  Solutions of an optimization problem are sensitive to changes caused by
approximations or parametric perturbations, especially in the nonconvex
setting. This paper investigates the ability of substitute problems,
constructed from Rockafellian functions, to provide robustness against such
perturbations. Unlike classical stability analysis focused on local changes
around (local) minimizers, we employ epi-convergence to examine whether the
approximating problems suitably approach the actual one globally. We show that
under natural assumptions the substitute problems can be well-behaved in the
sense of epi-convergence even though the actual one is not. We further quantify
the rates of convergence that often lead to Lipschitz-kind stability properties
for the substitute problems.
