Alternative methods for representing the inverse of linear programming basis matrices

Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape

Three-point Green function of massless QED in position space to lowest order

The transverse part of the three-point Green function of massless QED is determined to the lowest order in position space. Taken together with the evaluation of the longitudinal part in arXiv:0803.2630, this gives a relation for QED which is analogous to the star-triangle relation. We relate our result to conformal-invariant three-point functions.Comment: 8 page

The Expected Duration of Gamma-Ray Bursts in the Impulsive Hydrodynamic Models

Depending upon the various models and assumptions, the existing literature on Gamma Ray Bursts (GRBs) mentions that the gross theoretical value of the duration of the burst in the hydrodynamical models is tau~r^2/(eta^2 c), where r is the radius at which the blastwave associated with the fireball (FB) becomes radiative and sufficiently strong. Here eta = E/Mc^2, c is the speed of light, E is initial lab frame energy of the FB, and M is the baryonic mass of the same (Rees and Meszaros 1992). However, within the same basic framework, some authors (like Katz and Piran) have given tau ~ r^2 /(eta c). We intend to remove this confusion by considering this problem at a level deeper than what has been considered so far. Our analysis shows that none of the previously quoted expressions are exactly correct and in case the FB is produced impulsively and the radiative processes responsible for the generation of the GRB are sufficiently fast, its expected duration would be tau ~ar^2/(eta^2 c), where a~O(10^1). We further discuss the probable change, if any, of this expression, in case the FB propagates in an anisotropic fashion. We also discuss some associated points in the context of the Meszaros and Rees scenario.Comment: 21 pages, LATEX (AAMS4.STY -enclosed), 1 ps. Fig. Accepted in Astrophysical Journa

A program to reorder and solve sparse unsymmetric linear systems using the envelope method

The envelope data structure and the Choleski based (bordering) method for the solution of symmetric sparse systems of linear equations have been extended by the authors to solve unsymmetric systems of linear equations. The data structures used in this general linear equation Solver and a set of FORTRAN 77 subroutines are described. Some test data (extracted from LP problems as basis matrices) together with experimental results are presented

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery

Experimental investigation of an interior search method within a simple framework

A steepest gradient method for solving Linear Programming (LP) problems, followed by a procedure for purifying a non-basic solution to an improved extreme point solution have been embedded within an otherwise simplex based optimiser. The algorithm is designed to be hybrid in nature and exploits many aspects of sparse matrix and revised simplex technology. The interior search step terminates at a boundary point which is usually non-basic. This is then followed by a series of minor pivotal steps which lead to a basic feasible solution with a superior objective function value. It is concluded that the procedures discussed in this paper are likely to have three possible applications, which are (i) improving a non-basic feasible solution to a superior extreme point solution, (iii) an improved starting point for the revised simplex method, and (iii) an efficient implementation of the multiple price strategy of the revised simplex method

Sampling Properties of the Spectrum and Coherency of Sequences of Action Potentials

The spectrum and coherency are useful quantities for characterizing the temporal correlations and functional relations within and between point processes. This paper begins with a review of these quantities, their interpretation and how they may be estimated. A discussion of how to assess the statistical significance of features in these measures is included. In addition, new work is presented which builds on the framework established in the review section. This work investigates how the estimates and their error bars are modified by finite sample sizes. Finite sample corrections are derived based on a doubly stochastic inhomogeneous Poisson process model in which the rate functions are drawn from a low variance Gaussian process. It is found that, in contrast to continuous processes, the variance of the estimators cannot be reduced by smoothing beyond a scale which is set by the number of point events in the interval. Alternatively, the degrees of freedom of the estimators can be thought of as bounded from above by the expected number of point events in the interval. Further new work describing and illustrating a method for detecting the presence of a line in a point process spectrum is also presented, corresponding to the detection of a periodic modulation of the underlying rate. This work demonstrates that a known statistical test, applicable to continuous processes, applies, with little modification, to point process spectra, and is of utility in studying a point process driven by a continuous stimulus. While the material discussed is of general applicability to point processes attention will be confined to sequences of neuronal action potentials (spike trains) which were the motivation for this work.Comment: 33 pages, 9 figure

Asset liability management using stochastic programming

This chapter sets out to explain an important financial planning model called asset liability management (ALM); in particular, it discusses why in practice, optimum planning models are used. The ability to build an integrated approach that combines liability models with that of asset allocation decisions has proved to be desirable and more efficient in that it can lead to better ALM decisions. The role of uncertainty and quantification of risk in these planning models is considered