Stable computation of search directions for near-degenerate linear programming problems

Abstract

In this paper, we examine stability issues that arise when computing search directions ({delta}x, {delta}y, {delta} s) for a primal-dual path-following interior point method for linear programming. The dual step {delta}y can be obtained by solving a weighted least-squares problem for which the weight matrix becomes extremely il conditioned near the boundary of the feasible region. Hough and Vavisis proposed using a type of complete orthogonal decomposition (the COD algorithm) to solve such a problem and presented stability results. The work presented here addresses the stable computation of the primal step {delta}x and the change in the dual slacks {delta}s. These directions can be obtained in a straight-forward manner, but near-degeneracy in the linear programming instance introduces ill-conditioning which can cause numerical problems in this approach. Therefore, we propose a new method of computing {delta}x and {delta}s. More specifically, this paper describes and orthogonal projection algorithm that extends the COD method. Unlike other algorithms, this method is stable for interior point methods without assuming nondegeneracy in the linear programming instance. Thus, it is more general than other algorithms on near-degenerate problems.