Many real-world engineering design prob- lems are naturally cast in the form of optimization programs with uncertainty-contaminated data. In this context, a reliable design must be able to cope in some way with the presence of uncertainty. In this paper, we consider two standard philosophies for finding optimal solutions for uncertain convex opti- mization problems. In the first approach, classical in the stochastic optimization literature, the optimal de- sign should minimize the expected value of the ob- jective function with respect to uncertainty (average approach), while in the second one it should mini- mize the worst-case objective (worst-case or min-max approach). Both approaches are briefly reviewed in this paper and are shown to lead to exact and nu- merically efficient solution schemes when the uncer- tainty enters the data in simple form. For general uncertainty dependence however, these problems are numerically hard. In this paper, we present two tech- niques based on uncertainty randomization that permit to solve efficiently some suitable probabilistic relax- ation of the indicated problems, with full generality with respect to the way in which the uncertainty en- ters the problem data. In the specific context of truss topology design, uncertainty in the problem arises,