Approximate Projections in a Projective Method for the Linear Feasibility Problem

The key issue in implementing a projective method is the projection operation. In order to cut down computations, several authors have suggested using approximations instead of the projection itself. However, these approximations are not directly compatible with the standard proofs of polynomial complexity. In this chapter we present a relaxation of the main convergence lemma which makes it possible to accommodate approximate projections. We propose several types of approximations that preserve polynomial complexity for the version of Karmarkar’s algorithm presented by de Ghellinck and Vial.