6,726 research outputs found
Computing Integral Solutions of Complementarity Problems
AMS classifications: 90C33, 90C26, 91B50.Discrete set;complementarity problem;algorithm;triangulation
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Solving Discrete Systems of Nonlinear Equations
In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zn of the n-dimensional Euclidean space IRn. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zn and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangu- lation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the `continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further adapt the main result to a discrete variant of the well-known Borsuk-Ulam theorem and to a theorem for the existence of a solution for the discrete nonlinear complementarity problem.integrally convex set;triangulation;simplicial algorithm;discrete zero point
The D1-triangulation in simplicial variable dimension algorithms for computing solutions of nonlinear equations
Nonlinear Equations
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