The scenario approach developed by Calafiore and Campi to attack
chance-constrained convex programs utilizes random sampling on the uncertainty
parameter to substitute the original problem with a representative 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.
This paper has two main contributions. First, we present a generalization of
the Calafiore-Campi results to both integer and mixed-integer variables. In
fact, we demonstrate that their sampling estimates work naturally for variables
restricted to some subset S of Rd. The key elements are
generalizations of Helly's theorem where the convex sets are required to
intersect S⊂Rd. The size of samples in both algorithms will
be directly determined by the S-Helly numbers.
Motivated by the first half of the paper, for any subset S⊂Rd, we introduce the notion of an S-optimization problem, where the
variables take on values over S. It generalizes continuous, integer, and
mixed-integer optimization. We illustrate with examples the expressive power of
S-optimization to capture sophisticated combinatorial optimization problems
with difficult modular constraints. We reinforce the evidence that
S-optimization is "the right concept" by showing that the 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.Comment: 16 pages, 0 figures. This paper has been revised and split into two
parts. This version is the second part of the original paper. The first part
of the original paper is arXiv:1508.02380 (the original article contained 24
pages, 3 figures