Related papers: Basic Polyhedral Theory
We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…
Consider a polyhedral convex cone which is given by a finite number of linear inequalities. We investigate the problem to project this cone into a subspace and show that this problem is closely related to linear vector optimization: We…
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…
This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges…
Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical…
We investigate a recent semantics for intermediate (and modal) logics in terms of polyhedra. The main result is a finite axiomatisation of the intermediate logic of the class of all polytopes -- i.e., compact convex polyhedra -- denoted PL.…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
The aim of this paper is twofold. On one hand the generalized Minkowski sets are defined and characterized. On the other hand, the Motzkin decomposable sets, along with their epigraphic versions are considered and characterized in new ways.…
This short note gives an elementary alternative proof for a theorem of Danilov and Koshevoy on Minkowski summation and unimodularity in discrete convex analysis. It is intended to disseminate this fundamental theorem and make its proof…
After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra…
Lecture hall partitions are a fundamental combinatorial structure which have been studied extensively over the past two decades. These objects have produced new results, as well as reinterpretations and generalizations of classicial…
Convex polyhedra are the basis for several abstractions used in static analysis and computer-aided verification of complex and sometimes mission critical systems. For such applications, the identification of an appropriate…
Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…
We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…
This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…
These lecture notes cover the theory of convex optimization, with a particular emphasis on first-order methods.
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…