Finding Zeros of Hölder Metrically Subregular Mappings via Globally Convergent Levenberg-Marquardt Methods
Abstract
We present 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 nonmonotone trust-region 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 $\mathcal{O}(\varepsilon^{-2})$ function and gradient evaluations, for an accuracy parameter $\varepsilon>0$. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1812.00818
- arXiv:
- arXiv:1812.00818
- Bibcode:
- 2018arXiv181200818A
- Keywords:
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- Mathematics - Optimization and Control;
- Quantitative Biology - Molecular Networks;
- 90C26;
- 68Q25;
- 65K05
- E-Print:
- 28 pages, 3 figures