Related papers: Mixed-integer convex representability

This article presents a new set representation named the hybrid zonotope that is equivalent to the union of $2^N$ constrained zonotopes -- convex polytopes -- through the addition of $N$ binary zonotope factors. The major contribution of…

A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the…

Mixed integer sets have a strong modeling capacity to describe practical systems. Nevertheless, incorporating a mixed integer set often renders an optimization formulation drastically more challenging to compute. In this paper, we study how…

In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…

We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances and uncertain initial conditions. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing…

Let $S =\{x\in \re^n: g_1(x)\geq 0, ..., g_m(x)\geq 0\}$ be a semialgebraic set defined by multivariate polynomials $g_i(x)$. Assume $S$ is convex, compact and has nonempty interior. Let $S_i =\{x\in \re^n: g_i(x)\geq 0\}$, and $\bdS$…

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…

Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the…

Describing the solutions of inverse problems arising in signal or image processing is an important issue both for theoretical and numerical purposes. We propose a principle which describes the solutions to convex variational problems…

In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the…

In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem…

We show that maximal $S$-free convex sets are polyhedra when $S$ is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lov\'asz characterizing maximal lattice-free convex sets. Our…

We consider a set of probabilistic functions of some input variables as a representation of the inputs. We present bounds on how informative a representation is about input data. We extend these bounds to hierarchical representations so…

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s<s'$. In this paper, we determine the…

We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…

We study sets defined as the intersection of a rank-1 constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-1 set is polyhedral…

Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first…

We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most…

Many systems exhibit a mixture of continuous and discrete dynamics. We consider a family of mixed-integer non-convex non-linear optimisation problems obtained in discretisations of optimal control of such systems. For this family, a…

Implicit variables of an optimization problem are used to model variationally challenging feasibility conditions in a tractable way while not entering the objective function. Hence, it is a standard approach to treat implicit variables as…