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|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||Serial title, monograph or event:||Optimization||Volume:||65||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:||http://hdl.handle.net/10316/45243||Other Identifiers:||10.1080/02331934.2015.1124271||DOI:||10.1080/02331934.2015.1124271||Rights:||embargoedAccess|
|Appears in Collections:||I&D CMUC - Artigos em Revistas Internacionais|
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