Computational Optimization and Applications

, Volume 72, Issue 3, pp 561–588 | Cite as

Efficient calculation of regular simplex gradients

  • Ian Coope
  • Rachael TappendenEmail author


Simplex gradients are an essential feature of many derivative free optimization algorithms, and can be employed, for example, as part of the process of defining a direction of search, or as part of a termination criterion. The calculation of a general simplex gradient in \(\mathbf {R}^n\) can be computationally expensive, and often requires an overhead operation count of \(\mathcal {O}(n^3)\) and in some algorithms a storage overhead of \(\mathcal {O}(n^2)\). In this work we demonstrate that the linear algebra overhead and storage costs can be reduced, both to \(\mathcal {O}(n)\), when the simplex employed is regular and appropriately aligned. We also demonstrate that a gradient approximation that is second order accurate can be obtained cheaply from a combination of two, first order accurate (appropriately aligned) regular simplex gradients. Moreover, we show that, for an arbitrarily aligned regular simplex, the gradient can be computed in \(\mathcal {O}(n^2)\) operations.


Positive bases Numerical optimization Derivative free optimization Regular simplex Simplex gradient Least squares Well poised 

Mathematics Subject Classification

52B12 65F20 65F35 90C56 



The authors thank Luis Vicente, and the anonymous referees for their helpful comments and suggestions, leading to improvements in an earlier version of this work.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

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