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The scenario approach developed by Calafiore and Campi to attackchance-constrained convex programs utilizes random sampling on the uncertaintyparameter to substitute the original problem with a representative continuousconvex optimization with $N$ convex constraints which is a relaxation of theoriginal. Calafiore and Campi provided an explicit estimate on the size $N$ ofthe sampling relaxation to yield high-likelihood feasible solutions of thechance-constrained problem. They measured the probability of the originalconstraints to be violated by the random optimal solution from the relaxationof size $N$.  This paper has two main contributions. First, we present a generalization ofthe Calafiore-Campi results to both integer and mixed-integer variables. Infact, we demonstrate that their sampling estimates work naturally for variablesrestricted to some subset $S$ of $\mathbb R^d$. The key elements aregeneralizations of Helly's theorem where the convex sets are required tointersect $S \subset \mathbb R^d$. The size of samples in both algorithms willbe directly determined by the $S$-Helly numbers.  Motivated by the first half of the paper, for any subset $S \subset \mathbbR^d$, we introduce the notion of an $S$-optimization problem, where thevariables take on values over $S$. It generalizes continuous, integer, andmixed-integer optimization. We illustrate with examples the expressive power of$S$-optimization to capture sophisticated combinatorial optimization problemswith difficult modular constraints. We reinforce the evidence that$S$-optimization is "the right concept" by showing that the well-knownrandomized sampling algorithm of K. Clarkson for low-dimensional convexoptimization problems can be extended to work with variables taking values over$S$.

Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of $S$-optimization