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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 |
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