Optimization Letters

, Volume 10, Issue 4, pp 699–708 | Cite as

On the optimal order of worst case complexity of direct search

Original Paper


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.


Direct search Worst case complexity Optimal order Sphere covering Positive spanning set Cosine measure 



We would like to thank Professors Károly Böröczky, Jr., Ilya Dumer, Gabor Fejes Tóth, and Tibor Tarnai, with whom we had helpful discussions on sphere covering.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  3. 3.CERFACS-IRIT Joint LabToulouseFrance

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