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The scenario approach is a general data-driven algorithm tochance-constrained optimization. It seeks the optimal solution that is feasibleto a carefully chosen number of scenarios. A crucial step in the scenarioapproach is to compute the cardinality of essential sets, which is the smallestsubset of scenarios that determine the optimal solution. This paper addressesthe challenge of efficiently identifying essential sets. For convex problems,we demonstrate that the sparsest dual solution of the scenario problem couldpinpoint the essential set. For non-convex problems, we show that two simplealgorithms return the essential set when the scenario problem isnon-degenerate. Finally, we illustrate the theoretical results andcomputational algorithms on security-constrained unit commitment (SCUC) inpower systems. In particular, case studies of chance-constrained SCUC areperformed in the IEEE 118-bus system. Numerical results suggest that thescenario approach could be an attractive solution to practical power systemapplications.

Computing Essential Sets for Convex and Non-convex Scenario Problems: Theory and Application