  The scenario approach is a general data-driven algorithm to
chance-constrained optimization. It seeks the optimal solution that is feasible
to a carefully chosen number of scenarios. A crucial step in the scenario
approach is to compute the cardinality of essential sets, which is the smallest
subset of scenarios that determine the optimal solution. This paper addresses
the challenge of efficiently identifying essential sets. For convex problems,
we demonstrate that the sparsest dual solution of the scenario problem could
pinpoint the essential set. For non-convex problems, we show that two simple
algorithms return the essential set when the scenario problem is
non-degenerate. Finally, we illustrate the theoretical results and
computational algorithms on security-constrained unit commitment (SCUC) in
power systems. In particular, case studies of chance-constrained SCUC are
performed in the IEEE 118-bus system. Numerical results suggest that the
scenario approach could be an attractive solution to practical power system
applications.
