Linear updates for a single-phase projective method

De Ghellinck and Vial developed a single-phase polynomial projective method for the primal linear programming problem. The linear lower-bound update that they give does not ensure convergence with nonfeasible iterates, and is therefore supplemented by a quadratic procedure. In this paper, linear updates are derived that guarantee convergence of the algorithm in a polynomial number of steps. In particular, it is shown that it suffices to add only a single extra variable to the standard linear update for feasible iterates in order to achieve this result.