Please use this identifier to cite or link to this item:
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.
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

Files in This Item:
File Description SizeFormat
DSdeterm2.pdf450.52 kBAdobe PDFView/Open
Show full item record


checked on Feb 18, 2020


checked on May 29, 2020

Page view(s)

checked on Jul 8, 2020


checked on Jul 8, 2020

Google ScholarTM




Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.