  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 assumes knowledge of the uncertainty set, as it is common in
robust optimization, nor of its probability distribution, as it is usually
required in stochastic optimization. Moreover, the scenario approach is
computationally efficient as its solution is based on a deterministic
optimization program that is canonically convex, even when the original
chance-constrained problem is not. Recently, researchers have obtained
theoretical foundations for the scenario approach, providing 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. However, this paper shows
that these bounds can be improved in situations where the constraints have a
limited `support rank', a new concept that is introduced for the first time.
This property is typically found in a large number of practical
applications---most importantly, if the problem originally contains multiple
chance constraints (e.g. multi-stage uncertain decision problems), or if a
chance constraint belongs to a special class of constraints (e.g. linear or
quadratic constraints). In these cases the quality of the scenario solution is
improved while the same bound on the constraint violation probability is
maintained, and also the computational complexity is reduced.
