Numerical applications: updating the product form of the inverse for the revised simplex method

COMPUTER CODES for solving linear programs by the simplex method usually use one of three forms in representing the problem during the course of solution. These are: (a) - the standard form or original simplex method; (b) - the revised simplex method with explicit inverse; and (c) - the revised simplex method with inverse in product form1. [For a comparison of the relative efficiencies of the three methods, see text by Wolfe and Cutler2.] It is hoped that the method to be proposed will at least partially alleviate one of the principal disadvantages of (c), the product form algorithm, namely the need for frequent reinversion of the basis to reduce the number of transformations, without sacrificing too much of some of its advantages, such as the sparseness of the inverse and the ease with which the inverse is kept current. The chief advantage of this proposal is that the number of non-zeros in the inverse representation is conserved and remains approximately constant after the initial buildup. The product form of an inverse with which we are concerned here is the indicated product of a number of elementary m×m transformation matrices, each such matrix being an identity matrix with the exception of one column, the so-called transformation column. Computer codes using the product form of the inverse need to store only these exceptional columns and their column indices; usually only the non-zero elements of these columns and their row positions are stored.