A stable primal-dual approach for linear programming under nondegeneracy assumptions

This paper studies a primal-dual interior/exterior-point path-following approach for linear programming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primal-dual optimality equations. Under nondegeneracy assumptions, this nonlinear system is well-posed, i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. Assuming that a basis matrix (easily factorizable and well-conditioned) can be found, we apply a simple preprocessing step to eliminate both the primal and dual feasibility equations. This results in a single bilinear equation that maintains the well-posedness property. Sparsity is mantained. We then apply either a direct solution method or an iterative solver (within an inexact Newton framework) to solve this equation. Since the linearization is well posed, we use affine scaling and do not maintain nonnegativity once we are close enough Research supported by Universidad Simón Boĺıvar (DID-GID001) and Conicit (project G97000592), Venezuela. E-mail [email protected] Research supported by The Natural Sciences and Engineering Research Council of Canada and Bell Canada. E-mail [email protected] Research supported by The Natural Sciences and Engineering Research Council of Canada. E-mail [email protected] URL for paper: orion.math.uwaterloo.ca/ ̃hwolkowi/henry/reports/ABSTRACTS.html This report is a revision of the earlier CORR 2001-66.