Efficient calculation of regular simplex gradients

Rachael Tappenden

Abstract: 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 can be computationally expensive, and often requires an overhead operation count of and in some algorithms a storage overhead of . In this work we demonstrate that the linear algebra overhead and storage costs can be reduced, both to , 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 operations.