Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods. (2nd January 2022)

• Record Type: Journal Article
• Authors: Ahookhosh, Masoud; Fleming, Ronan M. T.; Vuong, Phan T.
• Is Part Of: Optimization methods and software. Volume 37:Number 1(2022)
• Page Start: 113
• Page End: 149
• Publication Date: 2022-01-02
• Subjects: Nonlinear equation -- Hölder metric subregularity -- non-isolated solutions -- Levenberg–Marquardt methods -- global convergence -- worst-case global complexity -- biochemical reaction network kinetics; 90C26 -- 68Q25 -- 65K05; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2020.1712602
• Languages: English
• ISSNs: 1055-6788

Abstract

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.