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The properties of positive bases make them a useful tool in derivative-free optimization and an interesting concept in mathematics. The notion of the cosine measure helps to quantify the quality of a positive basis. It provides information on how well the vectors in the positive basis uniformly cover the space considered. The number of vectors in a positive basis is known to be between n + 1 and 2n inclusively. When the number of vectors is strictly between n + 1 and 2n, we say that it is an intermediate positive basis. In this paper, the structure of intermediate positive bases with maximal cosine measure is investigated. The structure of an intermediate positive basis with maximal cosine measure over a certain subset of positive bases is provided. This type of positive bases has a simple structure that makes them easy to generate with a computer software. [ABSTRACT FROM AUTHOR]

This work investigates the asymptotic behaviour of the gradient approximation method called the generalized simplex gradient (GSG). This method has an error bound that at first glance seems to tend to infinity as the number of sample points increases, but with some careful construction, we show that this is not the case. For functions in finite dimensions, we present two new error bounds ad infinitum depending on the position of the reference point. The error bounds are not a function of the number of sample points and thus remain finite. [ABSTRACT FROM AUTHOR]

An explicit formula based on matrix algebra to approximate the diagonal entries of a Hessian matrix with any number of sample points is introduced. When the derivative-free technique called generalized centered simplex gradient is used to approximate the gradient, then the formula can be computed for only one additional function evaluation. An error bound is introduced and provides information on the form of the sample set of points that should be used to approximate the diagonal of a Hessian matrix. If the sample set of points is built in a specific manner, it is shown that the technique is a O (Δ S 2) accurate approximation of the diagonal entries of the Hessian matrix, where ΔS is the radius of the sample set. [ABSTRACT FROM AUTHOR]

Originally developed in 1954, positive bases and positive spanning sets have been found to be a valuable concept in derivative-free optimization (DFO). The quality of a positive basis (or positive spanning set) can be quantified via the cosine measure and convergence properties of certain DFO algorithms are intimately linked to the value of this measure. However, it is unclear how to compute the cosine measure for a positive basis from the definition. In this paper, a deterministic algorithm to compute the cosine measure of any positive basis or finite positive spanning set is provided. The algorithm is proven to return the exact value of the cosine measure in finite time. [ABSTRACT FROM AUTHOR]

A simplex, the convex hull of a set of n + 1 affinely independent points, is a useful tool in derivative-free optimization. The term uniform simplex was used by Audet and Hare (Derivative-free and blackbox optimization. Springer series in operations research and financial engineering, Springer, Cham, 2017). The purpose of this paper is to provide a framework for constructing a uniform simplex in R n , which can then be aligned with a given vector. We prove that a uniform simplex has the greatest normalized volume of any simplex. Moreover, we show how to create a uniform minimal positive basis from a uniform simplex. [ABSTRACT FROM AUTHOR]