Computation of Generalized Averaged Gaussian Quadrature Rules

The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie [2] developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, the author derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. In [2], [5], [3] stable numerical procedures for computation of the corresponding averaged Gaussian rules are proposed. An analogous procedure can be applied also for a more general class of weighted averaged Gaussian rules introduced in [1]. Those results are presented in [4]. Here we we give a survey of the quoted results, which are obtained jointly with L. Reichel (Kent State Univ., OH (U.S.)

The Gomory-Chvátal closure : polyhedrality, complexity, and extensions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Vita. Cataloged from PDF version of thesis.Includes bibliographical references (p. 163-166).In this thesis, we examine theoretical aspects of the Gomory-Chvátal closure of polyhedra. A Gomory-Chvátal cutting plane for a polyhedron P is derived from any rational inequality that is valid for P by shifting the boundary of the associated half-space towards the polyhedron until it intersects an integer point. The Gomory-ChvAital closure of P is the intersection of all half-spaces defined by its Gomory-Chvátal cuts. While it is was known that the separation problem for the Gomory-Chvátal closure of a rational polyhedron is NP-hard, we show that this remains true for the family of Gomory-Chvátal cuts for which all coefficients are either 0 or 1. Several combinatorially derived cutting planes belong to this class. Furthermore, as the hyperplanes associated with these cuts have very dense and symmetric lattices of integer points, these cutting planes are in some- sense the "simplest" cuts in the set of all Gomory-Chvátal cuts. In the second part of this thesis, we answer a question raised by Schrijver (1980) and show that the Gomory-Chvátal closure of any non-rational polytope is a polytope. Schrijver (1980) had established the polyhedrality of the Gomory-Chvdtal closure for rational polyhedra. In essence, his proof relies on the fact that the set of integer points in a rational polyhedral cone is generated by a finite subset of these points. This is not true for non-rational polyhedral cones. Hence, we develop a completely different proof technique to show that the Gomory-Chvátal closure of a non-rational polytope can be described by a finite set of Gomory-Chvátal cuts. Our proof is geometrically motivated and applies classic results from polyhedral theory and the geometry of numbers. Last, we introduce a natural modification of Gomory-Chvaital cutting planes for the important class of 0/1 integer programming problems. If the hyperplane associated with a Gomory-Chvátal cut for a polytope P C [0, 1]' does not contain any 0/1 point, shifting the hyperplane further towards P until it intersects a 0/1 point guarantees that the resulting half-space contains all feasible solutions. We formalize this observation and introduce the class of M-cuts that arises by strengthening the family of Gomory- Chvátal cuts in this way. We study the polyhedral properties of the resulting closure, its complexity, and the associated cutting plane procedure.by Juliane DunkelPh.D