###### Abstract  
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. blackbox optimization; derivative-free algorithms; trust-region methods; composite objective functions  
problems. Early works on DFTR methods include those of [PERSON], [PERSON], and [PERSON] [29], [PERSON] [30], [PERSON] [31; 32], and [PERSON] [33], to name a few. In 2009, the convergence properties of general DFTR algorithms for unconstrained optimization problems were rigorously investigated [34]. The pseudo-code of an algorithm that converges to a first-order critical point and the pseudo-code of an algorithm that converges to a second-order critical point were provided. A complete review of the DFTR methods for unconstrained optimization problems is available in [35] (Chapters 10 and 11) and [36] (Chapter 11). There now exist several DFTR algorithms for solving unconstrained optimization problems, such as Advanced DFO-TRNS [37], BOOSTERS [38], CSV2 [39], DFO [29; 40], UOBYQA [31], and WEDGE [30]. In recent years, DFTR algorithms have also been developed for constrained optimization problems. When an optimization problem is bound-constrained, some of the algorithms available in the literature are BC-DFO [41], BOBYQA [42], ORBIT [43; 44], SNOBFIT [45], and TRB-POWELL [46]. Other DFTR algorithms dealing with more general constrained optimization problems include CONDOR [47], CONORBIT [48], DEFT-FUNNEL [49], LCOBYQA [50], LINCOA [51], and S [52]. We define a _blackbox_ as any process that returns an output whenever an input is provided, but where the inner mechanism of the process is not analytically available to the optimizer [36]. In this paper, we consider a situation in which the objective function, \(F\), is obtained by manipulating several blackboxes. For instance, \(F\) could be the product of two functions, say, \(f_{1}\) and \(f_{2}\), where the function values for \(f_{1}\) are obtained through one blackbox and the function values for \(f_{2}\) are obtained through a different blackbox. An objective function that is defined by manipulating more than one function is called a _composite objective function_. Composite objective functions have inspired a particular direction of research under the assumption that the functions involved do not have the same computational costs. For instance, [PERSON] et al. [53] developed an algorithm for minimizing \(F=\phi+h\circ f\), where \(\phi\) is smooth with known derivatives, \(h\) is a known nonsmooth piecewise linear function, and \(f\) is smooth but expensive to evaluate. In [54], [PERSON] et al. investigated the minimization problem \(F=h\circ f\), where \(h\) is nonsmooth and inexpensive to compute and \(f\) is smooth, but its Jacobian is not available. Recently, [PERSON] and [PERSON] developed algorithms for bound-constrained nonsmooth composite minimization problems in which the objective function \(F\) has the form \(F=h(f(x))\), where \(h\) is cheap to evaluate, and \(f\) requires considerable time to evaluate. These ideas have led to research on more general calculus-based approaches to approximating gradients or Hessians of composite functions. In [55; 56; 57], the authors provided calculus rules (integer power, product, quotient, and chain) for (_generalized_) _simplex gradients_. These results were advanced to the (_generalized_) _centered simplex gradient_ in [58] and to the (_generalized_) _simplex Hessian_ in [59].