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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]

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]