Efficient calculation of regular simplex gradients

Ian Coope () and Rachael Tappenden ()
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Ian Coope: University of Canterbury
Rachael Tappenden: University of Canterbury

Computational Optimization and Applications, 2019, vol. 72, issue 3, No 2, 588 pages

Abstract: 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 $$\mathbf {R}^n$$ R n can be computationally expensive, and often requires an overhead operation count of $$\mathcal {O}(n^3)$$ O ( n 3 ) and in some algorithms a storage overhead of $$\mathcal {O}(n^2)$$ O ( n 2 ) . In this work we demonstrate that the linear algebra overhead and storage costs can be reduced, both to $$\mathcal {O}(n)$$ 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 $$\mathcal {O}(n^2)$$ O ( n 2 ) operations.

Keywords: Positive bases; Numerical optimization; Derivative free optimization; Regular simplex; Simplex gradient; Least squares; Well poised; 52B12; 65F20; 65F35; 90C56 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10589-019-00063-3

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