Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Beyond Chance-Constrained Convex Mixed-Integer Optimization: A
  Generalized Calafiore-Campi Algorithm and the notion of $S$-optimization   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 $\mathbb R^d$. The key elements 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.
  Motivated by the first half of the paper, for any subset $S \subset \mathbb
R^d$, 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
$S$.

13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs