Search

Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive k -spanning sets with geometrical guarantees. Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive k -spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive k -spanning sets with guarantees on the value of their cosine measures. [ABSTRACT FROM AUTHOR]

This paper examines a calculus-based approach to building model functions in a derivative-free algorithm. This calculus-based approach can be used when the objective function considered is defined via more than one blackbox. Two versions of a derivative-free trust-region method are implemented. The first version builds model functions by using a calculus-based approach, and the second version builds model functions by directly considering the objective function. The numerical experiments demonstrate that the calculus-based approach provides better results in most situations and significantly better results in specific situations. [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]

Using the Moore–Penrose pseudoinverse this work generalizes the gradient approximation technique called the centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the generalized centred simplex gradient. We develop error bounds and, under a full-rank condition, show that the error bounds have |${\mathcal O}(\varDelta ^2)$|⁠ , where |$\varDelta $| is the radius of the sample set of points used. We establish calculus rules for generalized centred simplex gradients, introduce a calculus-based generalized centred simplex gradient and confirm that error bounds for this new approach are also |${\mathcal O}(\varDelta ^2)$|⁠. We provide several examples to illustrate the results and some benefits of these new methods. [ABSTRACT FROM AUTHOR]

Simplex gradients, essentially the gradient of a linear approximation, are a popular tool in derivative-free optimization (DFO). In 2015, a product rule, a quotient rule, and a sum rule for simplex gradients were introduced by Regis [Optim. Lett., 9 (2015), pp. 845--865]. Unfortunately, those calculus rules only work under a restrictive set of assumptions. The purpose of this paper is to provide new calculus rules that work in a wider setting. The rules place minimal assumptions on the functions involved and the interpolation sets. The rules further lead to an alternative approach to gradient approximation in situations where the rules could be applied. We analyze the new approach, provide error bounds, and include some preliminary testing on numerical stability and accuracy. [ABSTRACT FROM AUTHOR]

We consider the question of numerically approximating the derivative of a smooth function using only function evaluations. In particular, we examine the regression gradient , the generalized simplex gradient and the generalized centered simplex gradient , three numerical techniques based on using function values at a collection of sample points to construct 'best-fit' linear models. Under some conditions, these gradient approximations have error bounds dependent on the number of sample points used, the Lipschitz constant of the true gradient, and the geometry of the sample set. Perhaps counter-intuitively, as the number of sample points increases (to infinity) on a fixed domain, the error bounds can increase (to infinity). In this work, we first explore the behavior of the error bound for generalized simplex gradients of a single-variable function (f : R ↦ R). Thereafter, we investigate the behavior of the absolute error for these three gradient approximation techniques as the number of sample points tends to infinity. Under reasonable assumptions, we prove that the absolute errors remain bounded as the number of sample points increases to infinity on a fixed interval. [ABSTRACT FROM AUTHOR]