DOUBLE-BASIS SIMPLEX METHOD FOR LARGE SCALE LINEAR PROGRAMMING.

Abstract

The basis handling procedures of the simplex method are formulated in terms of a "double basis". That is, the basis is factored as (DIAGRAM OMITTED...PLEASE SEE DAI) where ‘B, the pseudobasis matrix, is the basis matrix at the last refactorization. P and Q are permutation matrices. Forward and backward transformations and update are presented for each of two implementations of the double-basis method. The first implementation utilizes an explicit G⁻¹ matrix. The second uses a sparse LU factorization of G. Both are based on Marsten's modularized XMP package, in which standard simplex method routines are replaced by corresponding double-basis method routines. XMP and the LU double-basis method implementation employ Reid's LA05 routines for handling sparse linear programming bases. All calculations are done without reference to the H matrix. Therefore, the update is restricted to G, which has dimension limited by the refactorization frequency, and P and Q, which are held as lists. This can lead to a saving in storage space and updating time. The cost is that time for transformations will be about double. Computational comparisons of storage and speed performance are made with the standard simplex method on problems of up to 1480 constraints. It is found that, generally, the double-basis method performs best on larger, denser problems. Density seems to be the more important factor, and the problems with large nonzero growth between refactorizations are the better ones for the double-basis method. Storage saving in the basis inverse representation versus the standard method is as high as 36%, whereas the double-basis run times are 1.2 or more times as great.