On the Simplex Method using an Artificial Basis

The use of an artificial basis for the simplex method was suggested in an early paper by Dantzig. The idea is based on an observation that certain bases, which differ only in a relatively few columns from the true basis, may be easily inverted. Such artificial bases can then be exploited when carrying out simplex iterations. This idea was originally suggested for solving structured linear programming problems, and several approaches, such as Beale's method of pseudo-basic variables, have indeed been presented in the literature. In this paper, we shall not consider the structure explicitly; rather its exploitation in our case is expected to result directly from the choice of an artificial basis. We shall consider this basis to remain unchanged over a number of simplex iterations. In particular, this basis may be chosen as the true basis which has been most recently reinverted. In such a case our approach yields an interpretation for a basis representation recently proposed by Bisschop and Meeraus who point out very favorable properties regarding the build-up of nonzero elements in the basis representation. Our approach utilizes an auxiliary basis, which is small relative to the true basis, and whose dimension may change from one iteration to another. We shall finally develop an updating scheme for a product form representation of the inverse of such an auxiliary basis

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

Bayesian analysis of multifidelity computer models with local features and non-nested experimental designs: Application to the WRF model

Motivated by a multi-fidelity Weather Research and Forecasting (WRF) climate model application where the available simulations are not generated based on hierarchically nested experimental design, we develop a new co-kriging procedure called Augmented Bayesian Treed Co-Kriging. The proposed procedure extends the scope of co-kriging in two major ways. We introduce a binary treed partition latent process in the multifidelity setting to account for non-stationary and potential discontinuities in the model outputs at different fidelity levels. Moreover, we introduce an efficient imputation mechanism which allows the practical implementation of co-kriging when the experimental design is non-hierarchically nested by enabling the specification of semi-conjugate priors. Our imputation strategy allows the design of an efficient RJ-MCMC implementation that involves collapsed blocks and direct simulation from conditional distributions. We develop the Monte Carlo recursive emulator which provides a Monte Carlo proxy for the full predictive distribution of the model output at each fidelity level, in a computationally feasible manner. The performance of our method is demonstrated on benchmark examples and used for the analysis of a large-scale climate modeling application which involves the WRF model

Computational methods and software systems for dynamics and control of large space structures

Two key areas of crucial importance to the computer-based simulation of large space structures are discussed. The first area involves multibody dynamics (MBD) of flexible space structures, with applications directed to deployment, construction, and maneuvering. The second area deals with advanced software systems, with emphasis on parallel processing. The latest research thrust in the second area involves massively parallel computers