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Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension…
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…
Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral…
We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
Representing a polygon using a set of simple shapes has numerous applications in different use-case scenarios. We consider the problem of covering the interior of a rectilinear polygon with holes by a set of area-weighted, axis-aligned…
Given a set of squares and a strip of bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are…
We consider a model that arises in integer programming, and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result…
We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints,…
Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…
Multilevel programming is the standard framework for modeling hierarchical decision-making. In this paper, we characterize the computational complexity of deciding the existence of feasible and optimal solutions, as well as computing the…
Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional…
Solving difficult mixed-integer nonlinear programs via spatial branch-and-bound requires effective convex outer-approximations of nonconvex sets. In this framework, complex problem formulations are decomposed into simpler library functions,…
The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way.…
Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension $n\geq-1$; now in {\rm dim}. 2, 3 and 4 there are \emph{extra} polytopes, while in general dimensions only the…
A new line of research on the lasso exploits the beautiful geometric fact that the lasso fit is the residual from projecting the response vector $y$ onto a certain convex polytope. This geometric picture also allows an exact geometric…
Following the seminal work of Padberg on the Boolean quadric polytope $BQP$ and its LP relaxation $BQP_{LP}$, we consider a natural extension: $SATP$ and $SATP_{LP}$ polytopes, with $BQP_{LP}$ being projection of the $SATP_{LP}$ face (and…
The lattice size of a lattice polytope is a geometric invariant which was formally introduced in the context of simplification of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics.…