  Model-based methods are popular in derivative-free optimization (DFO). In
most of them, a single model function is built to approximate the objective
function. This is generally based on the assumption that the objective function
is one blackbox. However, some real-life and theoretical problems show that the
objective function may consist of several blackboxes. In those problems, the
information provided by each blackbox may not be equal. In this situation, one
could build multiple sub-models that are then combined to become a final model.
In this paper, we analyze the relation between the accuracy of those sub-models
and the model constructed through their operations. We develop a broad
framework that can be used as a theoretical tool in model error analysis and
future research in DFO algorithms design.
