Chance-Constrained Convex Mixed-Integer Optimization and Beyond: Two Sampling Algorithms within S-optimization

This paper makes two contributions to optimization theory derived from new methods of discrete convex analysis. \par\medskip Our first contribution is to stochastic optimization: The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs (i.e., optimization problems with convex constraints that are parametrized by an uncertainty parameter) utilizes random sampling on the uncertainty parameter to substitute the original problem with a deterministic continuous convex optimization with $N$ convex constraints which is a relaxation of the original. Calafiore and Campi provided an explicit estimate on the size $N$ of the sampling relaxation to yield high-likelihood feasible solutions of the chance-constrained problem. They measured the probability of the original constraints to be violated by the random optimal solution from the relaxation of size $N$. We present a generalization of the Calafiore-Campi results to both integer and mixed-integer variables. We demonstrate that their sampling estimates work naturally even for variables that take on more sophisticated values restricted to some subset $S$ of $\mathbb{R}^d$. In this way, a sampling or scenario algorithm for chance-constrained convex mixed integer optimization algorithm is just a very special case of a stronger sampling result in convex analysis. \par\medskip Second, motivated by the first half of the paper, for a subset $S \subset \mathbb{R}^d$, we formally introduce the notion of an $S$-optimization problem, where the variables take on values over $S$. $S$-optimization generalizes continuous ($S=\mathbb{R}^d$), integer ($S=\mathbb{Z}^d$), and mixed-integer optimization ($S=\mathbb{R}^k \times \mathbb{Z}^{d-k}$). We illustrate with examples the expressive power of $S$-optimization to capture combinatorial and integer optimization problems with difficult modular constraints. We reinforce the evidence that $S$-optimization is ``the right concept'' by showing that a second well-known randomized sampling algorithm of K. Clarkson for low-dimensional convex optimization problems can be extended to work with variables taking values over $S$. The key element in all the proofs, are generalizations of Helly's theorem where the convex sets are required to intersect $S \subset \mathbb{R}^d$. The size of samples in both algorithms will be directly determined by the $S$-Helly numbers.