  Derivative-free optimization (DFO) is the mathematical study of the
optimization algorithms that do not use derivatives. One branch of DFO focuses
on model-based DFO methods, where an approximation of the objective function is
used to guide the optimization algorithm. Proving convergence of such methods
often applies an assumption that the approximations form {\em fully linear
models} -- an assumption that requires the true objective function to be
smooth. However, some recent methods have loosened this assumption and instead
worked with functions that are compositions of smooth functions with simple
convex functions (the max-function or the $\ell_1$ norm). In this paper, we
examine the error bounds resulting from the composition of a convex lower
semi-continuous function with a smooth vector-valued function when it is
possible to provide fully linear models for each component of the vector-valued
function. We derive error bounds for the resulting function values and
subgradient vectors.
