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Convergence and cycling in Walker-type saddle search algorithms
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Levitt, Antoine and Ortner, Christoph (2017) Convergence and cycling in Walker-type saddle search algorithms. SIAM Journal on Numerical Analysis, 55 (5). pp. 2204-2227. doi:10.1137/16M1087199 ISSN 0036-1429.
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Official URL: https://doi.org/10.1137/16M1087199
Abstract
Algorithms for computing local minima of smooth objective functions enjoy a mature theory as well as robust and efficient implementations. By comparison, the theory and practice of saddle search is destitute. In this paper we present results for idealized versions of the dimer and gentlest ascent (GAD) saddle search algorithms that show-case the limitations of what is theoretically achievable within the current class of saddle search algorithms: (1) we present an improved estimate on the region of attraction of saddles; (2) we give explicit examples of potential energy wells from which GAD-type dynamics are unable to escape; and (3) we present a local analysis of “singular points” around which the dynamics gets trapped, and prove the existence of quasi-periodic solutions. These results indicate that it is impossible to obtain globally convergent variants of dimer and GAD type algorithms.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Algorithms, Maxima and minima | ||||||
| Journal or Publication Title: | SIAM Journal on Numerical Analysis | ||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||
| ISSN: | 0036-1429 | ||||||
| Official Date: | 12 September 2017 | ||||||
| Dates: |
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| Volume: | 55 | ||||||
| Number: | 5 | ||||||
| Page Range: | pp. 2204-2227 | ||||||
| DOI: | 10.1137/16M1087199 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 19 April 2017 | ||||||
| Date of first compliant Open Access: | 1 May 2018 | ||||||
| RIOXX Funder/Project Grant: |
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