Year: 1979
Type: book
Abstract: 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. \n \nIn 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. \n \nOur 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.
Author: Markku Kallio
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2534768866