Experiments with Active-Set LP Algorithms Allowing Basis Deficiency

Abstract

An interesting question for linear programming (LP) algorithms is how to deal with solutions in which the number of nonzero variables is less than the number of rows of the matrix in standard form. An approach is that of basis deficiency-allowing (BDA) simplex variations, which work with a subset of independent columns of the coefficient matrix in standard form, wherein the basis is not necessarily represented by a square matrix. We describe one such algorithm with several variants. The research question deals with studying the computational behaviour by using small, extreme cases. For these instances, we must wonder which parameter setting or variants are more appropriate. We compare the setting of two nonsimplex active-set methods with Holmström’s TomLab LpSimplex v3.0 commercial sparse primal simplex commercial implementation. All of them update a sparse QR factorization in Matlab. The first two implementations require fewer iterations and provide better solution quality and running time.