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https://hdl.handle.net/10316/45243
Title: | A second-order globally convergent direct-search method and its worst-case complexity | Authors: | Gratton, S. Royer, C. W. Vicente, Luís Nunes |
Issue Date: | 2016 | Publisher: | Taylor & Francis | Project: | info:eu-repo/grantAgreement/FCT/5876/147205/PT | metadata.degois.publication.title: | Optimization | metadata.degois.publication.volume: | 65 | metadata.degois.publication.issue: | 6 | Abstract: | Direct-search algorithms form one of the main classes of algorithms for smooth unconstrained derivative-free optimization, due to their simplicity and their well-established convergence results. They proceed by iteratively looking for improvement along some vectors or directions. In the presence of smoothness, first-order global convergence comes from the ability of the vectors to approximate the steepest descent direction, which can be quantified by a first-order criticality (cosine) measure. The use of a set of vectors with a positive cosine measure together with the imposition of a sufficient decrease condition to accept new iterates leads to a convergence result as well as a worst-case complexity bound. In this paper, we present a second-order study of a general class of direct-search methods. We start by proving a weak second-order convergence result related to a criticality measure defined along the directions used throughout the iterations. Extensions of this result to obtain a true second-order optimality one are discussed, one possibility being a method using approximate Hessian eigenvectors as directions (which is proved to be truly second-order globally convergent). Numerically guaranteeing such a convergence can be rather expensive to ensure, as it is indicated by the worst-case complexity analysis provided in this paper, but turns out to be appropriate for some pathological examples. | URI: | https://hdl.handle.net/10316/45243 | DOI: | 10.1080/02331934.2015.1124271 10.1080/02331934.2015.1124271 |
Rights: | embargoedAccess |
Appears in Collections: | I&D CMUC - Artigos em Revistas Internacionais |
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DSdeterm2.pdf | 450.52 kB | Adobe PDF | View/Open |
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