On the Bartels - Golub decomposition for linear programming bases

Donald Goldfarb

Abstract: Many implementations of the simplex method require the row of the inverse of the basis matrixB corresponding to the pivot row at each iteration for updating either a pricing vector or the nonbasic reduced costs. In this note we show that if the Bartels—Golub algorithm [1, 2] or one of its variants is used to update theLU factorization ofB, then less computing is needed if one works with the factors of the updatedB than with those ofB. These results are discussed as they apply to the column selection algorithms recently proposed by Goldfarb and Reid [4, 5] and Harris [6].