 28 Apr 2022 05:51

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Contributors

Author: Masoud Ahookhosh
Author: Ronan T. Fleming
Author: Phan Tu Vuong ORCID iD

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