Related papers: Tropical Homotopy Continuation
We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…
We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On…
We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the…
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…
We consider multidimensional optimization problems, which are formulated and solved in terms of tropical mathematics. The problems are to minimize (maximize) a linear or nonlinear function defined on vectors over an idempotent semifield,…
We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resluting algorithm is a an extension of the algorithm in [4]. The…
For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…
An algorithm to give an explicit description of all the solutions to any tropical linear system $A\odot x=B\odot x$ is presented. The given system is converted into a finite (rather small) number $p$ of pairs $(S,T)$ of classical linear…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called…
An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the…
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…
Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its…
The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots, and track no unnecessary path. Up to now, algorithms for that…
The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When…
We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…