Randomized Solutions to Convex Programs with Multiple Chance Constraints

The scenario-based optimization approach (``scenario approach'') provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty (``scenarios''). A key merit of this approach is that it neither requires explicit knowledge of the uncertainty set, as in robust optimization, nor of its probability distribution, as in stochastic optimization. The scenario approach is also computationally efficient because it only requires the solution to a convex optimization program, even if the original chance-constrained problem is nonconvex. Recent research has obtained a rigorous foundation for the scenario approach, by establishing a direct link between the number of scenarios and bounds on the constraint violation probability. These bounds are tight in the general case of an uncertain optimization problem with a single chance constraint. This paper shows that the bounds can be improved...