Showing papers on "QR decomposition published in 1978"

A Basis Factorization Method for Block Triangular Linear Programs.

Abstract: : Time-staged and multi-staged linear programs usually have a structure that is block triangular. Basic solutions to such problems typically have the property that similar type activities persist in the basis over several consecutive time-periods. When this occurs the basis is close to being square block triangular. In 1955 Dantzig suggested a way of factorizing the basis to take advantage of this property. This paper discusses persistance in staircase models and then presents a considerable refinement of the above factorization algorithm. The method has been implemented in an experimental code, with use being made of LU and QR factorization and updating techniques for the solution of small sub-systems of equations. An in-depth analysis is made of the work involved, and computational experience on several dynamic models is reported. (Author)

Differential Geometric Methods for Solving Nonlinear Constrained Optimization Problems and a Related System of Nonlinear Equations: Global Analysis and Implementation

Abstract: Robust methods that can produce convergence from a very poor initial estimate of the optimal solution for larger problems are discussed. A differential geometric method is developed specifically to obtain robust algorithms without resorting to the penalty-type approach. In particular, a generic class of feasibility-improving gradient acute projection (FIGAP) methods and their Levenberg-Marquardt-type modifications is developed for solving the general nonlinear constrained minimization problems. Each method in this class is an amalgamation of a generalized gradient projection method and a generalized Newton-Raphson method which, respectively, take care of reducing the value of the objective function and satisfying constraint equations at the same time. The class of FIGAP methods contains various new methods as well as many of the existing methods. A unified theory is developed for the methods by using extensively the concept of various generalized inverses and related projectors, which facilitates geometric interpretation of the FIGAP methods. Analysis is given to the continuous analogs of the methods to obtain robust algorithms, which also gives insight into the global behavior of the related algorithms. Various new algorithms are derived from the general theory that use the QR decomposition, the SVD decomposition and other decompositions of the Jacobian matrix of themore » constraint functions. Quasi-Newton algorithms which estimate projected Hessian matrix and require in some cases only approximations of nonnegative definite matrix of size n-m are developed to enhance the local convergence, where n and m are numbers of variables and constraint equations, respectively. 1 figure. (RWR)« less