Please use this identifier to cite or link to this item: https://hdl.handle.net/10316/45237
Title: On the optimal order of worst case complexity of direct search
Authors: Dodangeh, Mahdi 
Vicente, Luís Nunes 
Zhang, Zaikun 
Issue Date: 2016
Publisher: Springer Berlin Heidelberg
Project: info:eu-repo/grantAgreement/FCT/COMPETE/132981/PT 
Serial title, monograph or event: Optimization Letters
Volume: 10
Issue: 4
Abstract: The worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. For smooth unconstrained optimization, it is now known that such methods require at most \mathcal {O}(n^2\epsilon ^{-2}) function evaluations to compute a gradient of norm below \epsilon \in (0,1), where n is the dimension of the problem. Such a maximal effort is reduced to \mathcal {O}(n^2\epsilon ^{-1}) if the function is convex. The factor n^2 has been derived using the positive spanning set formed by the coordinate vectors and their negatives at all iterations. In this paper, we prove that such a factor of n^2 is optimal in these worst case complexity bounds, in the sense that no other positive spanning set will yield a better order of n. The proof is based on an observation that reveals the connection between cosine measure in positive spanning and sphere covering.
URI: https://hdl.handle.net/10316/45237
DOI: 10.1007/s11590-015-0908-1
10.1007/s11590-015-0908-1
Rights: embargoedAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais

Files in This Item:
File Description SizeFormat
oods.pdf119.28 kBAdobe PDFView/Open
Show full item record

Google ScholarTM

Check

Altmetric

Altmetric


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