Related papers: Some Algorithmic Problems in Polytope Theory
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear…
Convex quadratic objective functions are an important base case in state-of-the-art benchmark collections for single-objective optimization on continuous domains. Although often considered rather simple, they represent the highly relevant…
A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak…
Writing an uncomplicated, robust, and scalable three-dimensional convex hull algorithm is challenging and problematic. This includes, coplanar and collinear issues, numerical accuracy, performance, and complexity trade-offs. While there are…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
In this paper we analyze theoretical properties of bi-objective convex-quadratic problems. We give a complete description of their Pareto set and prove the convexity of their Pareto front. We show that the Pareto set is a line segment when…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of a quasihomogeneous singularity. One of them is a new conjecture on the characteristic polynomial. It is an…
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of…
In this work, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of linear optimization over the subset of vertices of P that are not contained in X. This…
The odd-red bipartite perfect matching problem asks to find a perfect matching containing an odd number of red edges in a given red-blue edge-colored bipartite graph. While this problem lies in $\mathsf{P}$, its polyhedral structure remains…
The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and…
We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many…
We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact…
Quadratic constrained quadratic programming problems often occur in various fields such as engineering practice, management science, and network communication. This article mainly studies a non convex quadratic programming problem with…
We introduce the problem of partitioning 2D regions (usually convex regions) into mutually congruent pieces ('tiles').
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…