Stable Computation of Search Directions for Near-Degenerate Linear Programming Problems

In this paper, we examine stability issues that arise when computing the search directions (Ax, Ay, As) for a primal-dual path-following interior point method for linear programming. The dual step Ay can be obtained by solving a weighted least-squares problem for which the weight matrix becomes extremely ill conditioned near the boundary of the feasible region. Hough and Vavasis 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 Ax and the change in the dual slacks As. These directions can be obtained in a straightforward 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 Ax and As. More specifically, this paper describes an 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 in this respect. Numerical tests indicate that it is more reliable than standard algorithms on near-degenerate problems. *This work was partially supported by ONR grant NOO014-96-1-0050 and NSF grant DMS9505155. f part of this work ~M done while the author was ~ member of the Center for Applied Mathematics at Cornell University.