Let ε - Argmin(Z) be the collection of all ε-optimal solutions for a stochastic process Z with locally bounded trajectories defined on a topological space. For sequences (Zn) of such stochastic processes and (ε_n) of nonnegative random variables we give sufficient conditions for the (closed) random sets ε_n - Argmin(Z _n) to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.