  We introduce a robust optimization model consisting in a family of
perturbation functions giving rise to certain pairs of dual optimization
problems in which the dual variable depends on the uncertainty parameter. The
interest of our approach is illustrated by some examples, including uncertain
conic optimization and infinite optimization via discretization. The main
results characterize desirable robust duality relations (as robust zero-duality
gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials
of the objective function. The two extreme cases, namely, the usual
perturbational duality (without uncertainty), and the duality for the supremum
of functions (duality parameter vanishing) are analyzed in detail.
