A direct orthogonal sparse static methodology for a finite continuation hybrid LP solver

Abstract

A finite continuation method for solving linear programs (LPs) has been recently put forward by K. Madsen and H. B. Nielsen which, to improve its performance, can be thought of as a Phase-I for a non-simplex active-set method (also known as basis-deficiency-allowing simplex variation); this avoids having to start the simplex method from a highly degenerate square basis. An efficient sparse implementation of this combined hybrid approach to solve LPs requires the use of the same sparse data structure in both phases, and a way to proceed in Phase-II when a non-square working matrix is obtained after Phase-I. In this paper a direct sparse orthogonalization methodology based on Givens rotations and a static sparsity data structure is proposed for both phases, with a LINPACK-like downdating without resorting to hyperbolic rotations. Its sparse implementation (recently put forward by us) is of reduced-gradient type, regularisation is not used in Phase-II, and occasional refactorizations can take advantage of row orderings and parallelizability issues to decrease the computational effort. © Springer-Verlag Berlin Heidelberg 2006.